# Properties

 Label 1200.4.f.t.49.2 Level $1200$ Weight $4$ Character 1200.49 Analytic conductor $70.802$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1200 = 2^{4} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1200.f (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$70.8022920069$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 120) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 49.2 Root $$1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 1200.49 Dual form 1200.4.f.t.49.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+3.00000i q^{3} +20.0000i q^{7} -9.00000 q^{9} +O(q^{10})$$ $$q+3.00000i q^{3} +20.0000i q^{7} -9.00000 q^{9} +56.0000 q^{11} -86.0000i q^{13} +106.000i q^{17} +4.00000 q^{19} -60.0000 q^{21} -136.000i q^{23} -27.0000i q^{27} +206.000 q^{29} +152.000 q^{31} +168.000i q^{33} -282.000i q^{37} +258.000 q^{39} -246.000 q^{41} -412.000i q^{43} +40.0000i q^{47} -57.0000 q^{49} -318.000 q^{51} -126.000i q^{53} +12.0000i q^{57} +56.0000 q^{59} -2.00000 q^{61} -180.000i q^{63} -388.000i q^{67} +408.000 q^{69} +672.000 q^{71} +1170.00i q^{73} +1120.00i q^{77} +408.000 q^{79} +81.0000 q^{81} -668.000i q^{83} +618.000i q^{87} -66.0000 q^{89} +1720.00 q^{91} +456.000i q^{93} +926.000i q^{97} -504.000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 18 q^{9}+O(q^{10})$$ 2 * q - 18 * q^9 $$2 q - 18 q^{9} + 112 q^{11} + 8 q^{19} - 120 q^{21} + 412 q^{29} + 304 q^{31} + 516 q^{39} - 492 q^{41} - 114 q^{49} - 636 q^{51} + 112 q^{59} - 4 q^{61} + 816 q^{69} + 1344 q^{71} + 816 q^{79} + 162 q^{81} - 132 q^{89} + 3440 q^{91} - 1008 q^{99}+O(q^{100})$$ 2 * q - 18 * q^9 + 112 * q^11 + 8 * q^19 - 120 * q^21 + 412 * q^29 + 304 * q^31 + 516 * q^39 - 492 * q^41 - 114 * q^49 - 636 * q^51 + 112 * q^59 - 4 * q^61 + 816 * q^69 + 1344 * q^71 + 816 * q^79 + 162 * q^81 - 132 * q^89 + 3440 * q^91 - 1008 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1200\mathbb{Z}\right)^\times$$.

 $$n$$ $$401$$ $$577$$ $$751$$ $$901$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$ $$1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 3.00000i 0.577350i
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 20.0000i 1.07990i 0.841698 + 0.539949i $$0.181557\pi$$
−0.841698 + 0.539949i $$0.818443\pi$$
$$8$$ 0 0
$$9$$ −9.00000 −0.333333
$$10$$ 0 0
$$11$$ 56.0000 1.53497 0.767483 0.641069i $$-0.221509\pi$$
0.767483 + 0.641069i $$0.221509\pi$$
$$12$$ 0 0
$$13$$ − 86.0000i − 1.83478i −0.397992 0.917389i $$-0.630293\pi$$
0.397992 0.917389i $$-0.369707\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 106.000i 1.51228i 0.654409 + 0.756140i $$0.272917\pi$$
−0.654409 + 0.756140i $$0.727083\pi$$
$$18$$ 0 0
$$19$$ 4.00000 0.0482980 0.0241490 0.999708i $$-0.492312\pi$$
0.0241490 + 0.999708i $$0.492312\pi$$
$$20$$ 0 0
$$21$$ −60.0000 −0.623480
$$22$$ 0 0
$$23$$ − 136.000i − 1.23295i −0.787373 0.616477i $$-0.788559\pi$$
0.787373 0.616477i $$-0.211441\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ − 27.0000i − 0.192450i
$$28$$ 0 0
$$29$$ 206.000 1.31908 0.659539 0.751671i $$-0.270752\pi$$
0.659539 + 0.751671i $$0.270752\pi$$
$$30$$ 0 0
$$31$$ 152.000 0.880645 0.440323 0.897840i $$-0.354864\pi$$
0.440323 + 0.897840i $$0.354864\pi$$
$$32$$ 0 0
$$33$$ 168.000i 0.886214i
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ − 282.000i − 1.25299i −0.779427 0.626493i $$-0.784490\pi$$
0.779427 0.626493i $$-0.215510\pi$$
$$38$$ 0 0
$$39$$ 258.000 1.05931
$$40$$ 0 0
$$41$$ −246.000 −0.937043 −0.468521 0.883452i $$-0.655213\pi$$
−0.468521 + 0.883452i $$0.655213\pi$$
$$42$$ 0 0
$$43$$ − 412.000i − 1.46115i −0.682833 0.730575i $$-0.739252\pi$$
0.682833 0.730575i $$-0.260748\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 40.0000i 0.124140i 0.998072 + 0.0620702i $$0.0197703\pi$$
−0.998072 + 0.0620702i $$0.980230\pi$$
$$48$$ 0 0
$$49$$ −57.0000 −0.166181
$$50$$ 0 0
$$51$$ −318.000 −0.873116
$$52$$ 0 0
$$53$$ − 126.000i − 0.326555i −0.986580 0.163278i $$-0.947793\pi$$
0.986580 0.163278i $$-0.0522066\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 12.0000i 0.0278849i
$$58$$ 0 0
$$59$$ 56.0000 0.123569 0.0617846 0.998090i $$-0.480321\pi$$
0.0617846 + 0.998090i $$0.480321\pi$$
$$60$$ 0 0
$$61$$ −2.00000 −0.00419793 −0.00209897 0.999998i $$-0.500668\pi$$
−0.00209897 + 0.999998i $$0.500668\pi$$
$$62$$ 0 0
$$63$$ − 180.000i − 0.359966i
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ − 388.000i − 0.707489i −0.935342 0.353744i $$-0.884908\pi$$
0.935342 0.353744i $$-0.115092\pi$$
$$68$$ 0 0
$$69$$ 408.000 0.711847
$$70$$ 0 0
$$71$$ 672.000 1.12326 0.561632 0.827387i $$-0.310174\pi$$
0.561632 + 0.827387i $$0.310174\pi$$
$$72$$ 0 0
$$73$$ 1170.00i 1.87586i 0.346818 + 0.937932i $$0.387262\pi$$
−0.346818 + 0.937932i $$0.612738\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 1120.00i 1.65761i
$$78$$ 0 0
$$79$$ 408.000 0.581058 0.290529 0.956866i $$-0.406169\pi$$
0.290529 + 0.956866i $$0.406169\pi$$
$$80$$ 0 0
$$81$$ 81.0000 0.111111
$$82$$ 0 0
$$83$$ − 668.000i − 0.883404i −0.897162 0.441702i $$-0.854375\pi$$
0.897162 0.441702i $$-0.145625\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 618.000i 0.761570i
$$88$$ 0 0
$$89$$ −66.0000 −0.0786066 −0.0393033 0.999227i $$-0.512514\pi$$
−0.0393033 + 0.999227i $$0.512514\pi$$
$$90$$ 0 0
$$91$$ 1720.00 1.98137
$$92$$ 0 0
$$93$$ 456.000i 0.508441i
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 926.000i 0.969289i 0.874711 + 0.484645i $$0.161051\pi$$
−0.874711 + 0.484645i $$0.838949\pi$$
$$98$$ 0 0
$$99$$ −504.000 −0.511656
$$100$$ 0 0
$$101$$ −198.000 −0.195067 −0.0975333 0.995232i $$-0.531095\pi$$
−0.0975333 + 0.995232i $$0.531095\pi$$
$$102$$ 0 0
$$103$$ 1532.00i 1.46556i 0.680467 + 0.732779i $$0.261777\pi$$
−0.680467 + 0.732779i $$0.738223\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ − 444.000i − 0.401150i −0.979678 0.200575i $$-0.935719\pi$$
0.979678 0.200575i $$-0.0642811\pi$$
$$108$$ 0 0
$$109$$ −62.0000 −0.0544819 −0.0272409 0.999629i $$-0.508672\pi$$
−0.0272409 + 0.999629i $$0.508672\pi$$
$$110$$ 0 0
$$111$$ 846.000 0.723412
$$112$$ 0 0
$$113$$ 414.000i 0.344653i 0.985040 + 0.172327i $$0.0551285\pi$$
−0.985040 + 0.172327i $$0.944872\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 774.000i 0.611593i
$$118$$ 0 0
$$119$$ −2120.00 −1.63311
$$120$$ 0 0
$$121$$ 1805.00 1.35612
$$122$$ 0 0
$$123$$ − 738.000i − 0.541002i
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ − 996.000i − 0.695911i −0.937511 0.347956i $$-0.886876\pi$$
0.937511 0.347956i $$-0.113124\pi$$
$$128$$ 0 0
$$129$$ 1236.00 0.843595
$$130$$ 0 0
$$131$$ 264.000 0.176075 0.0880374 0.996117i $$-0.471941\pi$$
0.0880374 + 0.996117i $$0.471941\pi$$
$$132$$ 0 0
$$133$$ 80.0000i 0.0521570i
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ − 2278.00i − 1.42060i −0.703897 0.710302i $$-0.748558\pi$$
0.703897 0.710302i $$-0.251442\pi$$
$$138$$ 0 0
$$139$$ 1812.00 1.10570 0.552848 0.833282i $$-0.313541\pi$$
0.552848 + 0.833282i $$0.313541\pi$$
$$140$$ 0 0
$$141$$ −120.000 −0.0716725
$$142$$ 0 0
$$143$$ − 4816.00i − 2.81632i
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ − 171.000i − 0.0959445i
$$148$$ 0 0
$$149$$ 1534.00 0.843424 0.421712 0.906730i $$-0.361429\pi$$
0.421712 + 0.906730i $$0.361429\pi$$
$$150$$ 0 0
$$151$$ 3016.00 1.62542 0.812711 0.582668i $$-0.197991\pi$$
0.812711 + 0.582668i $$0.197991\pi$$
$$152$$ 0 0
$$153$$ − 954.000i − 0.504094i
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 1814.00i 0.922121i 0.887369 + 0.461060i $$0.152531\pi$$
−0.887369 + 0.461060i $$0.847469\pi$$
$$158$$ 0 0
$$159$$ 378.000 0.188537
$$160$$ 0 0
$$161$$ 2720.00 1.33147
$$162$$ 0 0
$$163$$ 1844.00i 0.886093i 0.896499 + 0.443047i $$0.146102\pi$$
−0.896499 + 0.443047i $$0.853898\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 3768.00i 1.74597i 0.487749 + 0.872984i $$0.337818\pi$$
−0.487749 + 0.872984i $$0.662182\pi$$
$$168$$ 0 0
$$169$$ −5199.00 −2.36641
$$170$$ 0 0
$$171$$ −36.0000 −0.0160993
$$172$$ 0 0
$$173$$ 938.000i 0.412224i 0.978528 + 0.206112i $$0.0660812\pi$$
−0.978528 + 0.206112i $$0.933919\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 168.000i 0.0713427i
$$178$$ 0 0
$$179$$ 3968.00 1.65688 0.828442 0.560075i $$-0.189228\pi$$
0.828442 + 0.560075i $$0.189228\pi$$
$$180$$ 0 0
$$181$$ −3514.00 −1.44306 −0.721529 0.692384i $$-0.756560\pi$$
−0.721529 + 0.692384i $$0.756560\pi$$
$$182$$ 0 0
$$183$$ − 6.00000i − 0.00242368i
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 5936.00i 2.32130i
$$188$$ 0 0
$$189$$ 540.000 0.207827
$$190$$ 0 0
$$191$$ 1480.00 0.560676 0.280338 0.959901i $$-0.409554\pi$$
0.280338 + 0.959901i $$0.409554\pi$$
$$192$$ 0 0
$$193$$ − 2774.00i − 1.03460i −0.855806 0.517298i $$-0.826938\pi$$
0.855806 0.517298i $$-0.173062\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 3806.00i 1.37648i 0.725484 + 0.688239i $$0.241616\pi$$
−0.725484 + 0.688239i $$0.758384\pi$$
$$198$$ 0 0
$$199$$ −856.000 −0.304926 −0.152463 0.988309i $$-0.548720\pi$$
−0.152463 + 0.988309i $$0.548720\pi$$
$$200$$ 0 0
$$201$$ 1164.00 0.408469
$$202$$ 0 0
$$203$$ 4120.00i 1.42447i
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 1224.00i 0.410985i
$$208$$ 0 0
$$209$$ 224.000 0.0741359
$$210$$ 0 0
$$211$$ −3020.00 −0.985334 −0.492667 0.870218i $$-0.663978\pi$$
−0.492667 + 0.870218i $$0.663978\pi$$
$$212$$ 0 0
$$213$$ 2016.00i 0.648517i
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 3040.00i 0.951008i
$$218$$ 0 0
$$219$$ −3510.00 −1.08303
$$220$$ 0 0
$$221$$ 9116.00 2.77470
$$222$$ 0 0
$$223$$ 1684.00i 0.505690i 0.967507 + 0.252845i $$0.0813664\pi$$
−0.967507 + 0.252845i $$0.918634\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 2004.00i 0.585948i 0.956120 + 0.292974i $$0.0946449\pi$$
−0.956120 + 0.292974i $$0.905355\pi$$
$$228$$ 0 0
$$229$$ 5042.00 1.45496 0.727478 0.686131i $$-0.240693\pi$$
0.727478 + 0.686131i $$0.240693\pi$$
$$230$$ 0 0
$$231$$ −3360.00 −0.957021
$$232$$ 0 0
$$233$$ − 3090.00i − 0.868810i −0.900718 0.434405i $$-0.856959\pi$$
0.900718 0.434405i $$-0.143041\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 1224.00i 0.335474i
$$238$$ 0 0
$$239$$ 2136.00 0.578102 0.289051 0.957314i $$-0.406660\pi$$
0.289051 + 0.957314i $$0.406660\pi$$
$$240$$ 0 0
$$241$$ 98.0000 0.0261939 0.0130970 0.999914i $$-0.495831\pi$$
0.0130970 + 0.999914i $$0.495831\pi$$
$$242$$ 0 0
$$243$$ 243.000i 0.0641500i
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ − 344.000i − 0.0886162i
$$248$$ 0 0
$$249$$ 2004.00 0.510033
$$250$$ 0 0
$$251$$ 5040.00 1.26742 0.633709 0.773571i $$-0.281532\pi$$
0.633709 + 0.773571i $$0.281532\pi$$
$$252$$ 0 0
$$253$$ − 7616.00i − 1.89254i
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 1986.00i 0.482036i 0.970521 + 0.241018i $$0.0774813\pi$$
−0.970521 + 0.241018i $$0.922519\pi$$
$$258$$ 0 0
$$259$$ 5640.00 1.35310
$$260$$ 0 0
$$261$$ −1854.00 −0.439692
$$262$$ 0 0
$$263$$ − 1416.00i − 0.331994i −0.986126 0.165997i $$-0.946916\pi$$
0.986126 0.165997i $$-0.0530841\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ − 198.000i − 0.0453835i
$$268$$ 0 0
$$269$$ 6670.00 1.51181 0.755905 0.654681i $$-0.227197\pi$$
0.755905 + 0.654681i $$0.227197\pi$$
$$270$$ 0 0
$$271$$ −48.0000 −0.0107594 −0.00537969 0.999986i $$-0.501712\pi$$
−0.00537969 + 0.999986i $$0.501712\pi$$
$$272$$ 0 0
$$273$$ 5160.00i 1.14395i
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ − 6938.00i − 1.50492i −0.658636 0.752462i $$-0.728866\pi$$
0.658636 0.752462i $$-0.271134\pi$$
$$278$$ 0 0
$$279$$ −1368.00 −0.293548
$$280$$ 0 0
$$281$$ −1694.00 −0.359628 −0.179814 0.983701i $$-0.557550\pi$$
−0.179814 + 0.983701i $$0.557550\pi$$
$$282$$ 0 0
$$283$$ 6364.00i 1.33675i 0.743824 + 0.668376i $$0.233010\pi$$
−0.743824 + 0.668376i $$0.766990\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ − 4920.00i − 1.01191i
$$288$$ 0 0
$$289$$ −6323.00 −1.28699
$$290$$ 0 0
$$291$$ −2778.00 −0.559619
$$292$$ 0 0
$$293$$ − 3134.00i − 0.624881i −0.949937 0.312441i $$-0.898853\pi$$
0.949937 0.312441i $$-0.101147\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ − 1512.00i − 0.295405i
$$298$$ 0 0
$$299$$ −11696.0 −2.26220
$$300$$ 0 0
$$301$$ 8240.00 1.57789
$$302$$ 0 0
$$303$$ − 594.000i − 0.112622i
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ − 236.000i − 0.0438737i −0.999759 0.0219369i $$-0.993017\pi$$
0.999759 0.0219369i $$-0.00698328\pi$$
$$308$$ 0 0
$$309$$ −4596.00 −0.846140
$$310$$ 0 0
$$311$$ −3776.00 −0.688480 −0.344240 0.938882i $$-0.611863\pi$$
−0.344240 + 0.938882i $$0.611863\pi$$
$$312$$ 0 0
$$313$$ − 7918.00i − 1.42988i −0.699187 0.714939i $$-0.746454\pi$$
0.699187 0.714939i $$-0.253546\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ − 4362.00i − 0.772853i −0.922320 0.386426i $$-0.873709\pi$$
0.922320 0.386426i $$-0.126291\pi$$
$$318$$ 0 0
$$319$$ 11536.0 2.02474
$$320$$ 0 0
$$321$$ 1332.00 0.231604
$$322$$ 0 0
$$323$$ 424.000i 0.0730402i
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ − 186.000i − 0.0314551i
$$328$$ 0 0
$$329$$ −800.000 −0.134059
$$330$$ 0 0
$$331$$ −7980.00 −1.32514 −0.662569 0.749001i $$-0.730534\pi$$
−0.662569 + 0.749001i $$0.730534\pi$$
$$332$$ 0 0
$$333$$ 2538.00i 0.417662i
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 8294.00i 1.34066i 0.742062 + 0.670331i $$0.233848\pi$$
−0.742062 + 0.670331i $$0.766152\pi$$
$$338$$ 0 0
$$339$$ −1242.00 −0.198986
$$340$$ 0 0
$$341$$ 8512.00 1.35176
$$342$$ 0 0
$$343$$ 5720.00i 0.900440i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ − 964.000i − 0.149136i −0.997216 0.0745681i $$-0.976242\pi$$
0.997216 0.0745681i $$-0.0237578\pi$$
$$348$$ 0 0
$$349$$ −8670.00 −1.32978 −0.664892 0.746940i $$-0.731522\pi$$
−0.664892 + 0.746940i $$0.731522\pi$$
$$350$$ 0 0
$$351$$ −2322.00 −0.353103
$$352$$ 0 0
$$353$$ − 2314.00i − 0.348900i −0.984666 0.174450i $$-0.944185\pi$$
0.984666 0.174450i $$-0.0558148\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ − 6360.00i − 0.942876i
$$358$$ 0 0
$$359$$ −1896.00 −0.278738 −0.139369 0.990240i $$-0.544507\pi$$
−0.139369 + 0.990240i $$0.544507\pi$$
$$360$$ 0 0
$$361$$ −6843.00 −0.997667
$$362$$ 0 0
$$363$$ 5415.00i 0.782958i
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 1484.00i 0.211074i 0.994415 + 0.105537i $$0.0336562\pi$$
−0.994415 + 0.105537i $$0.966344\pi$$
$$368$$ 0 0
$$369$$ 2214.00 0.312348
$$370$$ 0 0
$$371$$ 2520.00 0.352647
$$372$$ 0 0
$$373$$ 12370.0i 1.71714i 0.512694 + 0.858571i $$0.328648\pi$$
−0.512694 + 0.858571i $$0.671352\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ − 17716.0i − 2.42021i
$$378$$ 0 0
$$379$$ 5620.00 0.761689 0.380844 0.924639i $$-0.375633\pi$$
0.380844 + 0.924639i $$0.375633\pi$$
$$380$$ 0 0
$$381$$ 2988.00 0.401784
$$382$$ 0 0
$$383$$ − 5880.00i − 0.784475i −0.919864 0.392238i $$-0.871701\pi$$
0.919864 0.392238i $$-0.128299\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 3708.00i 0.487050i
$$388$$ 0 0
$$389$$ −2082.00 −0.271367 −0.135683 0.990752i $$-0.543323\pi$$
−0.135683 + 0.990752i $$0.543323\pi$$
$$390$$ 0 0
$$391$$ 14416.0 1.86457
$$392$$ 0 0
$$393$$ 792.000i 0.101657i
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 1742.00i 0.220223i 0.993919 + 0.110111i $$0.0351208\pi$$
−0.993919 + 0.110111i $$0.964879\pi$$
$$398$$ 0 0
$$399$$ −240.000 −0.0301129
$$400$$ 0 0
$$401$$ −3270.00 −0.407222 −0.203611 0.979052i $$-0.565268\pi$$
−0.203611 + 0.979052i $$0.565268\pi$$
$$402$$ 0 0
$$403$$ − 13072.0i − 1.61579i
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ − 15792.0i − 1.92329i
$$408$$ 0 0
$$409$$ 6134.00 0.741581 0.370791 0.928716i $$-0.379087\pi$$
0.370791 + 0.928716i $$0.379087\pi$$
$$410$$ 0 0
$$411$$ 6834.00 0.820186
$$412$$ 0 0
$$413$$ 1120.00i 0.133442i
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 5436.00i 0.638374i
$$418$$ 0 0
$$419$$ −10392.0 −1.21165 −0.605826 0.795597i $$-0.707157\pi$$
−0.605826 + 0.795597i $$0.707157\pi$$
$$420$$ 0 0
$$421$$ −12690.0 −1.46906 −0.734528 0.678578i $$-0.762596\pi$$
−0.734528 + 0.678578i $$0.762596\pi$$
$$422$$ 0 0
$$423$$ − 360.000i − 0.0413801i
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ − 40.0000i − 0.00453334i
$$428$$ 0 0
$$429$$ 14448.0 1.62600
$$430$$ 0 0
$$431$$ 7408.00 0.827914 0.413957 0.910297i $$-0.364146\pi$$
0.413957 + 0.910297i $$0.364146\pi$$
$$432$$ 0 0
$$433$$ − 5062.00i − 0.561811i −0.959735 0.280906i $$-0.909365\pi$$
0.959735 0.280906i $$-0.0906348\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ − 544.000i − 0.0595493i
$$438$$ 0 0
$$439$$ −7160.00 −0.778424 −0.389212 0.921148i $$-0.627253\pi$$
−0.389212 + 0.921148i $$0.627253\pi$$
$$440$$ 0 0
$$441$$ 513.000 0.0553936
$$442$$ 0 0
$$443$$ − 17100.0i − 1.83396i −0.398930 0.916981i $$-0.630618\pi$$
0.398930 0.916981i $$-0.369382\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 4602.00i 0.486951i
$$448$$ 0 0
$$449$$ −8634.00 −0.907491 −0.453746 0.891131i $$-0.649913\pi$$
−0.453746 + 0.891131i $$0.649913\pi$$
$$450$$ 0 0
$$451$$ −13776.0 −1.43833
$$452$$ 0 0
$$453$$ 9048.00i 0.938437i
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ − 2986.00i − 0.305644i −0.988254 0.152822i $$-0.951164\pi$$
0.988254 0.152822i $$-0.0488361\pi$$
$$458$$ 0 0
$$459$$ 2862.00 0.291039
$$460$$ 0 0
$$461$$ −2406.00 −0.243077 −0.121539 0.992587i $$-0.538783\pi$$
−0.121539 + 0.992587i $$0.538783\pi$$
$$462$$ 0 0
$$463$$ 14316.0i 1.43698i 0.695538 + 0.718489i $$0.255166\pi$$
−0.695538 + 0.718489i $$0.744834\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ − 292.000i − 0.0289339i −0.999895 0.0144670i $$-0.995395\pi$$
0.999895 0.0144670i $$-0.00460514\pi$$
$$468$$ 0 0
$$469$$ 7760.00 0.764016
$$470$$ 0 0
$$471$$ −5442.00 −0.532387
$$472$$ 0 0
$$473$$ − 23072.0i − 2.24282i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 1134.00i 0.108852i
$$478$$ 0 0
$$479$$ 14056.0 1.34078 0.670391 0.742008i $$-0.266126\pi$$
0.670391 + 0.742008i $$0.266126\pi$$
$$480$$ 0 0
$$481$$ −24252.0 −2.29895
$$482$$ 0 0
$$483$$ 8160.00i 0.768722i
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ − 11204.0i − 1.04251i −0.853401 0.521254i $$-0.825464\pi$$
0.853401 0.521254i $$-0.174536\pi$$
$$488$$ 0 0
$$489$$ −5532.00 −0.511586
$$490$$ 0 0
$$491$$ −4608.00 −0.423536 −0.211768 0.977320i $$-0.567922\pi$$
−0.211768 + 0.977320i $$0.567922\pi$$
$$492$$ 0 0
$$493$$ 21836.0i 1.99482i
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 13440.0i 1.21301i
$$498$$ 0 0
$$499$$ 2468.00 0.221409 0.110704 0.993853i $$-0.464689\pi$$
0.110704 + 0.993853i $$0.464689\pi$$
$$500$$ 0 0
$$501$$ −11304.0 −1.00803
$$502$$ 0 0
$$503$$ − 12192.0i − 1.08074i −0.841426 0.540372i $$-0.818283\pi$$
0.841426 0.540372i $$-0.181717\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ − 15597.0i − 1.36625i
$$508$$ 0 0
$$509$$ −1714.00 −0.149257 −0.0746284 0.997211i $$-0.523777\pi$$
−0.0746284 + 0.997211i $$0.523777\pi$$
$$510$$ 0 0
$$511$$ −23400.0 −2.02574
$$512$$ 0 0
$$513$$ − 108.000i − 0.00929496i
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 2240.00i 0.190551i
$$518$$ 0 0
$$519$$ −2814.00 −0.237998
$$520$$ 0 0
$$521$$ −18014.0 −1.51479 −0.757397 0.652955i $$-0.773529\pi$$
−0.757397 + 0.652955i $$0.773529\pi$$
$$522$$ 0 0
$$523$$ 16748.0i 1.40027i 0.714013 + 0.700133i $$0.246876\pi$$
−0.714013 + 0.700133i $$0.753124\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 16112.0i 1.33178i
$$528$$ 0 0
$$529$$ −6329.00 −0.520178
$$530$$ 0 0
$$531$$ −504.000 −0.0411897
$$532$$ 0 0
$$533$$ 21156.0i 1.71926i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 11904.0i 0.956602i
$$538$$ 0 0
$$539$$ −3192.00 −0.255082
$$540$$ 0 0
$$541$$ −14018.0 −1.11401 −0.557006 0.830508i $$-0.688050\pi$$
−0.557006 + 0.830508i $$0.688050\pi$$
$$542$$ 0 0
$$543$$ − 10542.0i − 0.833150i
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ − 412.000i − 0.0322045i −0.999870 0.0161022i $$-0.994874\pi$$
0.999870 0.0161022i $$-0.00512572\pi$$
$$548$$ 0 0
$$549$$ 18.0000 0.00139931
$$550$$ 0 0
$$551$$ 824.000 0.0637089
$$552$$ 0 0
$$553$$ 8160.00i 0.627484i
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ − 18218.0i − 1.38586i −0.721007 0.692928i $$-0.756321\pi$$
0.721007 0.692928i $$-0.243679\pi$$
$$558$$ 0 0
$$559$$ −35432.0 −2.68088
$$560$$ 0 0
$$561$$ −17808.0 −1.34020
$$562$$ 0 0
$$563$$ − 23524.0i − 1.76096i −0.474087 0.880478i $$-0.657222\pi$$
0.474087 0.880478i $$-0.342778\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 1620.00i 0.119989i
$$568$$ 0 0
$$569$$ −23330.0 −1.71888 −0.859442 0.511234i $$-0.829189\pi$$
−0.859442 + 0.511234i $$0.829189\pi$$
$$570$$ 0 0
$$571$$ 13124.0 0.961860 0.480930 0.876759i $$-0.340299\pi$$
0.480930 + 0.876759i $$0.340299\pi$$
$$572$$ 0 0
$$573$$ 4440.00i 0.323706i
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ − 11714.0i − 0.845165i −0.906324 0.422582i $$-0.861124\pi$$
0.906324 0.422582i $$-0.138876\pi$$
$$578$$ 0 0
$$579$$ 8322.00 0.597324
$$580$$ 0 0
$$581$$ 13360.0 0.953987
$$582$$ 0 0
$$583$$ − 7056.00i − 0.501252i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ − 17628.0i − 1.23950i −0.784800 0.619749i $$-0.787234\pi$$
0.784800 0.619749i $$-0.212766\pi$$
$$588$$ 0 0
$$589$$ 608.000 0.0425335
$$590$$ 0 0
$$591$$ −11418.0 −0.794710
$$592$$ 0 0
$$593$$ − 2802.00i − 0.194038i −0.995283 0.0970188i $$-0.969069\pi$$
0.995283 0.0970188i $$-0.0309307\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ − 2568.00i − 0.176049i
$$598$$ 0 0
$$599$$ −2664.00 −0.181716 −0.0908582 0.995864i $$-0.528961\pi$$
−0.0908582 + 0.995864i $$0.528961\pi$$
$$600$$ 0 0
$$601$$ 23962.0 1.62634 0.813170 0.582026i $$-0.197740\pi$$
0.813170 + 0.582026i $$0.197740\pi$$
$$602$$ 0 0
$$603$$ 3492.00i 0.235830i
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 11940.0i 0.798401i 0.916864 + 0.399201i $$0.130712\pi$$
−0.916864 + 0.399201i $$0.869288\pi$$
$$608$$ 0 0
$$609$$ −12360.0 −0.822418
$$610$$ 0 0
$$611$$ 3440.00 0.227770
$$612$$ 0 0
$$613$$ 16794.0i 1.10653i 0.833005 + 0.553265i $$0.186618\pi$$
−0.833005 + 0.553265i $$0.813382\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 20706.0i 1.35104i 0.737341 + 0.675520i $$0.236081\pi$$
−0.737341 + 0.675520i $$0.763919\pi$$
$$618$$ 0 0
$$619$$ 10724.0 0.696339 0.348170 0.937432i $$-0.386803\pi$$
0.348170 + 0.937432i $$0.386803\pi$$
$$620$$ 0 0
$$621$$ −3672.00 −0.237282
$$622$$ 0 0
$$623$$ − 1320.00i − 0.0848871i
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ 672.000i 0.0428024i
$$628$$ 0 0
$$629$$ 29892.0 1.89487
$$630$$ 0 0
$$631$$ 5744.00 0.362385 0.181193 0.983448i $$-0.442004\pi$$
0.181193 + 0.983448i $$0.442004\pi$$
$$632$$ 0 0
$$633$$ − 9060.00i − 0.568883i
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 4902.00i 0.304905i
$$638$$ 0 0
$$639$$ −6048.00 −0.374421
$$640$$ 0 0
$$641$$ 27906.0 1.71953 0.859767 0.510687i $$-0.170609\pi$$
0.859767 + 0.510687i $$0.170609\pi$$
$$642$$ 0 0
$$643$$ − 20556.0i − 1.26073i −0.776299 0.630365i $$-0.782905\pi$$
0.776299 0.630365i $$-0.217095\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ − 10224.0i − 0.621247i −0.950533 0.310624i $$-0.899462\pi$$
0.950533 0.310624i $$-0.100538\pi$$
$$648$$ 0 0
$$649$$ 3136.00 0.189675
$$650$$ 0 0
$$651$$ −9120.00 −0.549064
$$652$$ 0 0
$$653$$ − 12982.0i − 0.777986i −0.921241 0.388993i $$-0.872823\pi$$
0.921241 0.388993i $$-0.127177\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ − 10530.0i − 0.625288i
$$658$$ 0 0
$$659$$ −1512.00 −0.0893766 −0.0446883 0.999001i $$-0.514229\pi$$
−0.0446883 + 0.999001i $$0.514229\pi$$
$$660$$ 0 0
$$661$$ 16710.0 0.983273 0.491637 0.870800i $$-0.336399\pi$$
0.491637 + 0.870800i $$0.336399\pi$$
$$662$$ 0 0
$$663$$ 27348.0i 1.60197i
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ − 28016.0i − 1.62636i
$$668$$ 0 0
$$669$$ −5052.00 −0.291961
$$670$$ 0 0
$$671$$ −112.000 −0.00644368
$$672$$ 0 0
$$673$$ 7962.00i 0.456036i 0.973657 + 0.228018i $$0.0732246\pi$$
−0.973657 + 0.228018i $$0.926775\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ − 12226.0i − 0.694067i −0.937853 0.347033i $$-0.887189\pi$$
0.937853 0.347033i $$-0.112811\pi$$
$$678$$ 0 0
$$679$$ −18520.0 −1.04673
$$680$$ 0 0
$$681$$ −6012.00 −0.338297
$$682$$ 0 0
$$683$$ 8748.00i 0.490092i 0.969511 + 0.245046i $$0.0788031\pi$$
−0.969511 + 0.245046i $$0.921197\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 15126.0i 0.840019i
$$688$$ 0 0
$$689$$ −10836.0 −0.599156
$$690$$ 0 0
$$691$$ 7324.00 0.403210 0.201605 0.979467i $$-0.435384\pi$$
0.201605 + 0.979467i $$0.435384\pi$$
$$692$$ 0 0
$$693$$ − 10080.0i − 0.552536i
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ − 26076.0i − 1.41707i
$$698$$ 0 0
$$699$$ 9270.00 0.501607
$$700$$ 0 0
$$701$$ −21934.0 −1.18179 −0.590896 0.806748i $$-0.701226\pi$$
−0.590896 + 0.806748i $$0.701226\pi$$
$$702$$ 0 0
$$703$$ − 1128.00i − 0.0605168i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ − 3960.00i − 0.210652i
$$708$$ 0 0
$$709$$ 10690.0 0.566250 0.283125 0.959083i $$-0.408629\pi$$
0.283125 + 0.959083i $$0.408629\pi$$
$$710$$ 0 0
$$711$$ −3672.00 −0.193686
$$712$$ 0 0
$$713$$ − 20672.0i − 1.08580i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 6408.00i 0.333767i
$$718$$ 0 0
$$719$$ 13792.0 0.715375 0.357688 0.933841i $$-0.383565\pi$$
0.357688 + 0.933841i $$0.383565\pi$$
$$720$$ 0 0
$$721$$ −30640.0 −1.58265
$$722$$ 0 0
$$723$$ 294.000i 0.0151231i
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ − 24004.0i − 1.22457i −0.790639 0.612283i $$-0.790251\pi$$
0.790639 0.612283i $$-0.209749\pi$$
$$728$$ 0 0
$$729$$ −729.000 −0.0370370
$$730$$ 0 0
$$731$$ 43672.0 2.20967
$$732$$ 0 0
$$733$$ 8562.00i 0.431439i 0.976455 + 0.215719i $$0.0692097\pi$$
−0.976455 + 0.215719i $$0.930790\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ − 21728.0i − 1.08597i
$$738$$ 0 0
$$739$$ −13836.0 −0.688722 −0.344361 0.938837i $$-0.611904\pi$$
−0.344361 + 0.938837i $$0.611904\pi$$
$$740$$ 0 0
$$741$$ 1032.00 0.0511626
$$742$$ 0 0
$$743$$ 22224.0i 1.09733i 0.836041 + 0.548667i $$0.184865\pi$$
−0.836041 + 0.548667i $$0.815135\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 6012.00i 0.294468i
$$748$$ 0 0
$$749$$ 8880.00 0.433202
$$750$$ 0 0
$$751$$ −11544.0 −0.560914 −0.280457 0.959867i $$-0.590486\pi$$
−0.280457 + 0.959867i $$0.590486\pi$$
$$752$$ 0 0
$$753$$ 15120.0i 0.731744i
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 3814.00i 0.183120i 0.995800 + 0.0915602i $$0.0291854\pi$$
−0.995800 + 0.0915602i $$0.970815\pi$$
$$758$$ 0 0
$$759$$ 22848.0 1.09266
$$760$$ 0 0
$$761$$ −25662.0 −1.22240 −0.611200 0.791476i $$-0.709313\pi$$
−0.611200 + 0.791476i $$0.709313\pi$$
$$762$$ 0 0
$$763$$ − 1240.00i − 0.0588349i
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ − 4816.00i − 0.226722i
$$768$$ 0 0
$$769$$ −30658.0 −1.43765 −0.718827 0.695189i $$-0.755321\pi$$
−0.718827 + 0.695189i $$0.755321\pi$$
$$770$$ 0 0
$$771$$ −5958.00 −0.278304
$$772$$ 0 0
$$773$$ − 30894.0i − 1.43749i −0.695274 0.718745i $$-0.744717\pi$$
0.695274 0.718745i $$-0.255283\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 16920.0i 0.781212i
$$778$$ 0 0
$$779$$ −984.000 −0.0452573
$$780$$ 0 0
$$781$$ 37632.0 1.72417
$$782$$ 0 0
$$783$$ − 5562.00i − 0.253857i
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 21596.0i 0.978163i 0.872238 + 0.489081i $$0.162668\pi$$
−0.872238 + 0.489081i $$0.837332\pi$$
$$788$$ 0 0
$$789$$ 4248.00 0.191677
$$790$$ 0 0
$$791$$ −8280.00 −0.372191
$$792$$ 0 0
$$793$$ 172.000i 0.00770227i
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 8646.00i 0.384262i 0.981369 + 0.192131i $$0.0615399\pi$$
−0.981369 + 0.192131i $$0.938460\pi$$
$$798$$ 0 0
$$799$$ −4240.00 −0.187735
$$800$$ 0 0
$$801$$ 594.000 0.0262022
$$802$$ 0 0
$$803$$ 65520.0i 2.87939i
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 20010.0i 0.872844i
$$808$$ 0 0
$$809$$ −24954.0 −1.08447 −0.542235 0.840227i $$-0.682422\pi$$
−0.542235 + 0.840227i $$0.682422\pi$$
$$810$$ 0 0
$$811$$ −40004.0 −1.73210 −0.866048 0.499960i $$-0.833348\pi$$
−0.866048 + 0.499960i $$0.833348\pi$$
$$812$$ 0 0
$$813$$ − 144.000i − 0.00621193i
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ − 1648.00i − 0.0705707i
$$818$$ 0 0
$$819$$ −15480.0 −0.660458
$$820$$ 0 0
$$821$$ 16570.0 0.704381 0.352191 0.935928i $$-0.385437\pi$$
0.352191 + 0.935928i $$0.385437\pi$$
$$822$$ 0 0
$$823$$ 4388.00i 0.185852i 0.995673 + 0.0929259i $$0.0296220\pi$$
−0.995673 + 0.0929259i $$0.970378\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 14364.0i 0.603972i 0.953312 + 0.301986i $$0.0976497\pi$$
−0.953312 + 0.301986i $$0.902350\pi$$
$$828$$ 0 0
$$829$$ 21170.0 0.886929 0.443465 0.896292i $$-0.353749\pi$$
0.443465 + 0.896292i $$0.353749\pi$$
$$830$$ 0 0
$$831$$ 20814.0 0.868868
$$832$$ 0 0
$$833$$ − 6042.00i − 0.251312i
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ − 4104.00i − 0.169480i
$$838$$ 0 0
$$839$$ 10664.0 0.438811 0.219405 0.975634i $$-0.429588\pi$$
0.219405 + 0.975634i $$0.429588\pi$$
$$840$$ 0 0
$$841$$ 18047.0 0.739965
$$842$$ 0 0
$$843$$ − 5082.00i − 0.207632i
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 36100.0i 1.46448i
$$848$$ 0 0
$$849$$ −19092.0 −0.771774
$$850$$ 0 0
$$851$$ −38352.0 −1.54488
$$852$$ 0 0
$$853$$ − 3190.00i − 0.128046i −0.997948 0.0640232i $$-0.979607\pi$$
0.997948 0.0640232i $$-0.0203932\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ − 20814.0i − 0.829630i −0.909906 0.414815i $$-0.863846\pi$$
0.909906 0.414815i $$-0.136154\pi$$
$$858$$ 0 0
$$859$$ −18988.0 −0.754205 −0.377103 0.926172i $$-0.623080\pi$$
−0.377103 + 0.926172i $$0.623080\pi$$
$$860$$ 0 0
$$861$$ 14760.0 0.584227
$$862$$ 0 0
$$863$$ − 11664.0i − 0.460078i −0.973181 0.230039i $$-0.926115\pi$$
0.973181 0.230039i $$-0.0738854\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ − 18969.0i − 0.743046i
$$868$$ 0 0
$$869$$ 22848.0 0.891905
$$870$$ 0 0
$$871$$ −33368.0 −1.29808
$$872$$ 0 0
$$873$$ − 8334.00i − 0.323096i
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 8246.00i 0.317500i 0.987319 + 0.158750i $$0.0507464\pi$$
−0.987319 + 0.158750i $$0.949254\pi$$
$$878$$ 0 0
$$879$$ 9402.00 0.360775
$$880$$ 0 0
$$881$$ 22890.0 0.875350 0.437675 0.899133i $$-0.355802\pi$$
0.437675 + 0.899133i $$0.355802\pi$$
$$882$$ 0 0
$$883$$ − 33548.0i − 1.27857i −0.768969 0.639287i $$-0.779230\pi$$
0.768969 0.639287i $$-0.220770\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ − 32264.0i − 1.22133i −0.791889 0.610665i $$-0.790902\pi$$
0.791889 0.610665i $$-0.209098\pi$$
$$888$$ 0 0
$$889$$ 19920.0 0.751513
$$890$$ 0 0
$$891$$ 4536.00 0.170552
$$892$$ 0 0
$$893$$ 160.000i 0.00599574i
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ − 35088.0i − 1.30608i
$$898$$ 0 0
$$899$$ 31312.0 1.16164
$$900$$ 0 0
$$901$$ 13356.0 0.493843
$$902$$ 0 0
$$903$$ 24720.0i 0.910997i
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 51228.0i 1.87541i 0.347431 + 0.937706i $$0.387054\pi$$
−0.347431 + 0.937706i $$0.612946\pi$$
$$908$$ 0 0
$$909$$ 1782.00 0.0650222
$$910$$ 0 0
$$911$$ 2144.00 0.0779735 0.0389868 0.999240i $$-0.487587\pi$$
0.0389868 + 0.999240i $$0.487587\pi$$
$$912$$ 0 0
$$913$$ − 37408.0i − 1.35600i
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 5280.00i 0.190143i
$$918$$ 0 0
$$919$$ 33584.0 1.20548 0.602739 0.797939i $$-0.294076\pi$$
0.602739 + 0.797939i $$0.294076\pi$$
$$920$$ 0 0
$$921$$ 708.000 0.0253305
$$922$$ 0 0
$$923$$ − 57792.0i − 2.06094i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ − 13788.0i − 0.488519i
$$928$$ 0 0
$$929$$ 3590.00 0.126786 0.0633929 0.997989i $$-0.479808\pi$$
0.0633929 + 0.997989i $$0.479808\pi$$
$$930$$ 0 0
$$931$$ −228.000 −0.00802621
$$932$$ 0 0
$$933$$ − 11328.0i − 0.397494i
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 21686.0i 0.756084i 0.925788 + 0.378042i $$0.123403\pi$$
−0.925788 + 0.378042i $$0.876597\pi$$
$$938$$ 0 0
$$939$$ 23754.0 0.825540
$$940$$ 0 0
$$941$$ −5174.00 −0.179243 −0.0896215 0.995976i $$-0.528566\pi$$
−0.0896215 + 0.995976i $$0.528566\pi$$
$$942$$ 0 0
$$943$$ 33456.0i 1.15533i
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 35524.0i 1.21898i 0.792793 + 0.609490i $$0.208626\pi$$
−0.792793 + 0.609490i $$0.791374\pi$$
$$948$$ 0 0
$$949$$ 100620. 3.44179
$$950$$ 0 0
$$951$$ 13086.0 0.446207
$$952$$ 0 0
$$953$$ − 16122.0i − 0.547999i −0.961730 0.273999i $$-0.911653\pi$$
0.961730 0.273999i $$-0.0883466\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 34608.0i 1.16898i
$$958$$ 0 0
$$959$$ 45560.0 1.53411
$$960$$ 0 0
$$961$$ −6687.00 −0.224464
$$962$$ 0 0
$$963$$ 3996.00i 0.133717i
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 19188.0i 0.638102i 0.947738 + 0.319051i $$0.103364\pi$$
−0.947738 + 0.319051i $$0.896636\pi$$
$$968$$ 0 0
$$969$$ −1272.00 −0.0421698
$$970$$ 0 0
$$971$$ 38464.0 1.27123 0.635617 0.772004i $$-0.280746\pi$$
0.635617 + 0.772004i $$0.280746\pi$$
$$972$$ 0 0
$$973$$ 36240.0i 1.19404i
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 43930.0i 1.43853i 0.694735 + 0.719266i $$0.255522\pi$$
−0.694735 + 0.719266i $$0.744478\pi$$
$$978$$ 0 0
$$979$$ −3696.00 −0.120659
$$980$$ 0 0
$$981$$ 558.000 0.0181606
$$982$$ 0 0
$$983$$ − 17328.0i − 0.562235i −0.959673 0.281118i $$-0.909295\pi$$
0.959673 0.281118i $$-0.0907051\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ − 2400.00i − 0.0773990i
$$988$$ 0 0
$$989$$ −56032.0 −1.80153
$$990$$ 0 0
$$991$$ 18160.0 0.582110 0.291055 0.956706i $$-0.405994\pi$$
0.291055 + 0.956706i $$0.405994\pi$$
$$992$$ 0 0
$$993$$ − 23940.0i − 0.765068i
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 9102.00i 0.289131i 0.989495 + 0.144565i $$0.0461784\pi$$
−0.989495 + 0.144565i $$0.953822\pi$$
$$998$$ 0 0
$$999$$ −7614.00 −0.241137
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 1200.4.f.t.49.2 2
4.3 odd 2 600.4.f.a.49.1 2
5.2 odd 4 240.4.a.g.1.1 1
5.3 odd 4 1200.4.a.p.1.1 1
5.4 even 2 inner 1200.4.f.t.49.1 2
12.11 even 2 1800.4.f.v.649.1 2
15.2 even 4 720.4.a.q.1.1 1
20.3 even 4 600.4.a.i.1.1 1
20.7 even 4 120.4.a.b.1.1 1
20.19 odd 2 600.4.f.a.49.2 2
40.27 even 4 960.4.a.bj.1.1 1
40.37 odd 4 960.4.a.k.1.1 1
60.23 odd 4 1800.4.a.f.1.1 1
60.47 odd 4 360.4.a.n.1.1 1
60.59 even 2 1800.4.f.v.649.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
120.4.a.b.1.1 1 20.7 even 4
240.4.a.g.1.1 1 5.2 odd 4
360.4.a.n.1.1 1 60.47 odd 4
600.4.a.i.1.1 1 20.3 even 4
600.4.f.a.49.1 2 4.3 odd 2
600.4.f.a.49.2 2 20.19 odd 2
720.4.a.q.1.1 1 15.2 even 4
960.4.a.k.1.1 1 40.37 odd 4
960.4.a.bj.1.1 1 40.27 even 4
1200.4.a.p.1.1 1 5.3 odd 4
1200.4.f.t.49.1 2 5.4 even 2 inner
1200.4.f.t.49.2 2 1.1 even 1 trivial
1800.4.a.f.1.1 1 60.23 odd 4
1800.4.f.v.649.1 2 12.11 even 2
1800.4.f.v.649.2 2 60.59 even 2