# Properties

 Label 9408.2.a.dj.1.1 Level $9408$ Weight $2$ Character 9408.1 Self dual yes Analytic conductor $75.123$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$9408 = 2^{6} \cdot 3 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 9408.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$75.1232582216$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{57})$$ Defining polynomial: $$x^{2} - x - 14$$ x^2 - x - 14 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 168) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$4.27492$$ of defining polynomial Character $$\chi$$ $$=$$ 9408.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{3} -4.27492 q^{5} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{3} -4.27492 q^{5} +1.00000 q^{9} +4.27492 q^{11} -1.27492 q^{13} +4.27492 q^{15} +4.00000 q^{17} +1.27492 q^{19} +4.00000 q^{23} +13.2749 q^{25} -1.00000 q^{27} +2.27492 q^{29} +1.00000 q^{31} -4.27492 q^{33} -5.27492 q^{37} +1.27492 q^{39} -10.5498 q^{41} +7.27492 q^{43} -4.27492 q^{45} +6.00000 q^{47} -4.00000 q^{51} -1.72508 q^{53} -18.2749 q^{55} -1.27492 q^{57} +6.27492 q^{59} +10.0000 q^{61} +5.45017 q^{65} -7.27492 q^{67} -4.00000 q^{69} +2.00000 q^{71} -3.27492 q^{73} -13.2749 q^{75} -3.54983 q^{79} +1.00000 q^{81} -0.274917 q^{83} -17.0997 q^{85} -2.27492 q^{87} -4.54983 q^{89} -1.00000 q^{93} -5.45017 q^{95} -16.2749 q^{97} +4.27492 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{3} - q^{5} + 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^3 - q^5 + 2 * q^9 $$2 q - 2 q^{3} - q^{5} + 2 q^{9} + q^{11} + 5 q^{13} + q^{15} + 8 q^{17} - 5 q^{19} + 8 q^{23} + 19 q^{25} - 2 q^{27} - 3 q^{29} + 2 q^{31} - q^{33} - 3 q^{37} - 5 q^{39} - 6 q^{41} + 7 q^{43} - q^{45} + 12 q^{47} - 8 q^{51} - 11 q^{53} - 29 q^{55} + 5 q^{57} + 5 q^{59} + 20 q^{61} + 26 q^{65} - 7 q^{67} - 8 q^{69} + 4 q^{71} + q^{73} - 19 q^{75} + 8 q^{79} + 2 q^{81} + 7 q^{83} - 4 q^{85} + 3 q^{87} + 6 q^{89} - 2 q^{93} - 26 q^{95} - 25 q^{97} + q^{99}+O(q^{100})$$ 2 * q - 2 * q^3 - q^5 + 2 * q^9 + q^11 + 5 * q^13 + q^15 + 8 * q^17 - 5 * q^19 + 8 * q^23 + 19 * q^25 - 2 * q^27 - 3 * q^29 + 2 * q^31 - q^33 - 3 * q^37 - 5 * q^39 - 6 * q^41 + 7 * q^43 - q^45 + 12 * q^47 - 8 * q^51 - 11 * q^53 - 29 * q^55 + 5 * q^57 + 5 * q^59 + 20 * q^61 + 26 * q^65 - 7 * q^67 - 8 * q^69 + 4 * q^71 + q^73 - 19 * q^75 + 8 * q^79 + 2 * q^81 + 7 * q^83 - 4 * q^85 + 3 * q^87 + 6 * q^89 - 2 * q^93 - 26 * q^95 - 25 * q^97 + q^99

## 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
$$3$$ −1.00000 −0.577350
$$4$$ 0 0
$$5$$ −4.27492 −1.91180 −0.955901 0.293691i $$-0.905116\pi$$
−0.955901 + 0.293691i $$0.905116\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ 4.27492 1.28894 0.644468 0.764631i $$-0.277079\pi$$
0.644468 + 0.764631i $$0.277079\pi$$
$$12$$ 0 0
$$13$$ −1.27492 −0.353598 −0.176799 0.984247i $$-0.556574\pi$$
−0.176799 + 0.984247i $$0.556574\pi$$
$$14$$ 0 0
$$15$$ 4.27492 1.10378
$$16$$ 0 0
$$17$$ 4.00000 0.970143 0.485071 0.874475i $$-0.338794\pi$$
0.485071 + 0.874475i $$0.338794\pi$$
$$18$$ 0 0
$$19$$ 1.27492 0.292486 0.146243 0.989249i $$-0.453282\pi$$
0.146243 + 0.989249i $$0.453282\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 4.00000 0.834058 0.417029 0.908893i $$-0.363071\pi$$
0.417029 + 0.908893i $$0.363071\pi$$
$$24$$ 0 0
$$25$$ 13.2749 2.65498
$$26$$ 0 0
$$27$$ −1.00000 −0.192450
$$28$$ 0 0
$$29$$ 2.27492 0.422442 0.211221 0.977438i $$-0.432256\pi$$
0.211221 + 0.977438i $$0.432256\pi$$
$$30$$ 0 0
$$31$$ 1.00000 0.179605 0.0898027 0.995960i $$-0.471376\pi$$
0.0898027 + 0.995960i $$0.471376\pi$$
$$32$$ 0 0
$$33$$ −4.27492 −0.744168
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −5.27492 −0.867191 −0.433596 0.901108i $$-0.642755\pi$$
−0.433596 + 0.901108i $$0.642755\pi$$
$$38$$ 0 0
$$39$$ 1.27492 0.204150
$$40$$ 0 0
$$41$$ −10.5498 −1.64761 −0.823804 0.566875i $$-0.808152\pi$$
−0.823804 + 0.566875i $$0.808152\pi$$
$$42$$ 0 0
$$43$$ 7.27492 1.10941 0.554707 0.832046i $$-0.312830\pi$$
0.554707 + 0.832046i $$0.312830\pi$$
$$44$$ 0 0
$$45$$ −4.27492 −0.637267
$$46$$ 0 0
$$47$$ 6.00000 0.875190 0.437595 0.899172i $$-0.355830\pi$$
0.437595 + 0.899172i $$0.355830\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 0 0
$$51$$ −4.00000 −0.560112
$$52$$ 0 0
$$53$$ −1.72508 −0.236958 −0.118479 0.992957i $$-0.537802\pi$$
−0.118479 + 0.992957i $$0.537802\pi$$
$$54$$ 0 0
$$55$$ −18.2749 −2.46419
$$56$$ 0 0
$$57$$ −1.27492 −0.168867
$$58$$ 0 0
$$59$$ 6.27492 0.816925 0.408462 0.912775i $$-0.366065\pi$$
0.408462 + 0.912775i $$0.366065\pi$$
$$60$$ 0 0
$$61$$ 10.0000 1.28037 0.640184 0.768221i $$-0.278858\pi$$
0.640184 + 0.768221i $$0.278858\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 5.45017 0.676010
$$66$$ 0 0
$$67$$ −7.27492 −0.888773 −0.444386 0.895835i $$-0.646578\pi$$
−0.444386 + 0.895835i $$0.646578\pi$$
$$68$$ 0 0
$$69$$ −4.00000 −0.481543
$$70$$ 0 0
$$71$$ 2.00000 0.237356 0.118678 0.992933i $$-0.462134\pi$$
0.118678 + 0.992933i $$0.462134\pi$$
$$72$$ 0 0
$$73$$ −3.27492 −0.383300 −0.191650 0.981463i $$-0.561384\pi$$
−0.191650 + 0.981463i $$0.561384\pi$$
$$74$$ 0 0
$$75$$ −13.2749 −1.53286
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −3.54983 −0.399388 −0.199694 0.979858i $$-0.563995\pi$$
−0.199694 + 0.979858i $$0.563995\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ −0.274917 −0.0301761 −0.0150880 0.999886i $$-0.504803\pi$$
−0.0150880 + 0.999886i $$0.504803\pi$$
$$84$$ 0 0
$$85$$ −17.0997 −1.85472
$$86$$ 0 0
$$87$$ −2.27492 −0.243897
$$88$$ 0 0
$$89$$ −4.54983 −0.482281 −0.241141 0.970490i $$-0.577522\pi$$
−0.241141 + 0.970490i $$0.577522\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ −1.00000 −0.103695
$$94$$ 0 0
$$95$$ −5.45017 −0.559175
$$96$$ 0 0
$$97$$ −16.2749 −1.65247 −0.826234 0.563327i $$-0.809521\pi$$
−0.826234 + 0.563327i $$0.809521\pi$$
$$98$$ 0 0
$$99$$ 4.27492 0.429645
$$100$$ 0 0
$$101$$ −6.00000 −0.597022 −0.298511 0.954406i $$-0.596490\pi$$
−0.298511 + 0.954406i $$0.596490\pi$$
$$102$$ 0 0
$$103$$ 11.8248 1.16513 0.582564 0.812785i $$-0.302050\pi$$
0.582564 + 0.812785i $$0.302050\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −6.82475 −0.659774 −0.329887 0.944020i $$-0.607011\pi$$
−0.329887 + 0.944020i $$0.607011\pi$$
$$108$$ 0 0
$$109$$ 5.82475 0.557910 0.278955 0.960304i $$-0.410012\pi$$
0.278955 + 0.960304i $$0.410012\pi$$
$$110$$ 0 0
$$111$$ 5.27492 0.500673
$$112$$ 0 0
$$113$$ 10.5498 0.992445 0.496222 0.868195i $$-0.334720\pi$$
0.496222 + 0.868195i $$0.334720\pi$$
$$114$$ 0 0
$$115$$ −17.0997 −1.59455
$$116$$ 0 0
$$117$$ −1.27492 −0.117866
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 7.27492 0.661356
$$122$$ 0 0
$$123$$ 10.5498 0.951247
$$124$$ 0 0
$$125$$ −35.3746 −3.16400
$$126$$ 0 0
$$127$$ −21.5498 −1.91224 −0.956119 0.292978i $$-0.905354\pi$$
−0.956119 + 0.292978i $$0.905354\pi$$
$$128$$ 0 0
$$129$$ −7.27492 −0.640521
$$130$$ 0 0
$$131$$ −0.274917 −0.0240196 −0.0120098 0.999928i $$-0.503823\pi$$
−0.0120098 + 0.999928i $$0.503823\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ 4.27492 0.367926
$$136$$ 0 0
$$137$$ 16.5498 1.41395 0.706974 0.707240i $$-0.250060\pi$$
0.706974 + 0.707240i $$0.250060\pi$$
$$138$$ 0 0
$$139$$ 0.725083 0.0615007 0.0307504 0.999527i $$-0.490210\pi$$
0.0307504 + 0.999527i $$0.490210\pi$$
$$140$$ 0 0
$$141$$ −6.00000 −0.505291
$$142$$ 0 0
$$143$$ −5.45017 −0.455766
$$144$$ 0 0
$$145$$ −9.72508 −0.807624
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ −0.549834 −0.0450442 −0.0225221 0.999746i $$-0.507170\pi$$
−0.0225221 + 0.999746i $$0.507170\pi$$
$$150$$ 0 0
$$151$$ −15.3746 −1.25117 −0.625583 0.780158i $$-0.715139\pi$$
−0.625583 + 0.780158i $$0.715139\pi$$
$$152$$ 0 0
$$153$$ 4.00000 0.323381
$$154$$ 0 0
$$155$$ −4.27492 −0.343370
$$156$$ 0 0
$$157$$ 14.5498 1.16120 0.580602 0.814188i $$-0.302817\pi$$
0.580602 + 0.814188i $$0.302817\pi$$
$$158$$ 0 0
$$159$$ 1.72508 0.136808
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 12.0000 0.939913 0.469956 0.882690i $$-0.344270\pi$$
0.469956 + 0.882690i $$0.344270\pi$$
$$164$$ 0 0
$$165$$ 18.2749 1.42270
$$166$$ 0 0
$$167$$ 6.00000 0.464294 0.232147 0.972681i $$-0.425425\pi$$
0.232147 + 0.972681i $$0.425425\pi$$
$$168$$ 0 0
$$169$$ −11.3746 −0.874968
$$170$$ 0 0
$$171$$ 1.27492 0.0974954
$$172$$ 0 0
$$173$$ −7.45017 −0.566426 −0.283213 0.959057i $$-0.591400\pi$$
−0.283213 + 0.959057i $$0.591400\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ −6.27492 −0.471652
$$178$$ 0 0
$$179$$ 1.45017 0.108390 0.0541952 0.998530i $$-0.482741\pi$$
0.0541952 + 0.998530i $$0.482741\pi$$
$$180$$ 0 0
$$181$$ 3.82475 0.284292 0.142146 0.989846i $$-0.454600\pi$$
0.142146 + 0.989846i $$0.454600\pi$$
$$182$$ 0 0
$$183$$ −10.0000 −0.739221
$$184$$ 0 0
$$185$$ 22.5498 1.65790
$$186$$ 0 0
$$187$$ 17.0997 1.25045
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −10.5498 −0.763359 −0.381680 0.924295i $$-0.624654\pi$$
−0.381680 + 0.924295i $$0.624654\pi$$
$$192$$ 0 0
$$193$$ 15.5498 1.11930 0.559651 0.828729i $$-0.310935\pi$$
0.559651 + 0.828729i $$0.310935\pi$$
$$194$$ 0 0
$$195$$ −5.45017 −0.390294
$$196$$ 0 0
$$197$$ 16.5498 1.17913 0.589563 0.807722i $$-0.299300\pi$$
0.589563 + 0.807722i $$0.299300\pi$$
$$198$$ 0 0
$$199$$ −25.0997 −1.77927 −0.889634 0.456674i $$-0.849041\pi$$
−0.889634 + 0.456674i $$0.849041\pi$$
$$200$$ 0 0
$$201$$ 7.27492 0.513133
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 45.0997 3.14990
$$206$$ 0 0
$$207$$ 4.00000 0.278019
$$208$$ 0 0
$$209$$ 5.45017 0.376996
$$210$$ 0 0
$$211$$ −17.6495 −1.21504 −0.607521 0.794304i $$-0.707836\pi$$
−0.607521 + 0.794304i $$0.707836\pi$$
$$212$$ 0 0
$$213$$ −2.00000 −0.137038
$$214$$ 0 0
$$215$$ −31.0997 −2.12098
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ 3.27492 0.221298
$$220$$ 0 0
$$221$$ −5.09967 −0.343041
$$222$$ 0 0
$$223$$ −6.27492 −0.420200 −0.210100 0.977680i $$-0.567379\pi$$
−0.210100 + 0.977680i $$0.567379\pi$$
$$224$$ 0 0
$$225$$ 13.2749 0.884994
$$226$$ 0 0
$$227$$ −3.72508 −0.247242 −0.123621 0.992329i $$-0.539451\pi$$
−0.123621 + 0.992329i $$0.539451\pi$$
$$228$$ 0 0
$$229$$ 10.7251 0.708733 0.354367 0.935107i $$-0.384696\pi$$
0.354367 + 0.935107i $$0.384696\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 14.5498 0.953191 0.476596 0.879123i $$-0.341871\pi$$
0.476596 + 0.879123i $$0.341871\pi$$
$$234$$ 0 0
$$235$$ −25.6495 −1.67319
$$236$$ 0 0
$$237$$ 3.54983 0.230587
$$238$$ 0 0
$$239$$ 30.5498 1.97610 0.988052 0.154119i $$-0.0492540\pi$$
0.988052 + 0.154119i $$0.0492540\pi$$
$$240$$ 0 0
$$241$$ 12.8248 0.826115 0.413057 0.910705i $$-0.364461\pi$$
0.413057 + 0.910705i $$0.364461\pi$$
$$242$$ 0 0
$$243$$ −1.00000 −0.0641500
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −1.62541 −0.103423
$$248$$ 0 0
$$249$$ 0.274917 0.0174222
$$250$$ 0 0
$$251$$ 19.3746 1.22291 0.611457 0.791278i $$-0.290584\pi$$
0.611457 + 0.791278i $$0.290584\pi$$
$$252$$ 0 0
$$253$$ 17.0997 1.07505
$$254$$ 0 0
$$255$$ 17.0997 1.07082
$$256$$ 0 0
$$257$$ 19.0997 1.19140 0.595702 0.803205i $$-0.296874\pi$$
0.595702 + 0.803205i $$0.296874\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 2.27492 0.140814
$$262$$ 0 0
$$263$$ −24.5498 −1.51381 −0.756904 0.653526i $$-0.773289\pi$$
−0.756904 + 0.653526i $$0.773289\pi$$
$$264$$ 0 0
$$265$$ 7.37459 0.453017
$$266$$ 0 0
$$267$$ 4.54983 0.278445
$$268$$ 0 0
$$269$$ 28.2749 1.72395 0.861976 0.506949i $$-0.169227\pi$$
0.861976 + 0.506949i $$0.169227\pi$$
$$270$$ 0 0
$$271$$ 6.27492 0.381174 0.190587 0.981670i $$-0.438961\pi$$
0.190587 + 0.981670i $$0.438961\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 56.7492 3.42210
$$276$$ 0 0
$$277$$ 4.17525 0.250866 0.125433 0.992102i $$-0.459968\pi$$
0.125433 + 0.992102i $$0.459968\pi$$
$$278$$ 0 0
$$279$$ 1.00000 0.0598684
$$280$$ 0 0
$$281$$ −11.4502 −0.683060 −0.341530 0.939871i $$-0.610945\pi$$
−0.341530 + 0.939871i $$0.610945\pi$$
$$282$$ 0 0
$$283$$ 26.9244 1.60049 0.800245 0.599673i $$-0.204703\pi$$
0.800245 + 0.599673i $$0.204703\pi$$
$$284$$ 0 0
$$285$$ 5.45017 0.322840
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ −1.00000 −0.0588235
$$290$$ 0 0
$$291$$ 16.2749 0.954053
$$292$$ 0 0
$$293$$ −5.17525 −0.302341 −0.151171 0.988508i $$-0.548304\pi$$
−0.151171 + 0.988508i $$0.548304\pi$$
$$294$$ 0 0
$$295$$ −26.8248 −1.56180
$$296$$ 0 0
$$297$$ −4.27492 −0.248056
$$298$$ 0 0
$$299$$ −5.09967 −0.294921
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ 6.00000 0.344691
$$304$$ 0 0
$$305$$ −42.7492 −2.44781
$$306$$ 0 0
$$307$$ −26.3746 −1.50528 −0.752639 0.658434i $$-0.771219\pi$$
−0.752639 + 0.658434i $$0.771219\pi$$
$$308$$ 0 0
$$309$$ −11.8248 −0.672687
$$310$$ 0 0
$$311$$ −10.5498 −0.598226 −0.299113 0.954218i $$-0.596691\pi$$
−0.299113 + 0.954218i $$0.596691\pi$$
$$312$$ 0 0
$$313$$ 4.45017 0.251538 0.125769 0.992060i $$-0.459860\pi$$
0.125769 + 0.992060i $$0.459860\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 5.17525 0.290671 0.145335 0.989382i $$-0.453574\pi$$
0.145335 + 0.989382i $$0.453574\pi$$
$$318$$ 0 0
$$319$$ 9.72508 0.544500
$$320$$ 0 0
$$321$$ 6.82475 0.380920
$$322$$ 0 0
$$323$$ 5.09967 0.283753
$$324$$ 0 0
$$325$$ −16.9244 −0.938798
$$326$$ 0 0
$$327$$ −5.82475 −0.322110
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −23.8248 −1.30953 −0.654763 0.755834i $$-0.727232\pi$$
−0.654763 + 0.755834i $$0.727232\pi$$
$$332$$ 0 0
$$333$$ −5.27492 −0.289064
$$334$$ 0 0
$$335$$ 31.0997 1.69916
$$336$$ 0 0
$$337$$ 6.09967 0.332270 0.166135 0.986103i $$-0.446871\pi$$
0.166135 + 0.986103i $$0.446871\pi$$
$$338$$ 0 0
$$339$$ −10.5498 −0.572988
$$340$$ 0 0
$$341$$ 4.27492 0.231500
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ 17.0997 0.920615
$$346$$ 0 0
$$347$$ 30.1993 1.62119 0.810593 0.585610i $$-0.199145\pi$$
0.810593 + 0.585610i $$0.199145\pi$$
$$348$$ 0 0
$$349$$ −6.00000 −0.321173 −0.160586 0.987022i $$-0.551338\pi$$
−0.160586 + 0.987022i $$0.551338\pi$$
$$350$$ 0 0
$$351$$ 1.27492 0.0680500
$$352$$ 0 0
$$353$$ 5.45017 0.290083 0.145042 0.989426i $$-0.453668\pi$$
0.145042 + 0.989426i $$0.453668\pi$$
$$354$$ 0 0
$$355$$ −8.54983 −0.453778
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 25.6495 1.35373 0.676865 0.736108i $$-0.263338\pi$$
0.676865 + 0.736108i $$0.263338\pi$$
$$360$$ 0 0
$$361$$ −17.3746 −0.914452
$$362$$ 0 0
$$363$$ −7.27492 −0.381834
$$364$$ 0 0
$$365$$ 14.0000 0.732793
$$366$$ 0 0
$$367$$ 8.09967 0.422799 0.211400 0.977400i $$-0.432198\pi$$
0.211400 + 0.977400i $$0.432198\pi$$
$$368$$ 0 0
$$369$$ −10.5498 −0.549202
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ −1.27492 −0.0660127 −0.0330064 0.999455i $$-0.510508\pi$$
−0.0330064 + 0.999455i $$0.510508\pi$$
$$374$$ 0 0
$$375$$ 35.3746 1.82674
$$376$$ 0 0
$$377$$ −2.90033 −0.149375
$$378$$ 0 0
$$379$$ −35.8248 −1.84019 −0.920097 0.391691i $$-0.871890\pi$$
−0.920097 + 0.391691i $$0.871890\pi$$
$$380$$ 0 0
$$381$$ 21.5498 1.10403
$$382$$ 0 0
$$383$$ 4.54983 0.232486 0.116243 0.993221i $$-0.462915\pi$$
0.116243 + 0.993221i $$0.462915\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 7.27492 0.369805
$$388$$ 0 0
$$389$$ 2.00000 0.101404 0.0507020 0.998714i $$-0.483854\pi$$
0.0507020 + 0.998714i $$0.483854\pi$$
$$390$$ 0 0
$$391$$ 16.0000 0.809155
$$392$$ 0 0
$$393$$ 0.274917 0.0138677
$$394$$ 0 0
$$395$$ 15.1752 0.763550
$$396$$ 0 0
$$397$$ 34.3746 1.72521 0.862606 0.505877i $$-0.168831\pi$$
0.862606 + 0.505877i $$0.168831\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 24.0000 1.19850 0.599251 0.800561i $$-0.295465\pi$$
0.599251 + 0.800561i $$0.295465\pi$$
$$402$$ 0 0
$$403$$ −1.27492 −0.0635082
$$404$$ 0 0
$$405$$ −4.27492 −0.212422
$$406$$ 0 0
$$407$$ −22.5498 −1.11775
$$408$$ 0 0
$$409$$ −25.5498 −1.26336 −0.631679 0.775230i $$-0.717634\pi$$
−0.631679 + 0.775230i $$0.717634\pi$$
$$410$$ 0 0
$$411$$ −16.5498 −0.816343
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 1.17525 0.0576907
$$416$$ 0 0
$$417$$ −0.725083 −0.0355075
$$418$$ 0 0
$$419$$ 13.4502 0.657084 0.328542 0.944489i $$-0.393443\pi$$
0.328542 + 0.944489i $$0.393443\pi$$
$$420$$ 0 0
$$421$$ 13.8248 0.673777 0.336889 0.941545i $$-0.390625\pi$$
0.336889 + 0.941545i $$0.390625\pi$$
$$422$$ 0 0
$$423$$ 6.00000 0.291730
$$424$$ 0 0
$$425$$ 53.0997 2.57571
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ 5.45017 0.263136
$$430$$ 0 0
$$431$$ 27.6495 1.33183 0.665915 0.746028i $$-0.268041\pi$$
0.665915 + 0.746028i $$0.268041\pi$$
$$432$$ 0 0
$$433$$ −25.8248 −1.24106 −0.620529 0.784183i $$-0.713082\pi$$
−0.620529 + 0.784183i $$0.713082\pi$$
$$434$$ 0 0
$$435$$ 9.72508 0.466282
$$436$$ 0 0
$$437$$ 5.09967 0.243950
$$438$$ 0 0
$$439$$ −9.72508 −0.464153 −0.232076 0.972698i $$-0.574552\pi$$
−0.232076 + 0.972698i $$0.574552\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −31.3746 −1.49065 −0.745326 0.666700i $$-0.767706\pi$$
−0.745326 + 0.666700i $$0.767706\pi$$
$$444$$ 0 0
$$445$$ 19.4502 0.922026
$$446$$ 0 0
$$447$$ 0.549834 0.0260063
$$448$$ 0 0
$$449$$ 5.45017 0.257209 0.128605 0.991696i $$-0.458950\pi$$
0.128605 + 0.991696i $$0.458950\pi$$
$$450$$ 0 0
$$451$$ −45.0997 −2.12366
$$452$$ 0 0
$$453$$ 15.3746 0.722361
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −8.64950 −0.404607 −0.202303 0.979323i $$-0.564843\pi$$
−0.202303 + 0.979323i $$0.564843\pi$$
$$458$$ 0 0
$$459$$ −4.00000 −0.186704
$$460$$ 0 0
$$461$$ 41.6495 1.93981 0.969905 0.243482i $$-0.0782897\pi$$
0.969905 + 0.243482i $$0.0782897\pi$$
$$462$$ 0 0
$$463$$ 35.8248 1.66492 0.832459 0.554087i $$-0.186933\pi$$
0.832459 + 0.554087i $$0.186933\pi$$
$$464$$ 0 0
$$465$$ 4.27492 0.198245
$$466$$ 0 0
$$467$$ 25.4502 1.17769 0.588847 0.808245i $$-0.299582\pi$$
0.588847 + 0.808245i $$0.299582\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ −14.5498 −0.670421
$$472$$ 0 0
$$473$$ 31.0997 1.42996
$$474$$ 0 0
$$475$$ 16.9244 0.776546
$$476$$ 0 0
$$477$$ −1.72508 −0.0789861
$$478$$ 0 0
$$479$$ 20.5498 0.938946 0.469473 0.882947i $$-0.344444\pi$$
0.469473 + 0.882947i $$0.344444\pi$$
$$480$$ 0 0
$$481$$ 6.72508 0.306637
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 69.5739 3.15919
$$486$$ 0 0
$$487$$ 1.00000 0.0453143 0.0226572 0.999743i $$-0.492787\pi$$
0.0226572 + 0.999743i $$0.492787\pi$$
$$488$$ 0 0
$$489$$ −12.0000 −0.542659
$$490$$ 0 0
$$491$$ 15.9244 0.718659 0.359330 0.933211i $$-0.383005\pi$$
0.359330 + 0.933211i $$0.383005\pi$$
$$492$$ 0 0
$$493$$ 9.09967 0.409828
$$494$$ 0 0
$$495$$ −18.2749 −0.821396
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ −24.7251 −1.10685 −0.553423 0.832900i $$-0.686679\pi$$
−0.553423 + 0.832900i $$0.686679\pi$$
$$500$$ 0 0
$$501$$ −6.00000 −0.268060
$$502$$ 0 0
$$503$$ −7.64950 −0.341074 −0.170537 0.985351i $$-0.554550\pi$$
−0.170537 + 0.985351i $$0.554550\pi$$
$$504$$ 0 0
$$505$$ 25.6495 1.14139
$$506$$ 0 0
$$507$$ 11.3746 0.505163
$$508$$ 0 0
$$509$$ −3.72508 −0.165111 −0.0825557 0.996586i $$-0.526308\pi$$
−0.0825557 + 0.996586i $$0.526308\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ −1.27492 −0.0562890
$$514$$ 0 0
$$515$$ −50.5498 −2.22749
$$516$$ 0 0
$$517$$ 25.6495 1.12806
$$518$$ 0 0
$$519$$ 7.45017 0.327026
$$520$$ 0 0
$$521$$ −0.549834 −0.0240887 −0.0120443 0.999927i $$-0.503834\pi$$
−0.0120443 + 0.999927i $$0.503834\pi$$
$$522$$ 0 0
$$523$$ −25.2749 −1.10519 −0.552597 0.833448i $$-0.686363\pi$$
−0.552597 + 0.833448i $$0.686363\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 4.00000 0.174243
$$528$$ 0 0
$$529$$ −7.00000 −0.304348
$$530$$ 0 0
$$531$$ 6.27492 0.272308
$$532$$ 0 0
$$533$$ 13.4502 0.582591
$$534$$ 0 0
$$535$$ 29.1752 1.26136
$$536$$ 0 0
$$537$$ −1.45017 −0.0625793
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 0.725083 0.0311737 0.0155869 0.999879i $$-0.495038\pi$$
0.0155869 + 0.999879i $$0.495038\pi$$
$$542$$ 0 0
$$543$$ −3.82475 −0.164136
$$544$$ 0 0
$$545$$ −24.9003 −1.06661
$$546$$ 0 0
$$547$$ 28.0000 1.19719 0.598597 0.801050i $$-0.295725\pi$$
0.598597 + 0.801050i $$0.295725\pi$$
$$548$$ 0 0
$$549$$ 10.0000 0.426790
$$550$$ 0 0
$$551$$ 2.90033 0.123558
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ −22.5498 −0.957187
$$556$$ 0 0
$$557$$ 7.17525 0.304025 0.152013 0.988379i $$-0.451425\pi$$
0.152013 + 0.988379i $$0.451425\pi$$
$$558$$ 0 0
$$559$$ −9.27492 −0.392287
$$560$$ 0 0
$$561$$ −17.0997 −0.721949
$$562$$ 0 0
$$563$$ 7.72508 0.325573 0.162787 0.986661i $$-0.447952\pi$$
0.162787 + 0.986661i $$0.447952\pi$$
$$564$$ 0 0
$$565$$ −45.0997 −1.89736
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 26.5498 1.11303 0.556513 0.830839i $$-0.312139\pi$$
0.556513 + 0.830839i $$0.312139\pi$$
$$570$$ 0 0
$$571$$ −0.725083 −0.0303438 −0.0151719 0.999885i $$-0.504830\pi$$
−0.0151719 + 0.999885i $$0.504830\pi$$
$$572$$ 0 0
$$573$$ 10.5498 0.440726
$$574$$ 0 0
$$575$$ 53.0997 2.21441
$$576$$ 0 0
$$577$$ 25.0000 1.04076 0.520382 0.853934i $$-0.325790\pi$$
0.520382 + 0.853934i $$0.325790\pi$$
$$578$$ 0 0
$$579$$ −15.5498 −0.646229
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ −7.37459 −0.305424
$$584$$ 0 0
$$585$$ 5.45017 0.225337
$$586$$ 0 0
$$587$$ 1.72508 0.0712018 0.0356009 0.999366i $$-0.488665\pi$$
0.0356009 + 0.999366i $$0.488665\pi$$
$$588$$ 0 0
$$589$$ 1.27492 0.0525320
$$590$$ 0 0
$$591$$ −16.5498 −0.680769
$$592$$ 0 0
$$593$$ −14.5498 −0.597490 −0.298745 0.954333i $$-0.596568\pi$$
−0.298745 + 0.954333i $$0.596568\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 25.0997 1.02726
$$598$$ 0 0
$$599$$ 7.45017 0.304406 0.152203 0.988349i $$-0.451363\pi$$
0.152203 + 0.988349i $$0.451363\pi$$
$$600$$ 0 0
$$601$$ 26.0997 1.06463 0.532314 0.846547i $$-0.321323\pi$$
0.532314 + 0.846547i $$0.321323\pi$$
$$602$$ 0 0
$$603$$ −7.27492 −0.296258
$$604$$ 0 0
$$605$$ −31.0997 −1.26438
$$606$$ 0 0
$$607$$ 7.00000 0.284121 0.142061 0.989858i $$-0.454627\pi$$
0.142061 + 0.989858i $$0.454627\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −7.64950 −0.309466
$$612$$ 0 0
$$613$$ −6.54983 −0.264545 −0.132273 0.991213i $$-0.542227\pi$$
−0.132273 + 0.991213i $$0.542227\pi$$
$$614$$ 0 0
$$615$$ −45.0997 −1.81859
$$616$$ 0 0
$$617$$ −6.00000 −0.241551 −0.120775 0.992680i $$-0.538538\pi$$
−0.120775 + 0.992680i $$0.538538\pi$$
$$618$$ 0 0
$$619$$ 6.17525 0.248204 0.124102 0.992269i $$-0.460395\pi$$
0.124102 + 0.992269i $$0.460395\pi$$
$$620$$ 0 0
$$621$$ −4.00000 −0.160514
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 84.8488 3.39395
$$626$$ 0 0
$$627$$ −5.45017 −0.217659
$$628$$ 0 0
$$629$$ −21.0997 −0.841299
$$630$$ 0 0
$$631$$ −2.82475 −0.112452 −0.0562258 0.998418i $$-0.517907\pi$$
−0.0562258 + 0.998418i $$0.517907\pi$$
$$632$$ 0 0
$$633$$ 17.6495 0.701505
$$634$$ 0 0
$$635$$ 92.1238 3.65582
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ 2.00000 0.0791188
$$640$$ 0 0
$$641$$ 41.6495 1.64506 0.822528 0.568724i $$-0.192563\pi$$
0.822528 + 0.568724i $$0.192563\pi$$
$$642$$ 0 0
$$643$$ −32.3746 −1.27673 −0.638365 0.769734i $$-0.720389\pi$$
−0.638365 + 0.769734i $$0.720389\pi$$
$$644$$ 0 0
$$645$$ 31.0997 1.22455
$$646$$ 0 0
$$647$$ 34.0000 1.33668 0.668339 0.743857i $$-0.267006\pi$$
0.668339 + 0.743857i $$0.267006\pi$$
$$648$$ 0 0
$$649$$ 26.8248 1.05296
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −45.9244 −1.79716 −0.898581 0.438808i $$-0.855401\pi$$
−0.898581 + 0.438808i $$0.855401\pi$$
$$654$$ 0 0
$$655$$ 1.17525 0.0459208
$$656$$ 0 0
$$657$$ −3.27492 −0.127767
$$658$$ 0 0
$$659$$ −18.1993 −0.708946 −0.354473 0.935066i $$-0.615340\pi$$
−0.354473 + 0.935066i $$0.615340\pi$$
$$660$$ 0 0
$$661$$ 29.8248 1.16005 0.580024 0.814599i $$-0.303043\pi$$
0.580024 + 0.814599i $$0.303043\pi$$
$$662$$ 0 0
$$663$$ 5.09967 0.198055
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 9.09967 0.352341
$$668$$ 0 0
$$669$$ 6.27492 0.242602
$$670$$ 0 0
$$671$$ 42.7492 1.65031
$$672$$ 0 0
$$673$$ 26.4502 1.01958 0.509789 0.860299i $$-0.329723\pi$$
0.509789 + 0.860299i $$0.329723\pi$$
$$674$$ 0 0
$$675$$ −13.2749 −0.510952
$$676$$ 0 0
$$677$$ −5.72508 −0.220033 −0.110016 0.993930i $$-0.535090\pi$$
−0.110016 + 0.993930i $$0.535090\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 3.72508 0.142745
$$682$$ 0 0
$$683$$ −34.8248 −1.33253 −0.666266 0.745714i $$-0.732108\pi$$
−0.666266 + 0.745714i $$0.732108\pi$$
$$684$$ 0 0
$$685$$ −70.7492 −2.70319
$$686$$ 0 0
$$687$$ −10.7251 −0.409187
$$688$$ 0 0
$$689$$ 2.19934 0.0837881
$$690$$ 0 0
$$691$$ 11.8248 0.449835 0.224917 0.974378i $$-0.427789\pi$$
0.224917 + 0.974378i $$0.427789\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ −3.09967 −0.117577
$$696$$ 0 0
$$697$$ −42.1993 −1.59841
$$698$$ 0 0
$$699$$ −14.5498 −0.550325
$$700$$ 0 0
$$701$$ −39.9244 −1.50792 −0.753962 0.656918i $$-0.771860\pi$$
−0.753962 + 0.656918i $$0.771860\pi$$
$$702$$ 0 0
$$703$$ −6.72508 −0.253641
$$704$$ 0 0
$$705$$ 25.6495 0.966016
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ −32.1993 −1.20927 −0.604636 0.796502i $$-0.706681\pi$$
−0.604636 + 0.796502i $$0.706681\pi$$
$$710$$ 0 0
$$711$$ −3.54983 −0.133129
$$712$$ 0 0
$$713$$ 4.00000 0.149801
$$714$$ 0 0
$$715$$ 23.2990 0.871333
$$716$$ 0 0
$$717$$ −30.5498 −1.14090
$$718$$ 0 0
$$719$$ 32.1993 1.20083 0.600416 0.799688i $$-0.295002\pi$$
0.600416 + 0.799688i $$0.295002\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ −12.8248 −0.476958
$$724$$ 0 0
$$725$$ 30.1993 1.12158
$$726$$ 0 0
$$727$$ −16.4502 −0.610103 −0.305051 0.952336i $$-0.598674\pi$$
−0.305051 + 0.952336i $$0.598674\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ 29.0997 1.07629
$$732$$ 0 0
$$733$$ −46.9244 −1.73319 −0.866597 0.499010i $$-0.833697\pi$$
−0.866597 + 0.499010i $$0.833697\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −31.0997 −1.14557
$$738$$ 0 0
$$739$$ 36.3746 1.33806 0.669030 0.743235i $$-0.266710\pi$$
0.669030 + 0.743235i $$0.266710\pi$$
$$740$$ 0 0
$$741$$ 1.62541 0.0597111
$$742$$ 0 0
$$743$$ 16.1993 0.594296 0.297148 0.954831i $$-0.403965\pi$$
0.297148 + 0.954831i $$0.403965\pi$$
$$744$$ 0 0
$$745$$ 2.35050 0.0861155
$$746$$ 0 0
$$747$$ −0.274917 −0.0100587
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −25.5498 −0.932327 −0.466163 0.884699i $$-0.654364\pi$$
−0.466163 + 0.884699i $$0.654364\pi$$
$$752$$ 0 0
$$753$$ −19.3746 −0.706049
$$754$$ 0 0
$$755$$ 65.7251 2.39198
$$756$$ 0 0
$$757$$ 22.0000 0.799604 0.399802 0.916602i $$-0.369079\pi$$
0.399802 + 0.916602i $$0.369079\pi$$
$$758$$ 0 0
$$759$$ −17.0997 −0.620679
$$760$$ 0 0
$$761$$ −5.09967 −0.184863 −0.0924314 0.995719i $$-0.529464\pi$$
−0.0924314 + 0.995719i $$0.529464\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ −17.0997 −0.618240
$$766$$ 0 0
$$767$$ −8.00000 −0.288863
$$768$$ 0 0
$$769$$ −12.6495 −0.456153 −0.228076 0.973643i $$-0.573244\pi$$
−0.228076 + 0.973643i $$0.573244\pi$$
$$770$$ 0 0
$$771$$ −19.0997 −0.687858
$$772$$ 0 0
$$773$$ −27.0997 −0.974707 −0.487354 0.873205i $$-0.662038\pi$$
−0.487354 + 0.873205i $$0.662038\pi$$
$$774$$ 0 0
$$775$$ 13.2749 0.476849
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ −13.4502 −0.481902
$$780$$ 0 0
$$781$$ 8.54983 0.305937
$$782$$ 0 0
$$783$$ −2.27492 −0.0812989
$$784$$ 0 0
$$785$$ −62.1993 −2.21999
$$786$$ 0 0
$$787$$ −12.5498 −0.447353 −0.223677 0.974663i $$-0.571806\pi$$
−0.223677 + 0.974663i $$0.571806\pi$$
$$788$$ 0 0
$$789$$ 24.5498 0.873997
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ −12.7492 −0.452736
$$794$$ 0 0
$$795$$ −7.37459 −0.261550
$$796$$ 0 0
$$797$$ 36.4743 1.29198 0.645992 0.763344i $$-0.276444\pi$$
0.645992 + 0.763344i $$0.276444\pi$$
$$798$$ 0 0
$$799$$ 24.0000 0.849059
$$800$$ 0 0
$$801$$ −4.54983 −0.160760
$$802$$ 0 0
$$803$$ −14.0000 −0.494049
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ −28.2749 −0.995324
$$808$$ 0 0
$$809$$ 15.6495 0.550207 0.275104 0.961415i $$-0.411288\pi$$
0.275104 + 0.961415i $$0.411288\pi$$
$$810$$ 0 0
$$811$$ −27.4502 −0.963906 −0.481953 0.876197i $$-0.660072\pi$$
−0.481953 + 0.876197i $$0.660072\pi$$
$$812$$ 0 0
$$813$$ −6.27492 −0.220071
$$814$$ 0 0
$$815$$ −51.2990 −1.79693
$$816$$ 0 0
$$817$$ 9.27492 0.324488
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −20.2749 −0.707599 −0.353800 0.935321i $$-0.615111\pi$$
−0.353800 + 0.935321i $$0.615111\pi$$
$$822$$ 0 0
$$823$$ 26.1993 0.913252 0.456626 0.889659i $$-0.349058\pi$$
0.456626 + 0.889659i $$0.349058\pi$$
$$824$$ 0 0
$$825$$ −56.7492 −1.97575
$$826$$ 0 0
$$827$$ −39.0241 −1.35700 −0.678500 0.734600i $$-0.737370\pi$$
−0.678500 + 0.734600i $$0.737370\pi$$
$$828$$ 0 0
$$829$$ 22.7251 0.789275 0.394637 0.918837i $$-0.370870\pi$$
0.394637 + 0.918837i $$0.370870\pi$$
$$830$$ 0 0
$$831$$ −4.17525 −0.144838
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 0 0
$$835$$ −25.6495 −0.887638
$$836$$ 0 0
$$837$$ −1.00000 −0.0345651
$$838$$ 0 0
$$839$$ −30.1993 −1.04260 −0.521298 0.853374i $$-0.674552\pi$$
−0.521298 + 0.853374i $$0.674552\pi$$
$$840$$ 0 0
$$841$$ −23.8248 −0.821543
$$842$$ 0 0
$$843$$ 11.4502 0.394365
$$844$$ 0 0
$$845$$ 48.6254 1.67277
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ −26.9244 −0.924044
$$850$$ 0 0
$$851$$ −21.0997 −0.723287
$$852$$ 0 0
$$853$$ −24.3746 −0.834570 −0.417285 0.908776i $$-0.637018\pi$$
−0.417285 + 0.908776i $$0.637018\pi$$
$$854$$ 0 0
$$855$$ −5.45017 −0.186392
$$856$$ 0 0
$$857$$ 28.5498 0.975244 0.487622 0.873055i $$-0.337864\pi$$
0.487622 + 0.873055i $$0.337864\pi$$
$$858$$ 0 0
$$859$$ −10.3505 −0.353154 −0.176577 0.984287i $$-0.556503\pi$$
−0.176577 + 0.984287i $$0.556503\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ −11.0997 −0.377837 −0.188919 0.981993i $$-0.560498\pi$$
−0.188919 + 0.981993i $$0.560498\pi$$
$$864$$ 0 0
$$865$$ 31.8488 1.08289
$$866$$ 0 0
$$867$$ 1.00000 0.0339618
$$868$$ 0 0
$$869$$ −15.1752 −0.514785
$$870$$ 0 0
$$871$$ 9.27492 0.314269
$$872$$ 0 0
$$873$$ −16.2749 −0.550822
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 28.7492 0.970791 0.485395 0.874295i $$-0.338676\pi$$
0.485395 + 0.874295i $$0.338676\pi$$
$$878$$ 0 0
$$879$$ 5.17525 0.174557
$$880$$ 0 0
$$881$$ −24.5498 −0.827105 −0.413552 0.910480i $$-0.635712\pi$$
−0.413552 + 0.910480i $$0.635712\pi$$
$$882$$ 0 0
$$883$$ 30.3746 1.02219 0.511093 0.859525i $$-0.329241\pi$$
0.511093 + 0.859525i $$0.329241\pi$$
$$884$$ 0 0
$$885$$ 26.8248 0.901704
$$886$$ 0 0
$$887$$ 11.6495 0.391152 0.195576 0.980689i $$-0.437342\pi$$
0.195576 + 0.980689i $$0.437342\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ 4.27492 0.143215
$$892$$ 0 0
$$893$$ 7.64950 0.255981
$$894$$ 0 0
$$895$$ −6.19934 −0.207221
$$896$$ 0 0
$$897$$ 5.09967 0.170273
$$898$$ 0 0
$$899$$ 2.27492 0.0758727
$$900$$ 0 0
$$901$$ −6.90033 −0.229883
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ −16.3505 −0.543509
$$906$$ 0 0
$$907$$ 37.2749 1.23769 0.618847 0.785512i $$-0.287600\pi$$
0.618847 + 0.785512i $$0.287600\pi$$
$$908$$ 0 0
$$909$$ −6.00000 −0.199007
$$910$$ 0 0
$$911$$ −51.8488 −1.71783 −0.858914 0.512119i $$-0.828861\pi$$
−0.858914 + 0.512119i $$0.828861\pi$$
$$912$$ 0 0
$$913$$ −1.17525 −0.0388950
$$914$$ 0 0
$$915$$ 42.7492 1.41324
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ 19.8248 0.653958 0.326979 0.945032i $$-0.393969\pi$$
0.326979 + 0.945032i $$0.393969\pi$$
$$920$$ 0 0
$$921$$ 26.3746 0.869072
$$922$$ 0 0
$$923$$ −2.54983 −0.0839288
$$924$$ 0 0
$$925$$ −70.0241 −2.30238
$$926$$ 0 0
$$927$$ 11.8248 0.388376
$$928$$ 0 0
$$929$$ −20.1993 −0.662719 −0.331359 0.943505i $$-0.607507\pi$$
−0.331359 + 0.943505i $$0.607507\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 0 0
$$933$$ 10.5498 0.345386
$$934$$ 0 0
$$935$$ −73.0997 −2.39061
$$936$$ 0 0
$$937$$ −24.0997 −0.787302 −0.393651 0.919260i $$-0.628788\pi$$
−0.393651 + 0.919260i $$0.628788\pi$$
$$938$$ 0 0
$$939$$ −4.45017 −0.145226
$$940$$ 0 0
$$941$$ 29.1752 0.951086 0.475543 0.879693i $$-0.342252\pi$$
0.475543 + 0.879693i $$0.342252\pi$$
$$942$$ 0 0
$$943$$ −42.1993 −1.37420
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 34.5498 1.12272 0.561359 0.827572i $$-0.310279\pi$$
0.561359 + 0.827572i $$0.310279\pi$$
$$948$$ 0 0
$$949$$ 4.17525 0.135534
$$950$$ 0 0
$$951$$ −5.17525 −0.167819
$$952$$ 0 0
$$953$$ 10.3505 0.335285 0.167643 0.985848i $$-0.446384\pi$$
0.167643 + 0.985848i $$0.446384\pi$$
$$954$$ 0 0
$$955$$ 45.0997 1.45939
$$956$$ 0 0
$$957$$ −9.72508 −0.314367
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −30.0000 −0.967742
$$962$$ 0 0
$$963$$ −6.82475 −0.219925
$$964$$ 0 0
$$965$$ −66.4743 −2.13988
$$966$$ 0 0
$$967$$ −38.4502 −1.23647 −0.618237 0.785992i $$-0.712153\pi$$
−0.618237 + 0.785992i $$0.712153\pi$$
$$968$$ 0 0
$$969$$ −5.09967 −0.163825
$$970$$ 0 0
$$971$$ −41.5739 −1.33417 −0.667085 0.744981i $$-0.732458\pi$$
−0.667085 + 0.744981i $$0.732458\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 0 0
$$975$$ 16.9244 0.542015
$$976$$ 0 0
$$977$$ 39.0997 1.25091 0.625455 0.780261i $$-0.284914\pi$$
0.625455 + 0.780261i $$0.284914\pi$$
$$978$$ 0 0
$$979$$ −19.4502 −0.621630
$$980$$ 0 0
$$981$$ 5.82475 0.185970
$$982$$ 0 0
$$983$$ 51.2990 1.63618 0.818092 0.575087i $$-0.195032\pi$$
0.818092 + 0.575087i $$0.195032\pi$$
$$984$$ 0 0
$$985$$ −70.7492 −2.25426
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 29.0997 0.925316
$$990$$ 0 0
$$991$$ 16.0997 0.511423 0.255711 0.966753i $$-0.417690\pi$$
0.255711 + 0.966753i $$0.417690\pi$$
$$992$$ 0 0
$$993$$ 23.8248 0.756056
$$994$$ 0 0
$$995$$ 107.299 3.40161
$$996$$ 0 0
$$997$$ −22.1752 −0.702297 −0.351149 0.936320i $$-0.614209\pi$$
−0.351149 + 0.936320i $$0.614209\pi$$
$$998$$ 0 0
$$999$$ 5.27492 0.166891
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 9408.2.a.dj.1.1 2
4.3 odd 2 9408.2.a.dw.1.1 2
7.3 odd 6 1344.2.q.w.961.1 4
7.5 odd 6 1344.2.q.w.193.1 4
7.6 odd 2 9408.2.a.ec.1.2 2
8.3 odd 2 2352.2.a.ba.1.2 2
8.5 even 2 1176.2.a.n.1.2 2
24.5 odd 2 3528.2.a.bd.1.1 2
24.11 even 2 7056.2.a.ch.1.1 2
28.3 even 6 1344.2.q.x.961.1 4
28.19 even 6 1344.2.q.x.193.1 4
28.27 even 2 9408.2.a.dp.1.2 2
56.3 even 6 336.2.q.g.289.2 4
56.5 odd 6 168.2.q.c.25.2 4
56.11 odd 6 2352.2.q.bf.961.1 4
56.13 odd 2 1176.2.a.k.1.1 2
56.19 even 6 336.2.q.g.193.2 4
56.27 even 2 2352.2.a.bf.1.1 2
56.37 even 6 1176.2.q.l.361.1 4
56.45 odd 6 168.2.q.c.121.2 yes 4
56.51 odd 6 2352.2.q.bf.1537.1 4
56.53 even 6 1176.2.q.l.961.1 4
168.5 even 6 504.2.s.i.361.1 4
168.53 odd 6 3528.2.s.bk.3313.2 4
168.59 odd 6 1008.2.s.r.289.1 4
168.83 odd 2 7056.2.a.cu.1.2 2
168.101 even 6 504.2.s.i.289.1 4
168.125 even 2 3528.2.a.bk.1.2 2
168.131 odd 6 1008.2.s.r.865.1 4
168.149 odd 6 3528.2.s.bk.361.2 4

By twisted newform
Twist Min Dim Char Parity Ord Type
168.2.q.c.25.2 4 56.5 odd 6
168.2.q.c.121.2 yes 4 56.45 odd 6
336.2.q.g.193.2 4 56.19 even 6
336.2.q.g.289.2 4 56.3 even 6
504.2.s.i.289.1 4 168.101 even 6
504.2.s.i.361.1 4 168.5 even 6
1008.2.s.r.289.1 4 168.59 odd 6
1008.2.s.r.865.1 4 168.131 odd 6
1176.2.a.k.1.1 2 56.13 odd 2
1176.2.a.n.1.2 2 8.5 even 2
1176.2.q.l.361.1 4 56.37 even 6
1176.2.q.l.961.1 4 56.53 even 6
1344.2.q.w.193.1 4 7.5 odd 6
1344.2.q.w.961.1 4 7.3 odd 6
1344.2.q.x.193.1 4 28.19 even 6
1344.2.q.x.961.1 4 28.3 even 6
2352.2.a.ba.1.2 2 8.3 odd 2
2352.2.a.bf.1.1 2 56.27 even 2
2352.2.q.bf.961.1 4 56.11 odd 6
2352.2.q.bf.1537.1 4 56.51 odd 6
3528.2.a.bd.1.1 2 24.5 odd 2
3528.2.a.bk.1.2 2 168.125 even 2
3528.2.s.bk.361.2 4 168.149 odd 6
3528.2.s.bk.3313.2 4 168.53 odd 6
7056.2.a.ch.1.1 2 24.11 even 2
7056.2.a.cu.1.2 2 168.83 odd 2
9408.2.a.dj.1.1 2 1.1 even 1 trivial
9408.2.a.dp.1.2 2 28.27 even 2
9408.2.a.dw.1.1 2 4.3 odd 2
9408.2.a.ec.1.2 2 7.6 odd 2