# Properties

 Label 525.4.d.b.274.2 Level $525$ Weight $4$ Character 525.274 Analytic conductor $30.976$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$525 = 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 525.d (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$30.9760027530$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 21) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 274.2 Root $$1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 525.274 Dual form 525.4.d.b.274.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+4.00000i q^{2} +3.00000i q^{3} -8.00000 q^{4} -12.0000 q^{6} -7.00000i q^{7} -9.00000 q^{9} +O(q^{10})$$ $$q+4.00000i q^{2} +3.00000i q^{3} -8.00000 q^{4} -12.0000 q^{6} -7.00000i q^{7} -9.00000 q^{9} +62.0000 q^{11} -24.0000i q^{12} +62.0000i q^{13} +28.0000 q^{14} -64.0000 q^{16} +84.0000i q^{17} -36.0000i q^{18} -100.000 q^{19} +21.0000 q^{21} +248.000i q^{22} +42.0000i q^{23} -248.000 q^{26} -27.0000i q^{27} +56.0000i q^{28} +10.0000 q^{29} -48.0000 q^{31} -256.000i q^{32} +186.000i q^{33} -336.000 q^{34} +72.0000 q^{36} -246.000i q^{37} -400.000i q^{38} -186.000 q^{39} -248.000 q^{41} +84.0000i q^{42} -68.0000i q^{43} -496.000 q^{44} -168.000 q^{46} +324.000i q^{47} -192.000i q^{48} -49.0000 q^{49} -252.000 q^{51} -496.000i q^{52} -258.000i q^{53} +108.000 q^{54} -300.000i q^{57} +40.0000i q^{58} -120.000 q^{59} +622.000 q^{61} -192.000i q^{62} +63.0000i q^{63} +512.000 q^{64} -744.000 q^{66} +904.000i q^{67} -672.000i q^{68} -126.000 q^{69} -678.000 q^{71} +642.000i q^{73} +984.000 q^{74} +800.000 q^{76} -434.000i q^{77} -744.000i q^{78} -740.000 q^{79} +81.0000 q^{81} -992.000i q^{82} -468.000i q^{83} -168.000 q^{84} +272.000 q^{86} +30.0000i q^{87} -200.000 q^{89} +434.000 q^{91} -336.000i q^{92} -144.000i q^{93} -1296.00 q^{94} +768.000 q^{96} -1266.00i q^{97} -196.000i q^{98} -558.000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 16q^{4} - 24q^{6} - 18q^{9} + O(q^{10})$$ $$2q - 16q^{4} - 24q^{6} - 18q^{9} + 124q^{11} + 56q^{14} - 128q^{16} - 200q^{19} + 42q^{21} - 496q^{26} + 20q^{29} - 96q^{31} - 672q^{34} + 144q^{36} - 372q^{39} - 496q^{41} - 992q^{44} - 336q^{46} - 98q^{49} - 504q^{51} + 216q^{54} - 240q^{59} + 1244q^{61} + 1024q^{64} - 1488q^{66} - 252q^{69} - 1356q^{71} + 1968q^{74} + 1600q^{76} - 1480q^{79} + 162q^{81} - 336q^{84} + 544q^{86} - 400q^{89} + 868q^{91} - 2592q^{94} + 1536q^{96} - 1116q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/525\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$176$$ $$451$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 4.00000i 1.41421i 0.707107 + 0.707107i $$0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$3$$ 3.00000i 0.577350i
$$4$$ −8.00000 −1.00000
$$5$$ 0 0
$$6$$ −12.0000 −0.816497
$$7$$ − 7.00000i − 0.377964i
$$8$$ 0 0
$$9$$ −9.00000 −0.333333
$$10$$ 0 0
$$11$$ 62.0000 1.69943 0.849714 0.527244i $$-0.176775\pi$$
0.849714 + 0.527244i $$0.176775\pi$$
$$12$$ − 24.0000i − 0.577350i
$$13$$ 62.0000i 1.32275i 0.750057 + 0.661373i $$0.230026\pi$$
−0.750057 + 0.661373i $$0.769974\pi$$
$$14$$ 28.0000 0.534522
$$15$$ 0 0
$$16$$ −64.0000 −1.00000
$$17$$ 84.0000i 1.19841i 0.800595 + 0.599206i $$0.204517\pi$$
−0.800595 + 0.599206i $$0.795483\pi$$
$$18$$ − 36.0000i − 0.471405i
$$19$$ −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
$$22$$ 248.000i 2.40335i
$$23$$ 42.0000i 0.380765i 0.981710 + 0.190383i $$0.0609729\pi$$
−0.981710 + 0.190383i $$0.939027\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ −248.000 −1.87065
$$27$$ − 27.0000i − 0.192450i
$$28$$ 56.0000i 0.377964i
$$29$$ 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 $$0.544405\pi$$
$$32$$ − 256.000i − 1.41421i
$$33$$ 186.000i 0.981165i
$$34$$ −336.000 −1.69481
$$35$$ 0 0
$$36$$ 72.0000 0.333333
$$37$$ − 246.000i − 1.09303i −0.837449 0.546516i $$-0.815954\pi$$
0.837449 0.546516i $$-0.184046\pi$$
$$38$$ − 400.000i − 1.70759i
$$39$$ −186.000 −0.763688
$$40$$ 0 0
$$41$$ −248.000 −0.944661 −0.472330 0.881422i $$-0.656587\pi$$
−0.472330 + 0.881422i $$0.656587\pi$$
$$42$$ 84.0000i 0.308607i
$$43$$ − 68.0000i − 0.241161i −0.992704 0.120580i $$-0.961524\pi$$
0.992704 0.120580i $$-0.0384755\pi$$
$$44$$ −496.000 −1.69943
$$45$$ 0 0
$$46$$ −168.000 −0.538484
$$47$$ 324.000i 1.00554i 0.864421 + 0.502769i $$0.167685\pi$$
−0.864421 + 0.502769i $$0.832315\pi$$
$$48$$ − 192.000i − 0.577350i
$$49$$ −49.0000 −0.142857
$$50$$ 0 0
$$51$$ −252.000 −0.691903
$$52$$ − 496.000i − 1.32275i
$$53$$ − 258.000i − 0.668661i −0.942456 0.334330i $$-0.891490\pi$$
0.942456 0.334330i $$-0.108510\pi$$
$$54$$ 108.000 0.272166
$$55$$ 0 0
$$56$$ 0 0
$$57$$ − 300.000i − 0.697122i
$$58$$ 40.0000i 0.0905562i
$$59$$ −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 $$0.273603\pi$$
$$62$$ − 192.000i − 0.393291i
$$63$$ 63.0000i 0.125988i
$$64$$ 512.000 1.00000
$$65$$ 0 0
$$66$$ −744.000 −1.38758
$$67$$ 904.000i 1.64838i 0.566316 + 0.824188i $$0.308368\pi$$
−0.566316 + 0.824188i $$0.691632\pi$$
$$68$$ − 672.000i − 1.19841i
$$69$$ −126.000 −0.219835
$$70$$ 0 0
$$71$$ −678.000 −1.13329 −0.566646 0.823961i $$-0.691759\pi$$
−0.566646 + 0.823961i $$0.691759\pi$$
$$72$$ 0 0
$$73$$ 642.000i 1.02932i 0.857394 + 0.514660i $$0.172082\pi$$
−0.857394 + 0.514660i $$0.827918\pi$$
$$74$$ 984.000 1.54578
$$75$$ 0 0
$$76$$ 800.000 1.20745
$$77$$ − 434.000i − 0.642323i
$$78$$ − 744.000i − 1.08002i
$$79$$ −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
$$82$$ − 992.000i − 1.33595i
$$83$$ − 468.000i − 0.618912i −0.950914 0.309456i $$-0.899853\pi$$
0.950914 0.309456i $$-0.100147\pi$$
$$84$$ −168.000 −0.218218
$$85$$ 0 0
$$86$$ 272.000 0.341052
$$87$$ 30.0000i 0.0369694i
$$88$$ 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
$$92$$ − 336.000i − 0.380765i
$$93$$ − 144.000i − 0.160560i
$$94$$ −1296.00 −1.42204
$$95$$ 0 0
$$96$$ 768.000 0.816497
$$97$$ − 1266.00i − 1.32518i −0.748981 0.662592i $$-0.769456\pi$$
0.748981 0.662592i $$-0.230544\pi$$
$$98$$ − 196.000i − 0.202031i
$$99$$ −558.000 −0.566476
$$100$$ 0 0
$$101$$ 232.000 0.228563 0.114281 0.993448i $$-0.463543\pi$$
0.114281 + 0.993448i $$0.463543\pi$$
$$102$$ − 1008.00i − 0.978499i
$$103$$ 1792.00i 1.71428i 0.515082 + 0.857141i $$0.327761\pi$$
−0.515082 + 0.857141i $$0.672239\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 1032.00 0.945629
$$107$$ − 1906.00i − 1.72206i −0.508558 0.861028i $$-0.669821\pi$$
0.508558 0.861028i $$-0.330179\pi$$
$$108$$ 216.000i 0.192450i
$$109$$ 90.0000 0.0790866 0.0395433 0.999218i $$-0.487410\pi$$
0.0395433 + 0.999218i $$0.487410\pi$$
$$110$$ 0 0
$$111$$ 738.000 0.631062
$$112$$ 448.000i 0.377964i
$$113$$ − 458.000i − 0.381283i −0.981660 0.190642i $$-0.938943\pi$$
0.981660 0.190642i $$-0.0610569\pi$$
$$114$$ 1200.00 0.985880
$$115$$ 0 0
$$116$$ −80.0000 −0.0640329
$$117$$ − 558.000i − 0.440916i
$$118$$ − 480.000i − 0.374471i
$$119$$ 588.000 0.452957
$$120$$ 0 0
$$121$$ 2513.00 1.88805
$$122$$ 2488.00i 1.84634i
$$123$$ − 744.000i − 0.545400i
$$124$$ 384.000 0.278099
$$125$$ 0 0
$$126$$ −252.000 −0.178174
$$127$$ 804.000i 0.561760i 0.959743 + 0.280880i $$0.0906262\pi$$
−0.959743 + 0.280880i $$0.909374\pi$$
$$128$$ 0 0
$$129$$ 204.000 0.139234
$$130$$ 0 0
$$131$$ 812.000 0.541563 0.270782 0.962641i $$-0.412718\pi$$
0.270782 + 0.962641i $$0.412718\pi$$
$$132$$ − 1488.00i − 0.981165i
$$133$$ 700.000i 0.456374i
$$134$$ −3616.00 −2.33116
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 414.000i 0.258178i 0.991633 + 0.129089i $$0.0412053\pi$$
−0.991633 + 0.129089i $$0.958795\pi$$
$$138$$ − 504.000i − 0.310894i
$$139$$ 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
$$142$$ − 2712.00i − 1.60272i
$$143$$ 3844.00i 2.24791i
$$144$$ 576.000 0.333333
$$145$$ 0 0
$$146$$ −2568.00 −1.45568
$$147$$ − 147.000i − 0.0824786i
$$148$$ 1968.00i 1.09303i
$$149$$ −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
$$153$$ − 756.000i − 0.399470i
$$154$$ 1736.00 0.908382
$$155$$ 0 0
$$156$$ 1488.00 0.763688
$$157$$ − 266.000i − 0.135217i −0.997712 0.0676086i $$-0.978463\pi$$
0.997712 0.0676086i $$-0.0215369\pi$$
$$158$$ − 2960.00i − 1.49041i
$$159$$ 774.000 0.386052
$$160$$ 0 0
$$161$$ 294.000 0.143916
$$162$$ 324.000i 0.157135i
$$163$$ 272.000i 0.130704i 0.997862 + 0.0653518i $$0.0208170\pi$$
−0.997862 + 0.0653518i $$0.979183\pi$$
$$164$$ 1984.00 0.944661
$$165$$ 0 0
$$166$$ 1872.00 0.875273
$$167$$ − 1876.00i − 0.869277i −0.900605 0.434638i $$-0.856876\pi$$
0.900605 0.434638i $$-0.143124\pi$$
$$168$$ 0 0
$$169$$ −1647.00 −0.749659
$$170$$ 0 0
$$171$$ 900.000 0.402484
$$172$$ 544.000i 0.241161i
$$173$$ 152.000i 0.0667997i 0.999442 + 0.0333998i $$0.0106335\pi$$
−0.999442 + 0.0333998i $$0.989367\pi$$
$$174$$ −120.000 −0.0522826
$$175$$ 0 0
$$176$$ −3968.00 −1.69943
$$177$$ − 360.000i − 0.152877i
$$178$$ − 800.000i − 0.336868i
$$179$$ −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 $$0.431366\pi$$
$$182$$ 1736.00i 0.707038i
$$183$$ 1866.00i 0.753763i
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 576.000 0.227067
$$187$$ 5208.00i 2.03661i
$$188$$ − 2592.00i − 1.00554i
$$189$$ −189.000 −0.0727393
$$190$$ 0 0
$$191$$ −2038.00 −0.772065 −0.386033 0.922485i $$-0.626155\pi$$
−0.386033 + 0.922485i $$0.626155\pi$$
$$192$$ 1536.00i 0.577350i
$$193$$ 2602.00i 0.970446i 0.874390 + 0.485223i $$0.161262\pi$$
−0.874390 + 0.485223i $$0.838738\pi$$
$$194$$ 5064.00 1.87409
$$195$$ 0 0
$$196$$ 392.000 0.142857
$$197$$ 2354.00i 0.851348i 0.904877 + 0.425674i $$0.139963\pi$$
−0.904877 + 0.425674i $$0.860037\pi$$
$$198$$ − 2232.00i − 0.801118i
$$199$$ −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
$$202$$ 928.000i 0.323237i
$$203$$ − 70.0000i − 0.0242022i
$$204$$ 2016.00 0.691903
$$205$$ 0 0
$$206$$ −7168.00 −2.42436
$$207$$ − 378.000i − 0.126922i
$$208$$ − 3968.00i − 1.32275i
$$209$$ −6200.00 −2.05198
$$210$$ 0 0
$$211$$ −668.000 −0.217948 −0.108974 0.994045i $$-0.534757\pi$$
−0.108974 + 0.994045i $$0.534757\pi$$
$$212$$ 2064.00i 0.668661i
$$213$$ − 2034.00i − 0.654307i
$$214$$ 7624.00 2.43535
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 336.000i 0.105111i
$$218$$ 360.000i 0.111845i
$$219$$ −1926.00 −0.594279
$$220$$ 0 0
$$221$$ −5208.00 −1.58519
$$222$$ 2952.00i 0.892456i
$$223$$ 1832.00i 0.550134i 0.961425 + 0.275067i $$0.0887000\pi$$
−0.961425 + 0.275067i $$0.911300\pi$$
$$224$$ −1792.00 −0.534522
$$225$$ 0 0
$$226$$ 1832.00 0.539216
$$227$$ 4944.00i 1.44557i 0.691072 + 0.722786i $$0.257139\pi$$
−0.691072 + 0.722786i $$0.742861\pi$$
$$228$$ 2400.00i 0.697122i
$$229$$ 5470.00 1.57846 0.789231 0.614096i $$-0.210479\pi$$
0.789231 + 0.614096i $$0.210479\pi$$
$$230$$ 0 0
$$231$$ 1302.00 0.370846
$$232$$ 0 0
$$233$$ 2802.00i 0.787833i 0.919146 + 0.393917i $$0.128880\pi$$
−0.919146 + 0.393917i $$0.871120\pi$$
$$234$$ 2232.00 0.623549
$$235$$ 0 0
$$236$$ 960.000 0.264791
$$237$$ − 2220.00i − 0.608458i
$$238$$ 2352.00i 0.640578i
$$239$$ 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 $$0.601152\pi$$
$$242$$ 10052.0i 2.67011i
$$243$$ 243.000i 0.0641500i
$$244$$ −4976.00 −1.30556
$$245$$ 0 0
$$246$$ 2976.00 0.771312
$$247$$ − 6200.00i − 1.59715i
$$248$$ 0 0
$$249$$ 1404.00 0.357329
$$250$$ 0 0
$$251$$ 2792.00 0.702109 0.351055 0.936355i $$-0.385823\pi$$
0.351055 + 0.936355i $$0.385823\pi$$
$$252$$ − 504.000i − 0.125988i
$$253$$ 2604.00i 0.647083i
$$254$$ −3216.00 −0.794448
$$255$$ 0 0
$$256$$ 4096.00 1.00000
$$257$$ 7024.00i 1.70484i 0.522854 + 0.852422i $$0.324867\pi$$
−0.522854 + 0.852422i $$0.675133\pi$$
$$258$$ 816.000i 0.196907i
$$259$$ −1722.00 −0.413127
$$260$$ 0 0
$$261$$ −90.0000 −0.0213443
$$262$$ 3248.00i 0.765886i
$$263$$ − 2438.00i − 0.571610i −0.958288 0.285805i $$-0.907739\pi$$
0.958288 0.285805i $$-0.0922610\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ −2800.00 −0.645410
$$267$$ − 600.000i − 0.137526i
$$268$$ − 7232.00i − 1.64838i
$$269$$ 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 $$0.569329\pi$$
$$272$$ − 5376.00i − 1.19841i
$$273$$ 1302.00i 0.288647i
$$274$$ −1656.00 −0.365119
$$275$$ 0 0
$$276$$ 1008.00 0.219835
$$277$$ 5554.00i 1.20472i 0.798224 + 0.602360i $$0.205773\pi$$
−0.798224 + 0.602360i $$0.794227\pi$$
$$278$$ 6480.00i 1.39800i
$$279$$ 432.000 0.0926995
$$280$$ 0 0
$$281$$ 1942.00 0.412278 0.206139 0.978523i $$-0.433910\pi$$
0.206139 + 0.978523i $$0.433910\pi$$
$$282$$ − 3888.00i − 0.821018i
$$283$$ − 4828.00i − 1.01412i −0.861912 0.507058i $$-0.830733\pi$$
0.861912 0.507058i $$-0.169267\pi$$
$$284$$ 5424.00 1.13329
$$285$$ 0 0
$$286$$ −15376.0 −3.17903
$$287$$ 1736.00i 0.357048i
$$288$$ 2304.00i 0.471405i
$$289$$ −2143.00 −0.436190
$$290$$ 0 0
$$291$$ 3798.00 0.765095
$$292$$ − 5136.00i − 1.02932i
$$293$$ 6152.00i 1.22663i 0.789837 + 0.613317i $$0.210165\pi$$
−0.789837 + 0.613317i $$0.789835\pi$$
$$294$$ 588.000 0.116642
$$295$$ 0 0
$$296$$ 0 0
$$297$$ − 1674.00i − 0.327055i
$$298$$ − 9480.00i − 1.84282i
$$299$$ −2604.00 −0.503656
$$300$$ 0 0
$$301$$ −476.000 −0.0911501
$$302$$ − 2272.00i − 0.432910i
$$303$$ 696.000i 0.131961i
$$304$$ 6400.00 1.20745
$$305$$ 0 0
$$306$$ 3024.00 0.564937
$$307$$ 5884.00i 1.09387i 0.837176 + 0.546934i $$0.184205\pi$$
−0.837176 + 0.546934i $$0.815795\pi$$
$$308$$ 3472.00i 0.642323i
$$309$$ −5376.00 −0.989741
$$310$$ 0 0
$$311$$ 9132.00 1.66504 0.832521 0.553993i $$-0.186897\pi$$
0.832521 + 0.553993i $$0.186897\pi$$
$$312$$ 0 0
$$313$$ 9382.00i 1.69426i 0.531389 + 0.847128i $$0.321670\pi$$
−0.531389 + 0.847128i $$0.678330\pi$$
$$314$$ 1064.00 0.191226
$$315$$ 0 0
$$316$$ 5920.00 1.05388
$$317$$ 3114.00i 0.551734i 0.961196 + 0.275867i $$0.0889649\pi$$
−0.961196 + 0.275867i $$0.911035\pi$$
$$318$$ 3096.00i 0.545959i
$$319$$ 620.000 0.108819
$$320$$ 0 0
$$321$$ 5718.00 0.994229
$$322$$ 1176.00i 0.203528i
$$323$$ − 8400.00i − 1.44702i
$$324$$ −648.000 −0.111111
$$325$$ 0 0
$$326$$ −1088.00 −0.184843
$$327$$ 270.000i 0.0456607i
$$328$$ 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 $$0.459401\pi$$
$$332$$ 3744.00i 0.618912i
$$333$$ 2214.00i 0.364344i
$$334$$ 7504.00 1.22934
$$335$$ 0 0
$$336$$ −1344.00 −0.218218
$$337$$ − 4166.00i − 0.673402i −0.941612 0.336701i $$-0.890689\pi$$
0.941612 0.336701i $$-0.109311\pi$$
$$338$$ − 6588.00i − 1.06018i
$$339$$ 1374.00 0.220134
$$340$$ 0 0
$$341$$ −2976.00 −0.472608
$$342$$ 3600.00i 0.569198i
$$343$$ 343.000i 0.0539949i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ −608.000 −0.0944690
$$347$$ − 11366.0i − 1.75838i −0.476469 0.879191i $$-0.658083\pi$$
0.476469 0.879191i $$-0.341917\pi$$
$$348$$ − 240.000i − 0.0369694i
$$349$$ −9310.00 −1.42795 −0.713973 0.700174i $$-0.753106\pi$$
−0.713973 + 0.700174i $$0.753106\pi$$
$$350$$ 0 0
$$351$$ 1674.00 0.254563
$$352$$ − 15872.0i − 2.40335i
$$353$$ 8572.00i 1.29247i 0.763139 + 0.646234i $$0.223657\pi$$
−0.763139 + 0.646234i $$0.776343\pi$$
$$354$$ 1440.00 0.216201
$$355$$ 0 0
$$356$$ 1600.00 0.238202
$$357$$ 1764.00i 0.261515i
$$358$$ − 2440.00i − 0.360218i
$$359$$ 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
$$362$$ 4168.00i 0.605153i
$$363$$ 7539.00i 1.09007i
$$364$$ −3472.00 −0.499951
$$365$$ 0 0
$$366$$ −7464.00 −1.06598
$$367$$ 5424.00i 0.771473i 0.922609 + 0.385736i $$0.126053\pi$$
−0.922609 + 0.385736i $$0.873947\pi$$
$$368$$ − 2688.00i − 0.380765i
$$369$$ 2232.00 0.314887
$$370$$ 0 0
$$371$$ −1806.00 −0.252730
$$372$$ 1152.00i 0.160560i
$$373$$ − 1838.00i − 0.255142i −0.991829 0.127571i $$-0.959282\pi$$
0.991829 0.127571i $$-0.0407181\pi$$
$$374$$ −20832.0 −2.88021
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 620.000i 0.0846993i
$$378$$ − 756.000i − 0.102869i
$$379$$ 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
$$382$$ − 8152.00i − 1.09187i
$$383$$ − 9048.00i − 1.20713i −0.797313 0.603566i $$-0.793746\pi$$
0.797313 0.603566i $$-0.206254\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −10408.0 −1.37242
$$387$$ 612.000i 0.0803868i
$$388$$ 10128.0i 1.32518i
$$389$$ 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
$$393$$ 2436.00i 0.312672i
$$394$$ −9416.00 −1.20399
$$395$$ 0 0
$$396$$ 4464.00 0.566476
$$397$$ − 1866.00i − 0.235899i −0.993020 0.117949i $$-0.962368\pi$$
0.993020 0.117949i $$-0.0376321\pi$$
$$398$$ − 6720.00i − 0.846340i
$$399$$ −2100.00 −0.263487
$$400$$ 0 0
$$401$$ 13662.0 1.70137 0.850683 0.525679i $$-0.176189\pi$$
0.850683 + 0.525679i $$0.176189\pi$$
$$402$$ − 10848.0i − 1.34589i
$$403$$ − 2976.00i − 0.367854i
$$404$$ −1856.00 −0.228563
$$405$$ 0 0
$$406$$ 280.000 0.0342270
$$407$$ − 15252.0i − 1.85753i
$$408$$ 0 0
$$409$$ 13210.0 1.59705 0.798524 0.601963i $$-0.205615\pi$$
0.798524 + 0.601963i $$0.205615\pi$$
$$410$$ 0 0
$$411$$ −1242.00 −0.149059
$$412$$ − 14336.0i − 1.71428i
$$413$$ 840.000i 0.100082i
$$414$$ 1512.00 0.179495
$$415$$ 0 0
$$416$$ 15872.0 1.87065
$$417$$ 4860.00i 0.570732i
$$418$$ − 24800.0i − 2.90193i
$$419$$ −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 $$0.343375\pi$$
$$422$$ − 2672.00i − 0.308225i
$$423$$ − 2916.00i − 0.335179i
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 8136.00 0.925330
$$427$$ − 4354.00i − 0.493454i
$$428$$ 15248.0i 1.72206i
$$429$$ −11532.0 −1.29783
$$430$$ 0 0
$$431$$ 16602.0 1.85543 0.927715 0.373290i $$-0.121770\pi$$
0.927715 + 0.373290i $$0.121770\pi$$
$$432$$ 1728.00i 0.192450i
$$433$$ − 7738.00i − 0.858810i −0.903112 0.429405i $$-0.858723\pi$$
0.903112 0.429405i $$-0.141277\pi$$
$$434$$ −1344.00 −0.148650
$$435$$ 0 0
$$436$$ −720.000 −0.0790866
$$437$$ − 4200.00i − 0.459756i
$$438$$ − 7704.00i − 0.840437i
$$439$$ 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
$$442$$ − 20832.0i − 2.24180i
$$443$$ − 6618.00i − 0.709776i −0.934909 0.354888i $$-0.884519\pi$$
0.934909 0.354888i $$-0.115481\pi$$
$$444$$ −5904.00 −0.631062
$$445$$ 0 0
$$446$$ −7328.00 −0.778006
$$447$$ − 7110.00i − 0.752330i
$$448$$ − 3584.00i − 0.377964i
$$449$$ −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
$$452$$ 3664.00i 0.381283i
$$453$$ − 1704.00i − 0.176735i
$$454$$ −19776.0 −2.04435
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 5914.00i 0.605351i 0.953094 + 0.302675i $$0.0978798\pi$$
−0.953094 + 0.302675i $$0.902120\pi$$
$$458$$ 21880.0i 2.23228i
$$459$$ 2268.00 0.230634
$$460$$ 0 0
$$461$$ −15968.0 −1.61324 −0.806620 0.591070i $$-0.798706\pi$$
−0.806620 + 0.591070i $$0.798706\pi$$
$$462$$ 5208.00i 0.524455i
$$463$$ 1172.00i 0.117640i 0.998269 + 0.0588202i $$0.0187338\pi$$
−0.998269 + 0.0588202i $$0.981266\pi$$
$$464$$ −640.000 −0.0640329
$$465$$ 0 0
$$466$$ −11208.0 −1.11416
$$467$$ 5304.00i 0.525567i 0.964855 + 0.262784i $$0.0846405\pi$$
−0.964855 + 0.262784i $$0.915359\pi$$
$$468$$ 4464.00i 0.440916i
$$469$$ 6328.00 0.623027
$$470$$ 0 0
$$471$$ 798.000 0.0780677
$$472$$ 0 0
$$473$$ − 4216.00i − 0.409835i
$$474$$ 8880.00 0.860489
$$475$$ 0 0
$$476$$ −4704.00 −0.452957
$$477$$ 2322.00i 0.222887i
$$478$$ 4680.00i 0.447821i
$$479$$ −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
$$482$$ − 9352.00i − 0.883759i
$$483$$ 882.000i 0.0830898i
$$484$$ −20104.0 −1.88805
$$485$$ 0 0
$$486$$ −972.000 −0.0907218
$$487$$ 8944.00i 0.832220i 0.909314 + 0.416110i $$0.136607\pi$$
−0.909314 + 0.416110i $$0.863393\pi$$
$$488$$ 0 0
$$489$$ −816.000 −0.0754617
$$490$$ 0 0
$$491$$ −5558.00 −0.510853 −0.255427 0.966828i $$-0.582216\pi$$
−0.255427 + 0.966828i $$0.582216\pi$$
$$492$$ 5952.00i 0.545400i
$$493$$ 840.000i 0.0767377i
$$494$$ 24800.0 2.25871
$$495$$ 0 0
$$496$$ 3072.00 0.278099
$$497$$ 4746.00i 0.428344i
$$498$$ 5616.00i 0.505339i
$$499$$ 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
$$502$$ 11168.0i 0.992933i
$$503$$ − 1848.00i − 0.163814i −0.996640 0.0819068i $$-0.973899\pi$$
0.996640 0.0819068i $$-0.0261010\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ −10416.0 −0.915114
$$507$$ − 4941.00i − 0.432816i
$$508$$ − 6432.00i − 0.561760i
$$509$$ −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
$$512$$ 16384.0i 1.41421i
$$513$$ 2700.00i 0.232374i
$$514$$ −28096.0 −2.41101
$$515$$ 0 0
$$516$$ −1632.00 −0.139234
$$517$$ 20088.0i 1.70884i
$$518$$ − 6888.00i − 0.584250i
$$519$$ −456.000 −0.0385668
$$520$$ 0 0
$$521$$ 10212.0 0.858725 0.429363 0.903132i $$-0.358738\pi$$
0.429363 + 0.903132i $$0.358738\pi$$
$$522$$ − 360.000i − 0.0301854i
$$523$$ 9332.00i 0.780229i 0.920766 + 0.390115i $$0.127565\pi$$
−0.920766 + 0.390115i $$0.872435\pi$$
$$524$$ −6496.00 −0.541563
$$525$$ 0 0
$$526$$ 9752.00 0.808379
$$527$$ − 4032.00i − 0.333276i
$$528$$ − 11904.0i − 0.981165i
$$529$$ 10403.0 0.855018
$$530$$ 0 0
$$531$$ 1080.00 0.0882637
$$532$$ − 5600.00i − 0.456374i
$$533$$ − 15376.0i − 1.24955i
$$534$$ 2400.00 0.194491
$$535$$ 0 0
$$536$$ 0 0
$$537$$ − 1830.00i − 0.147058i
$$538$$ 27120.0i 2.17328i
$$539$$ −3038.00 −0.242775
$$540$$ 0 0
$$541$$ −8998.00 −0.715073 −0.357536 0.933899i $$-0.616383\pi$$
−0.357536 + 0.933899i $$0.616383\pi$$
$$542$$ − 7712.00i − 0.611179i
$$543$$ 3126.00i 0.247052i
$$544$$ 21504.0 1.69481
$$545$$ 0 0
$$546$$ −5208.00 −0.408208
$$547$$ − 3416.00i − 0.267016i −0.991048 0.133508i $$-0.957376\pi$$
0.991048 0.133508i $$-0.0426241\pi$$
$$548$$ − 3312.00i − 0.258178i
$$549$$ −5598.00 −0.435185
$$550$$ 0 0
$$551$$ −1000.00 −0.0773166
$$552$$ 0 0
$$553$$ 5180.00i 0.398329i
$$554$$ −22216.0 −1.70373
$$555$$ 0 0
$$556$$ −12960.0 −0.988537
$$557$$ − 526.000i − 0.0400132i −0.999800 0.0200066i $$-0.993631\pi$$
0.999800 0.0200066i $$-0.00636872\pi$$
$$558$$ 1728.00i 0.131097i
$$559$$ 4216.00 0.318994
$$560$$ 0 0
$$561$$ −15624.0 −1.17584
$$562$$ 7768.00i 0.583049i
$$563$$ 6712.00i 0.502446i 0.967929 + 0.251223i $$0.0808327\pi$$
−0.967929 + 0.251223i $$0.919167\pi$$
$$564$$ 7776.00 0.580547
$$565$$ 0 0
$$566$$ 19312.0 1.43418
$$567$$ − 567.000i − 0.0419961i
$$568$$ 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 $$0.464560\pi$$
$$572$$ − 30752.0i − 2.24791i
$$573$$ − 6114.00i − 0.445752i
$$574$$ −6944.00 −0.504942
$$575$$ 0 0
$$576$$ −4608.00 −0.333333
$$577$$ 5434.00i 0.392063i 0.980598 + 0.196032i $$0.0628055\pi$$
−0.980598 + 0.196032i $$0.937195\pi$$
$$578$$ − 8572.00i − 0.616865i
$$579$$ −7806.00 −0.560287
$$580$$ 0 0
$$581$$ −3276.00 −0.233927
$$582$$ 15192.0i 1.08201i
$$583$$ − 15996.0i − 1.13634i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ −24608.0 −1.73472
$$587$$ 464.000i 0.0326258i 0.999867 + 0.0163129i $$0.00519278\pi$$
−0.999867 + 0.0163129i $$0.994807\pi$$
$$588$$ 1176.00i 0.0824786i
$$589$$ 4800.00 0.335790
$$590$$ 0 0
$$591$$ −7062.00 −0.491526
$$592$$ 15744.0i 1.09303i
$$593$$ − 11748.0i − 0.813546i −0.913529 0.406773i $$-0.866654\pi$$
0.913529 0.406773i $$-0.133346\pi$$
$$594$$ 6696.00 0.462526
$$595$$ 0 0
$$596$$ 18960.0 1.30307
$$597$$ − 5040.00i − 0.345517i
$$598$$ − 10416.0i − 0.712277i
$$599$$ −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 $$0.782948\pi$$
$$602$$ − 1904.00i − 0.128906i
$$603$$ − 8136.00i − 0.549459i
$$604$$ 4544.00 0.306114
$$605$$ 0 0
$$606$$ −2784.00 −0.186621
$$607$$ 704.000i 0.0470749i 0.999723 + 0.0235375i $$0.00749290\pi$$
−0.999723 + 0.0235375i $$0.992507\pi$$
$$608$$ 25600.0i 1.70759i
$$609$$ 210.000 0.0139731
$$610$$ 0 0
$$611$$ −20088.0 −1.33007
$$612$$ 6048.00i 0.399470i
$$613$$ − 24958.0i − 1.64444i −0.569167 0.822222i $$-0.692734\pi$$
0.569167 0.822222i $$-0.307266\pi$$
$$614$$ −23536.0 −1.54696
$$615$$ 0 0
$$616$$ 0 0
$$617$$ − 8826.00i − 0.575886i −0.957648 0.287943i $$-0.907029\pi$$
0.957648 0.287943i $$-0.0929713\pi$$
$$618$$ − 21504.0i − 1.39971i
$$619$$ −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
$$622$$ 36528.0i 2.35473i
$$623$$ 1400.00i 0.0900318i
$$624$$ 11904.0 0.763688
$$625$$ 0 0
$$626$$ −37528.0 −2.39604
$$627$$ − 18600.0i − 1.18471i
$$628$$ 2128.00i 0.135217i
$$629$$ 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
$$633$$ − 2004.00i − 0.125832i
$$634$$ −12456.0 −0.780270
$$635$$ 0 0
$$636$$ −6192.00 −0.386052
$$637$$ − 3038.00i − 0.188964i
$$638$$ 2480.00i 0.153894i
$$639$$ 6102.00 0.377764
$$640$$ 0 0
$$641$$ 13062.0 0.804864 0.402432 0.915450i $$-0.368165\pi$$
0.402432 + 0.915450i $$0.368165\pi$$
$$642$$ 22872.0i 1.40605i
$$643$$ 28012.0i 1.71802i 0.511961 + 0.859009i $$0.328919\pi$$
−0.511961 + 0.859009i $$0.671081\pi$$
$$644$$ −2352.00 −0.143916
$$645$$ 0 0
$$646$$ 33600.0 2.04640
$$647$$ 3844.00i 0.233575i 0.993157 + 0.116788i $$0.0372597\pi$$
−0.993157 + 0.116788i $$0.962740\pi$$
$$648$$ 0 0
$$649$$ −7440.00 −0.449993
$$650$$ 0 0
$$651$$ −1008.00 −0.0606861
$$652$$ − 2176.00i − 0.130704i
$$653$$ 28482.0i 1.70687i 0.521198 + 0.853436i $$0.325485\pi$$
−0.521198 + 0.853436i $$0.674515\pi$$
$$654$$ −1080.00 −0.0645739
$$655$$ 0 0
$$656$$ 15872.0 0.944661
$$657$$ − 5778.00i − 0.343107i
$$658$$ 9072.00i 0.537482i
$$659$$ 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 $$0.416811\pi$$
$$662$$ 6128.00i 0.359776i
$$663$$ − 15624.0i − 0.915212i
$$664$$ 0 0
$$665$$ 0 0
$$666$$ −8856.00 −0.515260
$$667$$ 420.000i 0.0243815i
$$668$$ 15008.0i 0.869277i
$$669$$ −5496.00 −0.317620
$$670$$ 0 0
$$671$$ 38564.0 2.21870
$$672$$ − 5376.00i − 0.308607i
$$673$$ 10562.0i 0.604956i 0.953156 + 0.302478i $$0.0978138\pi$$
−0.953156 + 0.302478i $$0.902186\pi$$
$$674$$ 16664.0 0.952334
$$675$$ 0 0
$$676$$ 13176.0 0.749659
$$677$$ − 26016.0i − 1.47692i −0.674296 0.738461i $$-0.735553\pi$$
0.674296 0.738461i $$-0.264447\pi$$
$$678$$ 5496.00i 0.311317i
$$679$$ −8862.00 −0.500872
$$680$$ 0 0
$$681$$ −14832.0 −0.834601
$$682$$ − 11904.0i − 0.668369i
$$683$$ − 8898.00i − 0.498496i −0.968440 0.249248i $$-0.919817\pi$$
0.968440 0.249248i $$-0.0801834\pi$$
$$684$$ −7200.00 −0.402484
$$685$$ 0 0
$$686$$ −1372.00 −0.0763604
$$687$$ 16410.0i 0.911325i
$$688$$ 4352.00i 0.241161i
$$689$$ 15996.0 0.884469
$$690$$ 0 0
$$691$$ 30572.0 1.68309 0.841544 0.540189i $$-0.181647\pi$$
0.841544 + 0.540189i $$0.181647\pi$$
$$692$$ − 1216.00i − 0.0667997i
$$693$$ 3906.00i 0.214108i
$$694$$ 45464.0 2.48673
$$695$$ 0 0
$$696$$ 0 0
$$697$$ − 20832.0i − 1.13209i
$$698$$ − 37240.0i − 2.01942i
$$699$$ −8406.00 −0.454856
$$700$$ 0 0
$$701$$ −30618.0 −1.64968 −0.824840 0.565366i $$-0.808735\pi$$
−0.824840 + 0.565366i $$0.808735\pi$$
$$702$$ 6696.00i 0.360006i
$$703$$ 24600.0i 1.31978i
$$704$$ 31744.0 1.69943
$$705$$ 0 0
$$706$$ −34288.0 −1.82783
$$707$$ − 1624.00i − 0.0863887i
$$708$$ 2880.00i 0.152877i
$$709$$ 8130.00 0.430647 0.215323 0.976543i $$-0.430919\pi$$
0.215323 + 0.976543i $$0.430919\pi$$
$$710$$ 0 0
$$711$$ 6660.00 0.351293
$$712$$ 0 0
$$713$$ − 2016.00i − 0.105890i
$$714$$ −7056.00 −0.369838
$$715$$ 0 0
$$716$$ 4880.00 0.254713
$$717$$ 3510.00i 0.182822i
$$718$$ 19160.0i 0.995884i
$$719$$ 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
$$722$$ 12564.0i 0.647623i
$$723$$ − 7014.00i − 0.360793i
$$724$$ −8336.00 −0.427907
$$725$$ 0 0
$$726$$ −30156.0 −1.54159
$$727$$ 14624.0i 0.746044i 0.927822 + 0.373022i $$0.121678\pi$$
−0.927822 + 0.373022i $$0.878322\pi$$
$$728$$ 0 0
$$729$$ −729.000 −0.0370370
$$730$$ 0 0
$$731$$ 5712.00 0.289010
$$732$$ − 14928.0i − 0.753763i
$$733$$ 20862.0i 1.05124i 0.850721 + 0.525618i $$0.176166\pi$$
−0.850721 + 0.525618i $$0.823834\pi$$
$$734$$ −21696.0 −1.09103
$$735$$ 0 0
$$736$$ 10752.0 0.538484
$$737$$ 56048.0i 2.80130i
$$738$$ 8928.00i 0.445317i
$$739$$ 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
$$742$$ − 7224.00i − 0.357414i
$$743$$ − 25578.0i − 1.26294i −0.775400 0.631471i $$-0.782452\pi$$
0.775400 0.631471i $$-0.217548\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 7352.00 0.360826
$$747$$ 4212.00i 0.206304i
$$748$$ − 41664.0i − 2.03661i
$$749$$ −13342.0 −0.650876
$$750$$ 0 0
$$751$$ 33472.0 1.62638 0.813189 0.581999i $$-0.197729\pi$$
0.813189 + 0.581999i $$0.197729\pi$$
$$752$$ − 20736.0i − 1.00554i
$$753$$ 8376.00i 0.405363i
$$754$$ −2480.00 −0.119783
$$755$$ 0 0
$$756$$ 1512.00 0.0727393
$$757$$ 25934.0i 1.24516i 0.782556 + 0.622581i $$0.213916\pi$$
−0.782556 + 0.622581i $$0.786084\pi$$
$$758$$ 17040.0i 0.816518i
$$759$$ −7812.00 −0.373594
$$760$$ 0 0
$$761$$ 26952.0 1.28385 0.641925 0.766768i $$-0.278136\pi$$
0.641925 + 0.766768i $$0.278136\pi$$
$$762$$ − 9648.00i − 0.458675i
$$763$$ − 630.000i − 0.0298919i
$$764$$ 16304.0 0.772065
$$765$$ 0 0
$$766$$ 36192.0 1.70714
$$767$$ − 7440.00i − 0.350251i
$$768$$ 12288.0i 0.577350i
$$769$$ −23450.0 −1.09965 −0.549824 0.835281i $$-0.685305\pi$$
−0.549824 + 0.835281i $$0.685305\pi$$
$$770$$ 0 0
$$771$$ −21072.0 −0.984293
$$772$$ − 20816.0i − 0.970446i
$$773$$ − 39568.0i − 1.84109i −0.390637 0.920545i $$-0.627745\pi$$
0.390637 0.920545i $$-0.372255\pi$$
$$774$$ −2448.00 −0.113684
$$775$$ 0 0
$$776$$ 0 0
$$777$$ − 5166.00i − 0.238519i
$$778$$ 45960.0i 2.11793i
$$779$$ 24800.0 1.14063
$$780$$ 0 0
$$781$$ −42036.0 −1.92595
$$782$$ − 14112.0i − 0.645325i
$$783$$ − 270.000i − 0.0123231i
$$784$$ 3136.00 0.142857
$$785$$ 0 0
$$786$$ −9744.00 −0.442184
$$787$$ − 12356.0i − 0.559649i −0.960051 0.279825i $$-0.909724\pi$$
0.960051 0.279825i $$-0.0902763\pi$$
$$788$$ − 18832.0i − 0.851348i
$$789$$ 7314.00 0.330019
$$790$$ 0 0
$$791$$ −3206.00 −0.144112
$$792$$ 0 0
$$793$$ 38564.0i 1.72692i
$$794$$ 7464.00 0.333611
$$795$$ 0 0
$$796$$ 13440.0 0.598452
$$797$$ − 21736.0i − 0.966033i −0.875611 0.483017i $$-0.839541\pi$$
0.875611 0.483017i $$-0.160459\pi$$
$$798$$ − 8400.00i − 0.372628i
$$799$$ −27216.0 −1.20505
$$800$$ 0 0
$$801$$ 1800.00 0.0794006
$$802$$ 54648.0i 2.40609i
$$803$$ 39804.0i 1.74926i
$$804$$ 21696.0 0.951690
$$805$$ 0 0
$$806$$ 11904.0 0.520224
$$807$$ 20340.0i 0.887239i
$$808$$ 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 $$0.485303\pi$$
$$812$$ 560.000i 0.0242022i
$$813$$ − 5784.00i − 0.249513i
$$814$$ 61008.0 2.62694
$$815$$ 0 0
$$816$$ 16128.0 0.691903
$$817$$ 6800.00i 0.291190i
$$818$$ 52840.0i 2.25857i
$$819$$ −3906.00 −0.166650
$$820$$ 0 0
$$821$$ 5002.00 0.212632 0.106316 0.994332i $$-0.466094\pi$$
0.106316 + 0.994332i $$0.466094\pi$$
$$822$$ − 4968.00i − 0.210802i
$$823$$ 3612.00i 0.152985i 0.997070 + 0.0764923i $$0.0243721\pi$$
−0.997070 + 0.0764923i $$0.975628\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ −3360.00 −0.141537
$$827$$ − 27666.0i − 1.16329i −0.813443 0.581645i $$-0.802409\pi$$
0.813443 0.581645i $$-0.197591\pi$$
$$828$$ 3024.00i 0.126922i
$$829$$ −12890.0 −0.540034 −0.270017 0.962856i $$-0.587029\pi$$
−0.270017 + 0.962856i $$0.587029\pi$$
$$830$$ 0 0
$$831$$ −16662.0 −0.695546
$$832$$ 31744.0i 1.32275i
$$833$$ − 4116.00i − 0.171202i
$$834$$ −19440.0 −0.807137
$$835$$ 0 0
$$836$$ 49600.0 2.05198
$$837$$ 1296.00i 0.0535201i
$$838$$ − 27840.0i − 1.14763i
$$839$$ 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
$$842$$ 32648.0i 1.33625i
$$843$$ 5826.00i 0.238029i
$$844$$ 5344.00 0.217948
$$845$$ 0 0
$$846$$ 11664.0 0.474015
$$847$$ − 17591.0i − 0.713617i
$$848$$ 16512.0i 0.668661i
$$849$$ 14484.0 0.585500
$$850$$ 0 0
$$851$$ 10332.0 0.416188
$$852$$ 16272.0i 0.654307i
$$853$$ 33082.0i 1.32791i 0.747773 + 0.663954i $$0.231123\pi$$
−0.747773 + 0.663954i $$0.768877\pi$$
$$854$$ 17416.0 0.697849
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 7544.00i 0.300698i 0.988633 + 0.150349i $$0.0480397\pi$$
−0.988633 + 0.150349i $$0.951960\pi$$
$$858$$ − 46128.0i − 1.83541i
$$859$$ −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
$$862$$ 66408.0i 2.62397i
$$863$$ − 10518.0i − 0.414875i −0.978248 0.207437i $$-0.933488\pi$$
0.978248 0.207437i $$-0.0665123\pi$$
$$864$$ −6912.00 −0.272166
$$865$$ 0 0
$$866$$ 30952.0 1.21454
$$867$$ − 6429.00i − 0.251834i
$$868$$ − 2688.00i − 0.105111i
$$869$$ −45880.0 −1.79099
$$870$$ 0 0
$$871$$ −56048.0 −2.18038
$$872$$ 0 0
$$873$$ 11394.0i 0.441728i
$$874$$ 16800.0 0.650193
$$875$$ 0 0
$$876$$ 15408.0 0.594279
$$877$$ 14134.0i 0.544209i 0.962268 + 0.272104i $$0.0877196\pi$$
−0.962268 + 0.272104i $$0.912280\pi$$
$$878$$ 3360.00i 0.129151i
$$879$$ −18456.0 −0.708197
$$880$$ 0 0
$$881$$ 6492.00 0.248265 0.124132 0.992266i $$-0.460385\pi$$
0.124132 + 0.992266i $$0.460385\pi$$
$$882$$ 1764.00i 0.0673435i
$$883$$ − 38228.0i − 1.45694i −0.685080 0.728468i $$-0.740233\pi$$
0.685080 0.728468i $$-0.259767\pi$$
$$884$$ 41664.0 1.58519
$$885$$ 0 0
$$886$$ 26472.0 1.00377
$$887$$ − 43076.0i − 1.63061i −0.579032 0.815305i $$-0.696569\pi$$
0.579032 0.815305i $$-0.303431\pi$$
$$888$$ 0 0
$$889$$ 5628.00 0.212325
$$890$$ 0 0
$$891$$ 5022.00 0.188825
$$892$$ − 14656.0i − 0.550134i
$$893$$ − 32400.0i − 1.21414i
$$894$$ 28440.0 1.06396
$$895$$ 0 0
$$896$$ 0 0
$$897$$ − 7812.00i − 0.290786i
$$898$$ − 12360.0i − 0.459308i
$$899$$ −480.000 −0.0178074
$$900$$ 0 0
$$901$$ 21672.0 0.801331
$$902$$ − 61504.0i − 2.27035i
$$903$$ − 1428.00i − 0.0526255i
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 6816.00 0.249941
$$907$$ − 32236.0i − 1.18013i −0.807355 0.590065i $$-0.799102\pi$$
0.807355 0.590065i $$-0.200898\pi$$
$$908$$ − 39552.0i − 1.44557i
$$909$$ −2088.00 −0.0761877
$$910$$ 0 0
$$911$$ −46518.0 −1.69178 −0.845889 0.533359i $$-0.820930\pi$$
−0.845889 + 0.533359i $$0.820930\pi$$
$$912$$ 19200.0i 0.697122i
$$913$$ − 29016.0i − 1.05180i
$$914$$ −23656.0 −0.856095
$$915$$ 0 0
$$916$$ −43760.0 −1.57846
$$917$$ − 5684.00i − 0.204692i
$$918$$ 9072.00i 0.326166i
$$919$$ −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
$$922$$ − 63872.0i − 2.28147i
$$923$$ − 42036.0i − 1.49906i
$$924$$ −10416.0 −0.370846
$$925$$ 0 0
$$926$$ −4688.00 −0.166369
$$927$$ − 16128.0i − 0.571427i
$$928$$ − 2560.00i − 0.0905562i
$$929$$ −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
$$932$$ − 22416.0i − 0.787833i
$$933$$ 27396.0i 0.961313i
$$934$$ −21216.0 −0.743264
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 36114.0i 1.25912i 0.776953 + 0.629559i $$0.216764\pi$$
−0.776953 + 0.629559i $$0.783236\pi$$
$$938$$ 25312.0i 0.881094i
$$939$$ −28146.0 −0.978179
$$940$$ 0 0
$$941$$ −4748.00 −0.164485 −0.0822425 0.996612i $$-0.526208\pi$$
−0.0822425 + 0.996612i $$0.526208\pi$$
$$942$$ 3192.00i 0.110404i
$$943$$ − 10416.0i − 0.359694i
$$944$$ 7680.00 0.264791
$$945$$ 0 0
$$946$$ 16864.0 0.579594
$$947$$ 42694.0i 1.46501i 0.680759 + 0.732507i $$0.261650\pi$$
−0.680759 + 0.732507i $$0.738350\pi$$
$$948$$ 17760.0i 0.608458i
$$949$$ −39804.0 −1.36153
$$950$$ 0 0
$$951$$ −9342.00 −0.318544
$$952$$ 0 0
$$953$$ 16742.0i 0.569073i 0.958665 + 0.284537i $$0.0918397\pi$$
−0.958665 + 0.284537i $$0.908160\pi$$
$$954$$ −9288.00 −0.315210
$$955$$ 0 0
$$956$$ −9360.00 −0.316657
$$957$$ 1860.00i 0.0628268i
$$958$$ − 22960.0i − 0.774326i
$$959$$ 2898.00 0.0975822
$$960$$ 0 0
$$961$$ −27487.0 −0.922661
$$962$$ 61008.0i 2.04467i
$$963$$ 17154.0i 0.574019i
$$964$$ 18704.0 0.624912
$$965$$ 0 0
$$966$$ −3528.00 −0.117507
$$967$$ − 9956.00i − 0.331089i −0.986202 0.165545i $$-0.947062\pi$$
0.986202 0.165545i $$-0.0529382\pi$$
$$968$$ 0 0
$$969$$ 25200.0 0.835439
$$970$$ 0 0
$$971$$ −26388.0 −0.872123 −0.436061 0.899917i $$-0.643627\pi$$
−0.436061 + 0.899917i $$0.643627\pi$$
$$972$$ − 1944.00i − 0.0641500i
$$973$$ − 11340.0i − 0.373632i
$$974$$ −35776.0 −1.17694
$$975$$ 0 0
$$976$$ −39808.0 −1.30556
$$977$$ − 786.000i − 0.0257383i −0.999917 0.0128692i $$-0.995904\pi$$
0.999917 0.0128692i $$-0.00409650\pi$$
$$978$$ − 3264.00i − 0.106719i
$$979$$ −12400.0 −0.404807
$$980$$ 0 0
$$981$$ −810.000 −0.0263622
$$982$$ − 22232.0i − 0.722456i
$$983$$ − 51888.0i − 1.68359i −0.539796 0.841796i $$-0.681499\pi$$
0.539796 0.841796i $$-0.318501\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ −3360.00 −0.108524
$$987$$ 6804.00i 0.219426i
$$988$$ 49600.0i 1.59715i
$$989$$ 2856.00 0.0918256
$$990$$ 0 0
$$991$$ −51928.0 −1.66453 −0.832264 0.554379i $$-0.812956\pi$$
−0.832264 + 0.554379i $$0.812956\pi$$
$$992$$ 12288.0i 0.393291i
$$993$$ 4596.00i 0.146878i
$$994$$ −18984.0 −0.605771
$$995$$ 0 0
$$996$$ −11232.0 −0.357329
$$997$$ − 386.000i − 0.0122615i −0.999981 0.00613076i $$-0.998049\pi$$
0.999981 0.00613076i $$-0.00195149\pi$$
$$998$$ 79280.0i 2.51459i
$$999$$ −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 525.4.d.b.274.2 2
5.2 odd 4 525.4.a.b.1.1 1
5.3 odd 4 21.4.a.b.1.1 1
5.4 even 2 inner 525.4.d.b.274.1 2
15.2 even 4 1575.4.a.k.1.1 1
15.8 even 4 63.4.a.a.1.1 1
20.3 even 4 336.4.a.h.1.1 1
35.3 even 12 147.4.e.b.79.1 2
35.13 even 4 147.4.a.g.1.1 1
35.18 odd 12 147.4.e.c.79.1 2
35.23 odd 12 147.4.e.c.67.1 2
35.33 even 12 147.4.e.b.67.1 2
40.3 even 4 1344.4.a.i.1.1 1
40.13 odd 4 1344.4.a.w.1.1 1
60.23 odd 4 1008.4.a.m.1.1 1
105.23 even 12 441.4.e.m.361.1 2
105.38 odd 12 441.4.e.n.226.1 2
105.53 even 12 441.4.e.m.226.1 2
105.68 odd 12 441.4.e.n.361.1 2
105.83 odd 4 441.4.a.b.1.1 1
140.83 odd 4 2352.4.a.l.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
21.4.a.b.1.1 1 5.3 odd 4
63.4.a.a.1.1 1 15.8 even 4
147.4.a.g.1.1 1 35.13 even 4
147.4.e.b.67.1 2 35.33 even 12
147.4.e.b.79.1 2 35.3 even 12
147.4.e.c.67.1 2 35.23 odd 12
147.4.e.c.79.1 2 35.18 odd 12
336.4.a.h.1.1 1 20.3 even 4
441.4.a.b.1.1 1 105.83 odd 4
441.4.e.m.226.1 2 105.53 even 12
441.4.e.m.361.1 2 105.23 even 12
441.4.e.n.226.1 2 105.38 odd 12
441.4.e.n.361.1 2 105.68 odd 12
525.4.a.b.1.1 1 5.2 odd 4
525.4.d.b.274.1 2 5.4 even 2 inner
525.4.d.b.274.2 2 1.1 even 1 trivial
1008.4.a.m.1.1 1 60.23 odd 4
1344.4.a.i.1.1 1 40.3 even 4
1344.4.a.w.1.1 1 40.13 odd 4
1575.4.a.k.1.1 1 15.2 even 4
2352.4.a.l.1.1 1 140.83 odd 4