# Properties

 Label 2352.4.a.i.1.1 Level $2352$ Weight $4$ Character 2352.1 Self dual yes Analytic conductor $138.772$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2352 = 2^{4} \cdot 3 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 2352.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$138.772492334$$ 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 $$\chi$$ $$=$$ 2352.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-3.00000 q^{3} -3.00000 q^{5} +9.00000 q^{9} +O(q^{10})$$ $$q-3.00000 q^{3} -3.00000 q^{5} +9.00000 q^{9} +15.0000 q^{11} -64.0000 q^{13} +9.00000 q^{15} +84.0000 q^{17} +16.0000 q^{19} +84.0000 q^{23} -116.000 q^{25} -27.0000 q^{27} -297.000 q^{29} +253.000 q^{31} -45.0000 q^{33} -316.000 q^{37} +192.000 q^{39} +360.000 q^{41} -26.0000 q^{43} -27.0000 q^{45} +30.0000 q^{47} -252.000 q^{51} +363.000 q^{53} -45.0000 q^{55} -48.0000 q^{57} +15.0000 q^{59} -118.000 q^{61} +192.000 q^{65} +370.000 q^{67} -252.000 q^{69} +342.000 q^{71} +362.000 q^{73} +348.000 q^{75} -467.000 q^{79} +81.0000 q^{81} -477.000 q^{83} -252.000 q^{85} +891.000 q^{87} +906.000 q^{89} -759.000 q^{93} -48.0000 q^{95} +503.000 q^{97} +135.000 q^{99} +O(q^{100})$$

## 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
$$5$$ −3.00000 −0.268328 −0.134164 0.990959i $$-0.542835\pi$$
−0.134164 + 0.990959i $$0.542835\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 0 0
$$9$$ 9.00000 0.333333
$$10$$ 0 0
$$11$$ 15.0000 0.411152 0.205576 0.978641i $$-0.434093\pi$$
0.205576 + 0.978641i $$0.434093\pi$$
$$12$$ 0 0
$$13$$ −64.0000 −1.36542 −0.682708 0.730691i $$-0.739198\pi$$
−0.682708 + 0.730691i $$0.739198\pi$$
$$14$$ 0 0
$$15$$ 9.00000 0.154919
$$16$$ 0 0
$$17$$ 84.0000 1.19841 0.599206 0.800595i $$-0.295483\pi$$
0.599206 + 0.800595i $$0.295483\pi$$
$$18$$ 0 0
$$19$$ 16.0000 0.193192 0.0965961 0.995324i $$-0.469204\pi$$
0.0965961 + 0.995324i $$0.469204\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 84.0000 0.761531 0.380765 0.924672i $$-0.375661\pi$$
0.380765 + 0.924672i $$0.375661\pi$$
$$24$$ 0 0
$$25$$ −116.000 −0.928000
$$26$$ 0 0
$$27$$ −27.0000 −0.192450
$$28$$ 0 0
$$29$$ −297.000 −1.90178 −0.950888 0.309535i $$-0.899827\pi$$
−0.950888 + 0.309535i $$0.899827\pi$$
$$30$$ 0 0
$$31$$ 253.000 1.46581 0.732906 0.680330i $$-0.238164\pi$$
0.732906 + 0.680330i $$0.238164\pi$$
$$32$$ 0 0
$$33$$ −45.0000 −0.237379
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −316.000 −1.40406 −0.702028 0.712149i $$-0.747722\pi$$
−0.702028 + 0.712149i $$0.747722\pi$$
$$38$$ 0 0
$$39$$ 192.000 0.788323
$$40$$ 0 0
$$41$$ 360.000 1.37128 0.685641 0.727940i $$-0.259522\pi$$
0.685641 + 0.727940i $$0.259522\pi$$
$$42$$ 0 0
$$43$$ −26.0000 −0.0922084 −0.0461042 0.998937i $$-0.514681\pi$$
−0.0461042 + 0.998937i $$0.514681\pi$$
$$44$$ 0 0
$$45$$ −27.0000 −0.0894427
$$46$$ 0 0
$$47$$ 30.0000 0.0931053 0.0465527 0.998916i $$-0.485176\pi$$
0.0465527 + 0.998916i $$0.485176\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 0 0
$$51$$ −252.000 −0.691903
$$52$$ 0 0
$$53$$ 363.000 0.940790 0.470395 0.882456i $$-0.344111\pi$$
0.470395 + 0.882456i $$0.344111\pi$$
$$54$$ 0 0
$$55$$ −45.0000 −0.110324
$$56$$ 0 0
$$57$$ −48.0000 −0.111540
$$58$$ 0 0
$$59$$ 15.0000 0.0330989 0.0165494 0.999863i $$-0.494732\pi$$
0.0165494 + 0.999863i $$0.494732\pi$$
$$60$$ 0 0
$$61$$ −118.000 −0.247678 −0.123839 0.992302i $$-0.539521\pi$$
−0.123839 + 0.992302i $$0.539521\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 192.000 0.366380
$$66$$ 0 0
$$67$$ 370.000 0.674667 0.337334 0.941385i $$-0.390475\pi$$
0.337334 + 0.941385i $$0.390475\pi$$
$$68$$ 0 0
$$69$$ −252.000 −0.439670
$$70$$ 0 0
$$71$$ 342.000 0.571661 0.285831 0.958280i $$-0.407731\pi$$
0.285831 + 0.958280i $$0.407731\pi$$
$$72$$ 0 0
$$73$$ 362.000 0.580396 0.290198 0.956967i $$-0.406279\pi$$
0.290198 + 0.956967i $$0.406279\pi$$
$$74$$ 0 0
$$75$$ 348.000 0.535781
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −467.000 −0.665084 −0.332542 0.943089i $$-0.607906\pi$$
−0.332542 + 0.943089i $$0.607906\pi$$
$$80$$ 0 0
$$81$$ 81.0000 0.111111
$$82$$ 0 0
$$83$$ −477.000 −0.630814 −0.315407 0.948957i $$-0.602141\pi$$
−0.315407 + 0.948957i $$0.602141\pi$$
$$84$$ 0 0
$$85$$ −252.000 −0.321568
$$86$$ 0 0
$$87$$ 891.000 1.09799
$$88$$ 0 0
$$89$$ 906.000 1.07905 0.539527 0.841968i $$-0.318603\pi$$
0.539527 + 0.841968i $$0.318603\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ −759.000 −0.846286
$$94$$ 0 0
$$95$$ −48.0000 −0.0518389
$$96$$ 0 0
$$97$$ 503.000 0.526515 0.263257 0.964726i $$-0.415203\pi$$
0.263257 + 0.964726i $$0.415203\pi$$
$$98$$ 0 0
$$99$$ 135.000 0.137051
$$100$$ 0 0
$$101$$ −1086.00 −1.06991 −0.534956 0.844880i $$-0.679672\pi$$
−0.534956 + 0.844880i $$0.679672\pi$$
$$102$$ 0 0
$$103$$ −1736.00 −1.66071 −0.830355 0.557235i $$-0.811862\pi$$
−0.830355 + 0.557235i $$0.811862\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 1353.00 1.22242 0.611212 0.791467i $$-0.290682\pi$$
0.611212 + 0.791467i $$0.290682\pi$$
$$108$$ 0 0
$$109$$ −370.000 −0.325134 −0.162567 0.986698i $$-0.551977\pi$$
−0.162567 + 0.986698i $$0.551977\pi$$
$$110$$ 0 0
$$111$$ 948.000 0.810632
$$112$$ 0 0
$$113$$ −648.000 −0.539458 −0.269729 0.962936i $$-0.586934\pi$$
−0.269729 + 0.962936i $$0.586934\pi$$
$$114$$ 0 0
$$115$$ −252.000 −0.204340
$$116$$ 0 0
$$117$$ −576.000 −0.455139
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −1106.00 −0.830954
$$122$$ 0 0
$$123$$ −1080.00 −0.791710
$$124$$ 0 0
$$125$$ 723.000 0.517337
$$126$$ 0 0
$$127$$ −377.000 −0.263412 −0.131706 0.991289i $$-0.542046\pi$$
−0.131706 + 0.991289i $$0.542046\pi$$
$$128$$ 0 0
$$129$$ 78.0000 0.0532366
$$130$$ 0 0
$$131$$ 651.000 0.434184 0.217092 0.976151i $$-0.430343\pi$$
0.217092 + 0.976151i $$0.430343\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ 81.0000 0.0516398
$$136$$ 0 0
$$137$$ −1770.00 −1.10381 −0.551903 0.833909i $$-0.686098\pi$$
−0.551903 + 0.833909i $$0.686098\pi$$
$$138$$ 0 0
$$139$$ 1558.00 0.950704 0.475352 0.879796i $$-0.342321\pi$$
0.475352 + 0.879796i $$0.342321\pi$$
$$140$$ 0 0
$$141$$ −90.0000 −0.0537544
$$142$$ 0 0
$$143$$ −960.000 −0.561393
$$144$$ 0 0
$$145$$ 891.000 0.510300
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 2454.00 1.34926 0.674629 0.738157i $$-0.264304\pi$$
0.674629 + 0.738157i $$0.264304\pi$$
$$150$$ 0 0
$$151$$ −1259.00 −0.678516 −0.339258 0.940693i $$-0.610176\pi$$
−0.339258 + 0.940693i $$0.610176\pi$$
$$152$$ 0 0
$$153$$ 756.000 0.399470
$$154$$ 0 0
$$155$$ −759.000 −0.393318
$$156$$ 0 0
$$157$$ −196.000 −0.0996338 −0.0498169 0.998758i $$-0.515864\pi$$
−0.0498169 + 0.998758i $$0.515864\pi$$
$$158$$ 0 0
$$159$$ −1089.00 −0.543166
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 1252.00 0.601621 0.300810 0.953684i $$-0.402743\pi$$
0.300810 + 0.953684i $$0.402743\pi$$
$$164$$ 0 0
$$165$$ 135.000 0.0636954
$$166$$ 0 0
$$167$$ 2646.00 1.22607 0.613035 0.790056i $$-0.289949\pi$$
0.613035 + 0.790056i $$0.289949\pi$$
$$168$$ 0 0
$$169$$ 1899.00 0.864360
$$170$$ 0 0
$$171$$ 144.000 0.0643974
$$172$$ 0 0
$$173$$ −786.000 −0.345425 −0.172712 0.984972i $$-0.555253\pi$$
−0.172712 + 0.984972i $$0.555253\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ −45.0000 −0.0191096
$$178$$ 0 0
$$179$$ −2892.00 −1.20759 −0.603794 0.797140i $$-0.706345\pi$$
−0.603794 + 0.797140i $$0.706345\pi$$
$$180$$ 0 0
$$181$$ 1352.00 0.555212 0.277606 0.960695i $$-0.410459\pi$$
0.277606 + 0.960695i $$0.410459\pi$$
$$182$$ 0 0
$$183$$ 354.000 0.142997
$$184$$ 0 0
$$185$$ 948.000 0.376748
$$186$$ 0 0
$$187$$ 1260.00 0.492729
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −3912.00 −1.48200 −0.741001 0.671504i $$-0.765649\pi$$
−0.741001 + 0.671504i $$0.765649\pi$$
$$192$$ 0 0
$$193$$ 1493.00 0.556832 0.278416 0.960461i $$-0.410191\pi$$
0.278416 + 0.960461i $$0.410191\pi$$
$$194$$ 0 0
$$195$$ −576.000 −0.211529
$$196$$ 0 0
$$197$$ −4086.00 −1.47774 −0.738872 0.673846i $$-0.764641\pi$$
−0.738872 + 0.673846i $$0.764641\pi$$
$$198$$ 0 0
$$199$$ 3556.00 1.26672 0.633362 0.773855i $$-0.281674\pi$$
0.633362 + 0.773855i $$0.281674\pi$$
$$200$$ 0 0
$$201$$ −1110.00 −0.389519
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ −1080.00 −0.367954
$$206$$ 0 0
$$207$$ 756.000 0.253844
$$208$$ 0 0
$$209$$ 240.000 0.0794313
$$210$$ 0 0
$$211$$ −1250.00 −0.407837 −0.203918 0.978988i $$-0.565368\pi$$
−0.203918 + 0.978988i $$0.565368\pi$$
$$212$$ 0 0
$$213$$ −1026.00 −0.330049
$$214$$ 0 0
$$215$$ 78.0000 0.0247421
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ −1086.00 −0.335092
$$220$$ 0 0
$$221$$ −5376.00 −1.63633
$$222$$ 0 0
$$223$$ −425.000 −0.127624 −0.0638119 0.997962i $$-0.520326\pi$$
−0.0638119 + 0.997962i $$0.520326\pi$$
$$224$$ 0 0
$$225$$ −1044.00 −0.309333
$$226$$ 0 0
$$227$$ −3855.00 −1.12716 −0.563580 0.826061i $$-0.690576\pi$$
−0.563580 + 0.826061i $$0.690576\pi$$
$$228$$ 0 0
$$229$$ −2188.00 −0.631385 −0.315692 0.948862i $$-0.602237\pi$$
−0.315692 + 0.948862i $$0.602237\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 852.000 0.239555 0.119778 0.992801i $$-0.461782\pi$$
0.119778 + 0.992801i $$0.461782\pi$$
$$234$$ 0 0
$$235$$ −90.0000 −0.0249828
$$236$$ 0 0
$$237$$ 1401.00 0.383986
$$238$$ 0 0
$$239$$ −5508.00 −1.49072 −0.745362 0.666660i $$-0.767723\pi$$
−0.745362 + 0.666660i $$0.767723\pi$$
$$240$$ 0 0
$$241$$ 791.000 0.211422 0.105711 0.994397i $$-0.466288\pi$$
0.105711 + 0.994397i $$0.466288\pi$$
$$242$$ 0 0
$$243$$ −243.000 −0.0641500
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −1024.00 −0.263788
$$248$$ 0 0
$$249$$ 1431.00 0.364201
$$250$$ 0 0
$$251$$ −5265.00 −1.32400 −0.662000 0.749504i $$-0.730292\pi$$
−0.662000 + 0.749504i $$0.730292\pi$$
$$252$$ 0 0
$$253$$ 1260.00 0.313105
$$254$$ 0 0
$$255$$ 756.000 0.185657
$$256$$ 0 0
$$257$$ −6870.00 −1.66747 −0.833733 0.552168i $$-0.813801\pi$$
−0.833733 + 0.552168i $$0.813801\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ −2673.00 −0.633925
$$262$$ 0 0
$$263$$ 222.000 0.0520498 0.0260249 0.999661i $$-0.491715\pi$$
0.0260249 + 0.999661i $$0.491715\pi$$
$$264$$ 0 0
$$265$$ −1089.00 −0.252441
$$266$$ 0 0
$$267$$ −2718.00 −0.622992
$$268$$ 0 0
$$269$$ 7851.00 1.77949 0.889747 0.456454i $$-0.150881\pi$$
0.889747 + 0.456454i $$0.150881\pi$$
$$270$$ 0 0
$$271$$ −5183.00 −1.16179 −0.580895 0.813979i $$-0.697297\pi$$
−0.580895 + 0.813979i $$0.697297\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ −1740.00 −0.381549
$$276$$ 0 0
$$277$$ −4960.00 −1.07588 −0.537938 0.842985i $$-0.680796\pi$$
−0.537938 + 0.842985i $$0.680796\pi$$
$$278$$ 0 0
$$279$$ 2277.00 0.488604
$$280$$ 0 0
$$281$$ −774.000 −0.164317 −0.0821583 0.996619i $$-0.526181\pi$$
−0.0821583 + 0.996619i $$0.526181\pi$$
$$282$$ 0 0
$$283$$ −3698.00 −0.776761 −0.388380 0.921499i $$-0.626965\pi$$
−0.388380 + 0.921499i $$0.626965\pi$$
$$284$$ 0 0
$$285$$ 144.000 0.0299292
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ 2143.00 0.436190
$$290$$ 0 0
$$291$$ −1509.00 −0.303983
$$292$$ 0 0
$$293$$ −6273.00 −1.25076 −0.625380 0.780321i $$-0.715056\pi$$
−0.625380 + 0.780321i $$0.715056\pi$$
$$294$$ 0 0
$$295$$ −45.0000 −0.00888136
$$296$$ 0 0
$$297$$ −405.000 −0.0791262
$$298$$ 0 0
$$299$$ −5376.00 −1.03981
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ 3258.00 0.617714
$$304$$ 0 0
$$305$$ 354.000 0.0664590
$$306$$ 0 0
$$307$$ 1684.00 0.313065 0.156533 0.987673i $$-0.449968\pi$$
0.156533 + 0.987673i $$0.449968\pi$$
$$308$$ 0 0
$$309$$ 5208.00 0.958812
$$310$$ 0 0
$$311$$ 1320.00 0.240676 0.120338 0.992733i $$-0.461602\pi$$
0.120338 + 0.992733i $$0.461602\pi$$
$$312$$ 0 0
$$313$$ −8503.00 −1.53552 −0.767760 0.640737i $$-0.778629\pi$$
−0.767760 + 0.640737i $$0.778629\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −2577.00 −0.456589 −0.228295 0.973592i $$-0.573315\pi$$
−0.228295 + 0.973592i $$0.573315\pi$$
$$318$$ 0 0
$$319$$ −4455.00 −0.781919
$$320$$ 0 0
$$321$$ −4059.00 −0.705767
$$322$$ 0 0
$$323$$ 1344.00 0.231524
$$324$$ 0 0
$$325$$ 7424.00 1.26711
$$326$$ 0 0
$$327$$ 1110.00 0.187716
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 484.000 0.0803717 0.0401859 0.999192i $$-0.487205\pi$$
0.0401859 + 0.999192i $$0.487205\pi$$
$$332$$ 0 0
$$333$$ −2844.00 −0.468019
$$334$$ 0 0
$$335$$ −1110.00 −0.181032
$$336$$ 0 0
$$337$$ −8359.00 −1.35117 −0.675584 0.737283i $$-0.736109\pi$$
−0.675584 + 0.737283i $$0.736109\pi$$
$$338$$ 0 0
$$339$$ 1944.00 0.311456
$$340$$ 0 0
$$341$$ 3795.00 0.602671
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ 756.000 0.117976
$$346$$ 0 0
$$347$$ 1860.00 0.287752 0.143876 0.989596i $$-0.454043\pi$$
0.143876 + 0.989596i $$0.454043\pi$$
$$348$$ 0 0
$$349$$ −1918.00 −0.294178 −0.147089 0.989123i $$-0.546990\pi$$
−0.147089 + 0.989123i $$0.546990\pi$$
$$350$$ 0 0
$$351$$ 1728.00 0.262774
$$352$$ 0 0
$$353$$ −3048.00 −0.459571 −0.229786 0.973241i $$-0.573803\pi$$
−0.229786 + 0.973241i $$0.573803\pi$$
$$354$$ 0 0
$$355$$ −1026.00 −0.153393
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 30.0000 0.00441042 0.00220521 0.999998i $$-0.499298\pi$$
0.00220521 + 0.999998i $$0.499298\pi$$
$$360$$ 0 0
$$361$$ −6603.00 −0.962677
$$362$$ 0 0
$$363$$ 3318.00 0.479752
$$364$$ 0 0
$$365$$ −1086.00 −0.155737
$$366$$ 0 0
$$367$$ 11311.0 1.60880 0.804400 0.594088i $$-0.202487\pi$$
0.804400 + 0.594088i $$0.202487\pi$$
$$368$$ 0 0
$$369$$ 3240.00 0.457094
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 1208.00 0.167689 0.0838443 0.996479i $$-0.473280\pi$$
0.0838443 + 0.996479i $$0.473280\pi$$
$$374$$ 0 0
$$375$$ −2169.00 −0.298684
$$376$$ 0 0
$$377$$ 19008.0 2.59672
$$378$$ 0 0
$$379$$ −7640.00 −1.03546 −0.517731 0.855543i $$-0.673223\pi$$
−0.517731 + 0.855543i $$0.673223\pi$$
$$380$$ 0 0
$$381$$ 1131.00 0.152081
$$382$$ 0 0
$$383$$ −12750.0 −1.70103 −0.850515 0.525951i $$-0.823710\pi$$
−0.850515 + 0.525951i $$0.823710\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ −234.000 −0.0307361
$$388$$ 0 0
$$389$$ 3126.00 0.407441 0.203720 0.979029i $$-0.434697\pi$$
0.203720 + 0.979029i $$0.434697\pi$$
$$390$$ 0 0
$$391$$ 7056.00 0.912627
$$392$$ 0 0
$$393$$ −1953.00 −0.250676
$$394$$ 0 0
$$395$$ 1401.00 0.178461
$$396$$ 0 0
$$397$$ −5932.00 −0.749921 −0.374960 0.927041i $$-0.622344\pi$$
−0.374960 + 0.927041i $$0.622344\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 1608.00 0.200249 0.100124 0.994975i $$-0.468076\pi$$
0.100124 + 0.994975i $$0.468076\pi$$
$$402$$ 0 0
$$403$$ −16192.0 −2.00144
$$404$$ 0 0
$$405$$ −243.000 −0.0298142
$$406$$ 0 0
$$407$$ −4740.00 −0.577280
$$408$$ 0 0
$$409$$ −4465.00 −0.539805 −0.269902 0.962888i $$-0.586991\pi$$
−0.269902 + 0.962888i $$0.586991\pi$$
$$410$$ 0 0
$$411$$ 5310.00 0.637282
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 1431.00 0.169265
$$416$$ 0 0
$$417$$ −4674.00 −0.548889
$$418$$ 0 0
$$419$$ 1584.00 0.184686 0.0923430 0.995727i $$-0.470564\pi$$
0.0923430 + 0.995727i $$0.470564\pi$$
$$420$$ 0 0
$$421$$ −1330.00 −0.153967 −0.0769837 0.997032i $$-0.524529\pi$$
−0.0769837 + 0.997032i $$0.524529\pi$$
$$422$$ 0 0
$$423$$ 270.000 0.0310351
$$424$$ 0 0
$$425$$ −9744.00 −1.11213
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ 2880.00 0.324121
$$430$$ 0 0
$$431$$ −9588.00 −1.07155 −0.535775 0.844361i $$-0.679980\pi$$
−0.535775 + 0.844361i $$0.679980\pi$$
$$432$$ 0 0
$$433$$ 494.000 0.0548271 0.0274135 0.999624i $$-0.491273\pi$$
0.0274135 + 0.999624i $$0.491273\pi$$
$$434$$ 0 0
$$435$$ −2673.00 −0.294622
$$436$$ 0 0
$$437$$ 1344.00 0.147122
$$438$$ 0 0
$$439$$ 16009.0 1.74047 0.870237 0.492634i $$-0.163966\pi$$
0.870237 + 0.492634i $$0.163966\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −7773.00 −0.833649 −0.416824 0.908987i $$-0.636857\pi$$
−0.416824 + 0.908987i $$0.636857\pi$$
$$444$$ 0 0
$$445$$ −2718.00 −0.289541
$$446$$ 0 0
$$447$$ −7362.00 −0.778995
$$448$$ 0 0
$$449$$ 864.000 0.0908122 0.0454061 0.998969i $$-0.485542\pi$$
0.0454061 + 0.998969i $$0.485542\pi$$
$$450$$ 0 0
$$451$$ 5400.00 0.563805
$$452$$ 0 0
$$453$$ 3777.00 0.391742
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 2519.00 0.257842 0.128921 0.991655i $$-0.458849\pi$$
0.128921 + 0.991655i $$0.458849\pi$$
$$458$$ 0 0
$$459$$ −2268.00 −0.230634
$$460$$ 0 0
$$461$$ −342.000 −0.0345521 −0.0172761 0.999851i $$-0.505499\pi$$
−0.0172761 + 0.999851i $$0.505499\pi$$
$$462$$ 0 0
$$463$$ 4336.00 0.435229 0.217614 0.976035i $$-0.430172\pi$$
0.217614 + 0.976035i $$0.430172\pi$$
$$464$$ 0 0
$$465$$ 2277.00 0.227082
$$466$$ 0 0
$$467$$ −18636.0 −1.84662 −0.923310 0.384056i $$-0.874527\pi$$
−0.923310 + 0.384056i $$0.874527\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 588.000 0.0575236
$$472$$ 0 0
$$473$$ −390.000 −0.0379117
$$474$$ 0 0
$$475$$ −1856.00 −0.179282
$$476$$ 0 0
$$477$$ 3267.00 0.313597
$$478$$ 0 0
$$479$$ −15078.0 −1.43827 −0.719135 0.694870i $$-0.755462\pi$$
−0.719135 + 0.694870i $$0.755462\pi$$
$$480$$ 0 0
$$481$$ 20224.0 1.91712
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ −1509.00 −0.141279
$$486$$ 0 0
$$487$$ −6221.00 −0.578851 −0.289425 0.957201i $$-0.593464\pi$$
−0.289425 + 0.957201i $$0.593464\pi$$
$$488$$ 0 0
$$489$$ −3756.00 −0.347346
$$490$$ 0 0
$$491$$ 7371.00 0.677492 0.338746 0.940878i $$-0.389997\pi$$
0.338746 + 0.940878i $$0.389997\pi$$
$$492$$ 0 0
$$493$$ −24948.0 −2.27911
$$494$$ 0 0
$$495$$ −405.000 −0.0367745
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ −4274.00 −0.383428 −0.191714 0.981451i $$-0.561405\pi$$
−0.191714 + 0.981451i $$0.561405\pi$$
$$500$$ 0 0
$$501$$ −7938.00 −0.707872
$$502$$ 0 0
$$503$$ 2520.00 0.223382 0.111691 0.993743i $$-0.464373\pi$$
0.111691 + 0.993743i $$0.464373\pi$$
$$504$$ 0 0
$$505$$ 3258.00 0.287087
$$506$$ 0 0
$$507$$ −5697.00 −0.499039
$$508$$ 0 0
$$509$$ −14277.0 −1.24326 −0.621628 0.783313i $$-0.713528\pi$$
−0.621628 + 0.783313i $$0.713528\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ −432.000 −0.0371799
$$514$$ 0 0
$$515$$ 5208.00 0.445615
$$516$$ 0 0
$$517$$ 450.000 0.0382804
$$518$$ 0 0
$$519$$ 2358.00 0.199431
$$520$$ 0 0
$$521$$ −6306.00 −0.530270 −0.265135 0.964211i $$-0.585417\pi$$
−0.265135 + 0.964211i $$0.585417\pi$$
$$522$$ 0 0
$$523$$ −8072.00 −0.674883 −0.337442 0.941346i $$-0.609562\pi$$
−0.337442 + 0.941346i $$0.609562\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 21252.0 1.75664
$$528$$ 0 0
$$529$$ −5111.00 −0.420071
$$530$$ 0 0
$$531$$ 135.000 0.0110330
$$532$$ 0 0
$$533$$ −23040.0 −1.87237
$$534$$ 0 0
$$535$$ −4059.00 −0.328011
$$536$$ 0 0
$$537$$ 8676.00 0.697201
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −22858.0 −1.81653 −0.908264 0.418396i $$-0.862592\pi$$
−0.908264 + 0.418396i $$0.862592\pi$$
$$542$$ 0 0
$$543$$ −4056.00 −0.320552
$$544$$ 0 0
$$545$$ 1110.00 0.0872425
$$546$$ 0 0
$$547$$ 24724.0 1.93258 0.966291 0.257454i $$-0.0828835\pi$$
0.966291 + 0.257454i $$0.0828835\pi$$
$$548$$ 0 0
$$549$$ −1062.00 −0.0825593
$$550$$ 0 0
$$551$$ −4752.00 −0.367408
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ −2844.00 −0.217515
$$556$$ 0 0
$$557$$ −9843.00 −0.748764 −0.374382 0.927275i $$-0.622145\pi$$
−0.374382 + 0.927275i $$0.622145\pi$$
$$558$$ 0 0
$$559$$ 1664.00 0.125903
$$560$$ 0 0
$$561$$ −3780.00 −0.284477
$$562$$ 0 0
$$563$$ 13371.0 1.00092 0.500462 0.865758i $$-0.333163\pi$$
0.500462 + 0.865758i $$0.333163\pi$$
$$564$$ 0 0
$$565$$ 1944.00 0.144752
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −5232.00 −0.385478 −0.192739 0.981250i $$-0.561737\pi$$
−0.192739 + 0.981250i $$0.561737\pi$$
$$570$$ 0 0
$$571$$ 14398.0 1.05523 0.527616 0.849483i $$-0.323086\pi$$
0.527616 + 0.849483i $$0.323086\pi$$
$$572$$ 0 0
$$573$$ 11736.0 0.855634
$$574$$ 0 0
$$575$$ −9744.00 −0.706701
$$576$$ 0 0
$$577$$ 19871.0 1.43369 0.716846 0.697231i $$-0.245585\pi$$
0.716846 + 0.697231i $$0.245585\pi$$
$$578$$ 0 0
$$579$$ −4479.00 −0.321487
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 5445.00 0.386808
$$584$$ 0 0
$$585$$ 1728.00 0.122127
$$586$$ 0 0
$$587$$ 16137.0 1.13466 0.567330 0.823491i $$-0.307976\pi$$
0.567330 + 0.823491i $$0.307976\pi$$
$$588$$ 0 0
$$589$$ 4048.00 0.283183
$$590$$ 0 0
$$591$$ 12258.0 0.853176
$$592$$ 0 0
$$593$$ −21324.0 −1.47668 −0.738340 0.674428i $$-0.764390\pi$$
−0.738340 + 0.674428i $$0.764390\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ −10668.0 −0.731344
$$598$$ 0 0
$$599$$ 8646.00 0.589760 0.294880 0.955534i $$-0.404720\pi$$
0.294880 + 0.955534i $$0.404720\pi$$
$$600$$ 0 0
$$601$$ 11195.0 0.759823 0.379911 0.925023i $$-0.375954\pi$$
0.379911 + 0.925023i $$0.375954\pi$$
$$602$$ 0 0
$$603$$ 3330.00 0.224889
$$604$$ 0 0
$$605$$ 3318.00 0.222968
$$606$$ 0 0
$$607$$ 8971.00 0.599871 0.299935 0.953959i $$-0.403035\pi$$
0.299935 + 0.953959i $$0.403035\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −1920.00 −0.127127
$$612$$ 0 0
$$613$$ −12772.0 −0.841527 −0.420764 0.907170i $$-0.638238\pi$$
−0.420764 + 0.907170i $$0.638238\pi$$
$$614$$ 0 0
$$615$$ 3240.00 0.212438
$$616$$ 0 0
$$617$$ 12762.0 0.832705 0.416352 0.909203i $$-0.363308\pi$$
0.416352 + 0.909203i $$0.363308\pi$$
$$618$$ 0 0
$$619$$ −12842.0 −0.833867 −0.416933 0.908937i $$-0.636895\pi$$
−0.416933 + 0.908937i $$0.636895\pi$$
$$620$$ 0 0
$$621$$ −2268.00 −0.146557
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 12331.0 0.789184
$$626$$ 0 0
$$627$$ −720.000 −0.0458597
$$628$$ 0 0
$$629$$ −26544.0 −1.68264
$$630$$ 0 0
$$631$$ −21365.0 −1.34790 −0.673952 0.738775i $$-0.735404\pi$$
−0.673952 + 0.738775i $$0.735404\pi$$
$$632$$ 0 0
$$633$$ 3750.00 0.235465
$$634$$ 0 0
$$635$$ 1131.00 0.0706809
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ 3078.00 0.190554
$$640$$ 0 0
$$641$$ 8274.00 0.509834 0.254917 0.966963i $$-0.417952\pi$$
0.254917 + 0.966963i $$0.417952\pi$$
$$642$$ 0 0
$$643$$ −27998.0 −1.71716 −0.858580 0.512680i $$-0.828653\pi$$
−0.858580 + 0.512680i $$0.828653\pi$$
$$644$$ 0 0
$$645$$ −234.000 −0.0142849
$$646$$ 0 0
$$647$$ 17466.0 1.06130 0.530649 0.847592i $$-0.321948\pi$$
0.530649 + 0.847592i $$0.321948\pi$$
$$648$$ 0 0
$$649$$ 225.000 0.0136087
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 2157.00 0.129265 0.0646324 0.997909i $$-0.479413\pi$$
0.0646324 + 0.997909i $$0.479413\pi$$
$$654$$ 0 0
$$655$$ −1953.00 −0.116504
$$656$$ 0 0
$$657$$ 3258.00 0.193465
$$658$$ 0 0
$$659$$ −19944.0 −1.17892 −0.589460 0.807798i $$-0.700659\pi$$
−0.589460 + 0.807798i $$0.700659\pi$$
$$660$$ 0 0
$$661$$ 27506.0 1.61855 0.809273 0.587432i $$-0.199861\pi$$
0.809273 + 0.587432i $$0.199861\pi$$
$$662$$ 0 0
$$663$$ 16128.0 0.944735
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −24948.0 −1.44826
$$668$$ 0 0
$$669$$ 1275.00 0.0736836
$$670$$ 0 0
$$671$$ −1770.00 −0.101833
$$672$$ 0 0
$$673$$ −19123.0 −1.09530 −0.547650 0.836707i $$-0.684478\pi$$
−0.547650 + 0.836707i $$0.684478\pi$$
$$674$$ 0 0
$$675$$ 3132.00 0.178594
$$676$$ 0 0
$$677$$ 13857.0 0.786658 0.393329 0.919398i $$-0.371323\pi$$
0.393329 + 0.919398i $$0.371323\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 11565.0 0.650766
$$682$$ 0 0
$$683$$ 22245.0 1.24624 0.623120 0.782127i $$-0.285865\pi$$
0.623120 + 0.782127i $$0.285865\pi$$
$$684$$ 0 0
$$685$$ 5310.00 0.296182
$$686$$ 0 0
$$687$$ 6564.00 0.364530
$$688$$ 0 0
$$689$$ −23232.0 −1.28457
$$690$$ 0 0
$$691$$ 640.000 0.0352341 0.0176170 0.999845i $$-0.494392\pi$$
0.0176170 + 0.999845i $$0.494392\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ −4674.00 −0.255101
$$696$$ 0 0
$$697$$ 30240.0 1.64336
$$698$$ 0 0
$$699$$ −2556.00 −0.138307
$$700$$ 0 0
$$701$$ −15561.0 −0.838418 −0.419209 0.907890i $$-0.637693\pi$$
−0.419209 + 0.907890i $$0.637693\pi$$
$$702$$ 0 0
$$703$$ −5056.00 −0.271253
$$704$$ 0 0
$$705$$ 270.000 0.0144238
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 5534.00 0.293136 0.146568 0.989201i $$-0.453177\pi$$
0.146568 + 0.989201i $$0.453177\pi$$
$$710$$ 0 0
$$711$$ −4203.00 −0.221695
$$712$$ 0 0
$$713$$ 21252.0 1.11626
$$714$$ 0 0
$$715$$ 2880.00 0.150638
$$716$$ 0 0
$$717$$ 16524.0 0.860670
$$718$$ 0 0
$$719$$ 21846.0 1.13313 0.566564 0.824018i $$-0.308273\pi$$
0.566564 + 0.824018i $$0.308273\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ −2373.00 −0.122065
$$724$$ 0 0
$$725$$ 34452.0 1.76485
$$726$$ 0 0
$$727$$ 11089.0 0.565706 0.282853 0.959163i $$-0.408719\pi$$
0.282853 + 0.959163i $$0.408719\pi$$
$$728$$ 0 0
$$729$$ 729.000 0.0370370
$$730$$ 0 0
$$731$$ −2184.00 −0.110504
$$732$$ 0 0
$$733$$ 11762.0 0.592687 0.296343 0.955081i $$-0.404233\pi$$
0.296343 + 0.955081i $$0.404233\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 5550.00 0.277391
$$738$$ 0 0
$$739$$ 22726.0 1.13124 0.565622 0.824665i $$-0.308636\pi$$
0.565622 + 0.824665i $$0.308636\pi$$
$$740$$ 0 0
$$741$$ 3072.00 0.152298
$$742$$ 0 0
$$743$$ −6678.00 −0.329734 −0.164867 0.986316i $$-0.552719\pi$$
−0.164867 + 0.986316i $$0.552719\pi$$
$$744$$ 0 0
$$745$$ −7362.00 −0.362044
$$746$$ 0 0
$$747$$ −4293.00 −0.210271
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 19987.0 0.971153 0.485577 0.874194i $$-0.338610\pi$$
0.485577 + 0.874194i $$0.338610\pi$$
$$752$$ 0 0
$$753$$ 15795.0 0.764411
$$754$$ 0 0
$$755$$ 3777.00 0.182065
$$756$$ 0 0
$$757$$ 314.000 0.0150760 0.00753799 0.999972i $$-0.497601\pi$$
0.00753799 + 0.999972i $$0.497601\pi$$
$$758$$ 0 0
$$759$$ −3780.00 −0.180771
$$760$$ 0 0
$$761$$ −11496.0 −0.547608 −0.273804 0.961786i $$-0.588282\pi$$
−0.273804 + 0.961786i $$0.588282\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ −2268.00 −0.107189
$$766$$ 0 0
$$767$$ −960.000 −0.0451937
$$768$$ 0 0
$$769$$ 2765.00 0.129660 0.0648299 0.997896i $$-0.479350\pi$$
0.0648299 + 0.997896i $$0.479350\pi$$
$$770$$ 0 0
$$771$$ 20610.0 0.962712
$$772$$ 0 0
$$773$$ −14046.0 −0.653557 −0.326778 0.945101i $$-0.605963\pi$$
−0.326778 + 0.945101i $$0.605963\pi$$
$$774$$ 0 0
$$775$$ −29348.0 −1.36027
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 5760.00 0.264921
$$780$$ 0 0
$$781$$ 5130.00 0.235039
$$782$$ 0 0
$$783$$ 8019.00 0.365997
$$784$$ 0 0
$$785$$ 588.000 0.0267345
$$786$$ 0 0
$$787$$ 18514.0 0.838568 0.419284 0.907855i $$-0.362281\pi$$
0.419284 + 0.907855i $$0.362281\pi$$
$$788$$ 0 0
$$789$$ −666.000 −0.0300510
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 7552.00 0.338183
$$794$$ 0 0
$$795$$ 3267.00 0.145747
$$796$$ 0 0
$$797$$ −27495.0 −1.22199 −0.610993 0.791636i $$-0.709230\pi$$
−0.610993 + 0.791636i $$0.709230\pi$$
$$798$$ 0 0
$$799$$ 2520.00 0.111578
$$800$$ 0 0
$$801$$ 8154.00 0.359685
$$802$$ 0 0
$$803$$ 5430.00 0.238631
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ −23553.0 −1.02739
$$808$$ 0 0
$$809$$ −7944.00 −0.345236 −0.172618 0.984989i $$-0.555223\pi$$
−0.172618 + 0.984989i $$0.555223\pi$$
$$810$$ 0 0
$$811$$ 28942.0 1.25313 0.626567 0.779368i $$-0.284460\pi$$
0.626567 + 0.779368i $$0.284460\pi$$
$$812$$ 0 0
$$813$$ 15549.0 0.670759
$$814$$ 0 0
$$815$$ −3756.00 −0.161432
$$816$$ 0 0
$$817$$ −416.000 −0.0178140
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 8187.00 0.348025 0.174012 0.984743i $$-0.444327\pi$$
0.174012 + 0.984743i $$0.444327\pi$$
$$822$$ 0 0
$$823$$ 280.000 0.0118593 0.00592964 0.999982i $$-0.498113\pi$$
0.00592964 + 0.999982i $$0.498113\pi$$
$$824$$ 0 0
$$825$$ 5220.00 0.220287
$$826$$ 0 0
$$827$$ −25317.0 −1.06452 −0.532260 0.846581i $$-0.678657\pi$$
−0.532260 + 0.846581i $$0.678657\pi$$
$$828$$ 0 0
$$829$$ 15320.0 0.641840 0.320920 0.947106i $$-0.396008\pi$$
0.320920 + 0.947106i $$0.396008\pi$$
$$830$$ 0 0
$$831$$ 14880.0 0.621157
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 0 0
$$835$$ −7938.00 −0.328989
$$836$$ 0 0
$$837$$ −6831.00 −0.282095
$$838$$ 0 0
$$839$$ −34092.0 −1.40284 −0.701422 0.712746i $$-0.747451\pi$$
−0.701422 + 0.712746i $$0.747451\pi$$
$$840$$ 0 0
$$841$$ 63820.0 2.61675
$$842$$ 0 0
$$843$$ 2322.00 0.0948682
$$844$$ 0 0
$$845$$ −5697.00 −0.231932
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ 11094.0 0.448463
$$850$$ 0 0
$$851$$ −26544.0 −1.06923
$$852$$ 0 0
$$853$$ −7378.00 −0.296152 −0.148076 0.988976i $$-0.547308\pi$$
−0.148076 + 0.988976i $$0.547308\pi$$
$$854$$ 0 0
$$855$$ −432.000 −0.0172796
$$856$$ 0 0
$$857$$ −15594.0 −0.621565 −0.310782 0.950481i $$-0.600591\pi$$
−0.310782 + 0.950481i $$0.600591\pi$$
$$858$$ 0 0
$$859$$ 30538.0 1.21297 0.606486 0.795094i $$-0.292579\pi$$
0.606486 + 0.795094i $$0.292579\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 822.000 0.0324232 0.0162116 0.999869i $$-0.494839\pi$$
0.0162116 + 0.999869i $$0.494839\pi$$
$$864$$ 0 0
$$865$$ 2358.00 0.0926872
$$866$$ 0 0
$$867$$ −6429.00 −0.251834
$$868$$ 0 0
$$869$$ −7005.00 −0.273450
$$870$$ 0 0
$$871$$ −23680.0 −0.921201
$$872$$ 0 0
$$873$$ 4527.00 0.175505
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −41824.0 −1.61037 −0.805186 0.593022i $$-0.797935\pi$$
−0.805186 + 0.593022i $$0.797935\pi$$
$$878$$ 0 0
$$879$$ 18819.0 0.722126
$$880$$ 0 0
$$881$$ −46098.0 −1.76286 −0.881431 0.472313i $$-0.843419\pi$$
−0.881431 + 0.472313i $$0.843419\pi$$
$$882$$ 0 0
$$883$$ −21008.0 −0.800652 −0.400326 0.916373i $$-0.631103\pi$$
−0.400326 + 0.916373i $$0.631103\pi$$
$$884$$ 0 0
$$885$$ 135.000 0.00512766
$$886$$ 0 0
$$887$$ −24036.0 −0.909865 −0.454932 0.890526i $$-0.650337\pi$$
−0.454932 + 0.890526i $$0.650337\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ 1215.00 0.0456835
$$892$$ 0 0
$$893$$ 480.000 0.0179872
$$894$$ 0 0
$$895$$ 8676.00 0.324030
$$896$$ 0 0
$$897$$ 16128.0 0.600332
$$898$$ 0 0
$$899$$ −75141.0 −2.78764
$$900$$ 0 0
$$901$$ 30492.0 1.12745
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ −4056.00 −0.148979
$$906$$ 0 0
$$907$$ −13292.0 −0.486608 −0.243304 0.969950i $$-0.578231\pi$$
−0.243304 + 0.969950i $$0.578231\pi$$
$$908$$ 0 0
$$909$$ −9774.00 −0.356637
$$910$$ 0 0
$$911$$ 9306.00 0.338443 0.169221 0.985578i $$-0.445875\pi$$
0.169221 + 0.985578i $$0.445875\pi$$
$$912$$ 0 0
$$913$$ −7155.00 −0.259360
$$914$$ 0 0
$$915$$ −1062.00 −0.0383701
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ −16496.0 −0.592114 −0.296057 0.955170i $$-0.595672\pi$$
−0.296057 + 0.955170i $$0.595672\pi$$
$$920$$ 0 0
$$921$$ −5052.00 −0.180748
$$922$$ 0 0
$$923$$ −21888.0 −0.780555
$$924$$ 0 0
$$925$$ 36656.0 1.30296
$$926$$ 0 0
$$927$$ −15624.0 −0.553570
$$928$$ 0 0
$$929$$ 14154.0 0.499868 0.249934 0.968263i $$-0.419591\pi$$
0.249934 + 0.968263i $$0.419591\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 0 0
$$933$$ −3960.00 −0.138955
$$934$$ 0 0
$$935$$ −3780.00 −0.132213
$$936$$ 0 0
$$937$$ −3781.00 −0.131825 −0.0659124 0.997825i $$-0.520996\pi$$
−0.0659124 + 0.997825i $$0.520996\pi$$
$$938$$ 0 0
$$939$$ 25509.0 0.886533
$$940$$ 0 0
$$941$$ −25863.0 −0.895972 −0.447986 0.894041i $$-0.647859\pi$$
−0.447986 + 0.894041i $$0.647859\pi$$
$$942$$ 0 0
$$943$$ 30240.0 1.04427
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 42384.0 1.45438 0.727188 0.686438i $$-0.240827\pi$$
0.727188 + 0.686438i $$0.240827\pi$$
$$948$$ 0 0
$$949$$ −23168.0 −0.792482
$$950$$ 0 0
$$951$$ 7731.00 0.263612
$$952$$ 0 0
$$953$$ 10530.0 0.357923 0.178961 0.983856i $$-0.442726\pi$$
0.178961 + 0.983856i $$0.442726\pi$$
$$954$$ 0 0
$$955$$ 11736.0 0.397663
$$956$$ 0 0
$$957$$ 13365.0 0.451441
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ 34218.0 1.14860
$$962$$ 0 0
$$963$$ 12177.0 0.407475
$$964$$ 0 0
$$965$$ −4479.00 −0.149414
$$966$$ 0 0
$$967$$ 38341.0 1.27504 0.637520 0.770434i $$-0.279960\pi$$
0.637520 + 0.770434i $$0.279960\pi$$
$$968$$ 0 0
$$969$$ −4032.00 −0.133670
$$970$$ 0 0
$$971$$ −1923.00 −0.0635551 −0.0317776 0.999495i $$-0.510117\pi$$
−0.0317776 + 0.999495i $$0.510117\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 0 0
$$975$$ −22272.0 −0.731564
$$976$$ 0 0
$$977$$ 57090.0 1.86947 0.934734 0.355347i $$-0.115637\pi$$
0.934734 + 0.355347i $$0.115637\pi$$
$$978$$ 0 0
$$979$$ 13590.0 0.443655
$$980$$ 0 0
$$981$$ −3330.00 −0.108378
$$982$$ 0 0
$$983$$ −5484.00 −0.177937 −0.0889687 0.996034i $$-0.528357\pi$$
−0.0889687 + 0.996034i $$0.528357\pi$$
$$984$$ 0 0
$$985$$ 12258.0 0.396520
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −2184.00 −0.0702196
$$990$$ 0 0
$$991$$ 22465.0 0.720105 0.360053 0.932932i $$-0.382759\pi$$
0.360053 + 0.932932i $$0.382759\pi$$
$$992$$ 0 0
$$993$$ −1452.00 −0.0464026
$$994$$ 0 0
$$995$$ −10668.0 −0.339898
$$996$$ 0 0
$$997$$ 29366.0 0.932829 0.466415 0.884566i $$-0.345546\pi$$
0.466415 + 0.884566i $$0.345546\pi$$
$$998$$ 0 0
$$999$$ 8532.00 0.270211
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 2352.4.a.i.1.1 1
4.3 odd 2 147.4.a.b.1.1 1
7.2 even 3 336.4.q.e.193.1 2
7.4 even 3 336.4.q.e.289.1 2
7.6 odd 2 2352.4.a.bd.1.1 1
12.11 even 2 441.4.a.l.1.1 1
28.3 even 6 147.4.e.h.79.1 2
28.11 odd 6 21.4.e.a.16.1 yes 2
28.19 even 6 147.4.e.h.67.1 2
28.23 odd 6 21.4.e.a.4.1 2
28.27 even 2 147.4.a.a.1.1 1
84.11 even 6 63.4.e.a.37.1 2
84.23 even 6 63.4.e.a.46.1 2
84.47 odd 6 441.4.e.c.361.1 2
84.59 odd 6 441.4.e.c.226.1 2
84.83 odd 2 441.4.a.k.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
21.4.e.a.4.1 2 28.23 odd 6
21.4.e.a.16.1 yes 2 28.11 odd 6
63.4.e.a.37.1 2 84.11 even 6
63.4.e.a.46.1 2 84.23 even 6
147.4.a.a.1.1 1 28.27 even 2
147.4.a.b.1.1 1 4.3 odd 2
147.4.e.h.67.1 2 28.19 even 6
147.4.e.h.79.1 2 28.3 even 6
336.4.q.e.193.1 2 7.2 even 3
336.4.q.e.289.1 2 7.4 even 3
441.4.a.k.1.1 1 84.83 odd 2
441.4.a.l.1.1 1 12.11 even 2
441.4.e.c.226.1 2 84.59 odd 6
441.4.e.c.361.1 2 84.47 odd 6
2352.4.a.i.1.1 1 1.1 even 1 trivial
2352.4.a.bd.1.1 1 7.6 odd 2