# Properties

 Label 7056.2.a.cu.1.2 Level 7056 Weight 2 Character 7056.1 Self dual yes Analytic conductor 56.342 Analytic rank 0 Dimension 2 CM no Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$7056 = 2^{4} \cdot 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 7056.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$56.3424436662$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{57})$$ Defining polynomial: $$x^{2} - x - 14$$ 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 1.2 Root $$4.27492$$ of defining polynomial Character $$\chi$$ $$=$$ 7056.1

## $q$-expansion

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

## 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
$$3$$ 0 0
$$4$$ 0 0
$$5$$ 4.27492 1.91180 0.955901 0.293691i $$-0.0948835\pi$$
0.955901 + 0.293691i $$0.0948835\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ −4.27492 −1.28894 −0.644468 0.764631i $$-0.722921\pi$$
−0.644468 + 0.764631i $$0.722921\pi$$
$$12$$ 0 0
$$13$$ −1.27492 −0.353598 −0.176799 0.984247i $$-0.556574\pi$$
−0.176799 + 0.984247i $$0.556574\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 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 $$-0.546718\pi$$
−0.146243 + 0.989249i $$0.546718\pi$$
$$20$$ 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
$$27$$ 0 0
$$28$$ 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
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 5.27492 0.867191 0.433596 0.901108i $$-0.357245\pi$$
0.433596 + 0.901108i $$0.357245\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 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
$$45$$ 0 0
$$46$$ 0 0
$$47$$ −6.00000 −0.875190 −0.437595 0.899172i $$-0.644170\pi$$
−0.437595 + 0.899172i $$0.644170\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 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
$$57$$ 0 0
$$58$$ 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
$$69$$ 0 0
$$70$$ 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 $$-0.438616\pi$$
0.191650 + 0.981463i $$0.438616\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 3.54983 0.399388 0.199694 0.979858i $$-0.436005\pi$$
0.199694 + 0.979858i $$0.436005\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 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
$$87$$ 0 0
$$88$$ 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
$$93$$ 0 0
$$94$$ 0 0
$$95$$ −5.45017 −0.559175
$$96$$ 0 0
$$97$$ 16.2749 1.65247 0.826234 0.563327i $$-0.190479\pi$$
0.826234 + 0.563327i $$0.190479\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 6.00000 0.597022 0.298511 0.954406i $$-0.403510\pi$$
0.298511 + 0.954406i $$0.403510\pi$$
$$102$$ 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 $$-0.392989\pi$$
0.329887 + 0.944020i $$0.392989\pi$$
$$108$$ 0 0
$$109$$ −5.82475 −0.557910 −0.278955 0.960304i $$-0.589988\pi$$
−0.278955 + 0.960304i $$0.589988\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ −10.5498 −0.992445 −0.496222 0.868195i $$-0.665280\pi$$
−0.496222 + 0.868195i $$0.665280\pi$$
$$114$$ 0 0
$$115$$ 17.0997 1.59455
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 7.27492 0.661356
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 35.3746 3.16400
$$126$$ 0 0
$$127$$ 21.5498 1.91224 0.956119 0.292978i $$-0.0946462\pi$$
0.956119 + 0.292978i $$0.0946462\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 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
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −16.5498 −1.41395 −0.706974 0.707240i $$-0.749940\pi$$
−0.706974 + 0.707240i $$0.749940\pi$$
$$138$$ 0 0
$$139$$ −0.725083 −0.0615007 −0.0307504 0.999527i $$-0.509790\pi$$
−0.0307504 + 0.999527i $$0.509790\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 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 $$-0.284861\pi$$
0.625583 + 0.780158i $$0.284861\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 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
$$159$$ 0 0
$$160$$ 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
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −6.00000 −0.464294 −0.232147 0.972681i $$-0.574575\pi$$
−0.232147 + 0.972681i $$0.574575\pi$$
$$168$$ 0 0
$$169$$ −11.3746 −0.874968
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ 7.45017 0.566426 0.283213 0.959057i $$-0.408600\pi$$
0.283213 + 0.959057i $$0.408600\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ −1.45017 −0.108390 −0.0541952 0.998530i $$-0.517259\pi$$
−0.0541952 + 0.998530i $$0.517259\pi$$
$$180$$ 0 0
$$181$$ 3.82475 0.284292 0.142146 0.989846i $$-0.454600\pi$$
0.142146 + 0.989846i $$0.454600\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 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
$$195$$ 0 0
$$196$$ 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
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ −45.0997 −3.14990
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 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
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 31.0997 2.12098
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 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
$$225$$ 0 0
$$226$$ 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 $$-0.658129\pi$$
−0.476596 + 0.879123i $$0.658129\pi$$
$$234$$ 0 0
$$235$$ −25.6495 −1.67319
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 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 $$-0.635539\pi$$
−0.413057 + 0.910705i $$0.635539\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 1.62541 0.103423
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 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
$$255$$ 0 0
$$256$$ 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
$$261$$ 0 0
$$262$$ 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
$$267$$ 0 0
$$268$$ 0 0
$$269$$ −28.2749 −1.72395 −0.861976 0.506949i $$-0.830773\pi$$
−0.861976 + 0.506949i $$0.830773\pi$$
$$270$$ 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 $$-0.540032\pi$$
−0.125433 + 0.992102i $$0.540032\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 11.4502 0.683060 0.341530 0.939871i $$-0.389055\pi$$
0.341530 + 0.939871i $$0.389055\pi$$
$$282$$ 0 0
$$283$$ −26.9244 −1.60049 −0.800245 0.599673i $$-0.795297\pi$$
−0.800245 + 0.599673i $$0.795297\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ −1.00000 −0.0588235
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ 5.17525 0.302341 0.151171 0.988508i $$-0.451696\pi$$
0.151171 + 0.988508i $$0.451696\pi$$
$$294$$ 0 0
$$295$$ 26.8248 1.56180
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ −5.09967 −0.294921
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 42.7492 2.44781
$$306$$ 0 0
$$307$$ 26.3746 1.50528 0.752639 0.658434i $$-0.228781\pi$$
0.752639 + 0.658434i $$0.228781\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 10.5498 0.598226 0.299113 0.954218i $$-0.403309\pi$$
0.299113 + 0.954218i $$0.403309\pi$$
$$312$$ 0 0
$$313$$ −4.45017 −0.251538 −0.125769 0.992060i $$-0.540140\pi$$
−0.125769 + 0.992060i $$0.540140\pi$$
$$314$$ 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
$$321$$ 0 0
$$322$$ 0 0
$$323$$ −5.09967 −0.283753
$$324$$ 0 0
$$325$$ −16.9244 −0.938798
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 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
$$333$$ 0 0
$$334$$ 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
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −4.27492 −0.231500
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −30.1993 −1.62119 −0.810593 0.585610i $$-0.800855\pi$$
−0.810593 + 0.585610i $$0.800855\pi$$
$$348$$ 0 0
$$349$$ −6.00000 −0.321173 −0.160586 0.987022i $$-0.551338\pi$$
−0.160586 + 0.987022i $$0.551338\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 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
$$363$$ 0 0
$$364$$ 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
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 1.27492 0.0660127 0.0330064 0.999455i $$-0.489492\pi$$
0.0330064 + 0.999455i $$0.489492\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 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
$$381$$ 0 0
$$382$$ 0 0
$$383$$ −4.54983 −0.232486 −0.116243 0.993221i $$-0.537085\pi$$
−0.116243 + 0.993221i $$0.537085\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 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 0
$$394$$ 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 $$-0.704535\pi$$
−0.599251 + 0.800561i $$0.704535\pi$$
$$402$$ 0 0
$$403$$ −1.27492 −0.0635082
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −22.5498 −1.11775
$$408$$ 0 0
$$409$$ 25.5498 1.26336 0.631679 0.775230i $$-0.282366\pi$$
0.631679 + 0.775230i $$0.282366\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ −1.17525 −0.0576907
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 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 $$-0.609375\pi$$
−0.336889 + 0.941545i $$0.609375\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 53.0997 2.57571
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 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 $$-0.286918\pi$$
0.620529 + 0.784183i $$0.286918\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 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 $$-0.232294\pi$$
0.745326 + 0.666700i $$0.232294\pi$$
$$444$$ 0 0
$$445$$ −19.4502 −0.922026
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ −5.45017 −0.257209 −0.128605 0.991696i $$-0.541050\pi$$
−0.128605 + 0.991696i $$0.541050\pi$$
$$450$$ 0 0
$$451$$ 45.0997 2.12366
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 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
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −41.6495 −1.93981 −0.969905 0.243482i $$-0.921710\pi$$
−0.969905 + 0.243482i $$0.921710\pi$$
$$462$$ 0 0
$$463$$ −35.8248 −1.66492 −0.832459 0.554087i $$-0.813067\pi$$
−0.832459 + 0.554087i $$0.813067\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 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
$$471$$ 0 0
$$472$$ 0 0
$$473$$ −31.0997 −1.42996
$$474$$ 0 0
$$475$$ −16.9244 −0.776546
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ −20.5498 −0.938946 −0.469473 0.882947i $$-0.655556\pi$$
−0.469473 + 0.882947i $$0.655556\pi$$
$$480$$ 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 $$-0.507213\pi$$
−0.0226572 + 0.999743i $$0.507213\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −15.9244 −0.718659 −0.359330 0.933211i $$-0.616995\pi$$
−0.359330 + 0.933211i $$0.616995\pi$$
$$492$$ 0 0
$$493$$ 9.09967 0.409828
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 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
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 7.64950 0.341074 0.170537 0.985351i $$-0.445450\pi$$
0.170537 + 0.985351i $$0.445450\pi$$
$$504$$ 0 0
$$505$$ 25.6495 1.14139
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ 3.72508 0.165111 0.0825557 0.996586i $$-0.473692\pi$$
0.0825557 + 0.996586i $$0.473692\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 50.5498 2.22749
$$516$$ 0 0
$$517$$ 25.6495 1.12806
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 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 $$-0.313637\pi$$
0.552597 + 0.833448i $$0.313637\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 4.00000 0.174243
$$528$$ 0 0
$$529$$ −7.00000 −0.304348
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 13.4502 0.582591
$$534$$ 0 0
$$535$$ 29.1752 1.26136
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −0.725083 −0.0311737 −0.0155869 0.999879i $$-0.504962\pi$$
−0.0155869 + 0.999879i $$0.504962\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 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
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −2.90033 −0.123558
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 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
$$561$$ 0 0
$$562$$ 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 $$-0.687861\pi$$
−0.556513 + 0.830839i $$0.687861\pi$$
$$570$$ 0 0
$$571$$ −0.725083 −0.0303438 −0.0151719 0.999885i $$-0.504830\pi$$
−0.0151719 + 0.999885i $$0.504830\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 53.0997 2.21441
$$576$$ 0 0
$$577$$ −25.0000 −1.04076 −0.520382 0.853934i $$-0.674210\pi$$
−0.520382 + 0.853934i $$0.674210\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 7.37459 0.305424
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 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
$$591$$ 0 0
$$592$$ 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
$$597$$ 0 0
$$598$$ 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 $$-0.678677\pi$$
−0.532314 + 0.846547i $$0.678677\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 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 $$-0.457773\pi$$
0.132273 + 0.991213i $$0.457773\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 6.00000 0.241551 0.120775 0.992680i $$-0.461462\pi$$
0.120775 + 0.992680i $$0.461462\pi$$
$$618$$ 0 0
$$619$$ −6.17525 −0.248204 −0.124102 0.992269i $$-0.539605\pi$$
−0.124102 + 0.992269i $$0.539605\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 84.8488 3.39395
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 21.0997 0.841299
$$630$$ 0 0
$$631$$ 2.82475 0.112452 0.0562258 0.998418i $$-0.482093\pi$$
0.0562258 + 0.998418i $$0.482093\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ 92.1238 3.65582
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −41.6495 −1.64506 −0.822528 0.568724i $$-0.807437\pi$$
−0.822528 + 0.568724i $$0.807437\pi$$
$$642$$ 0 0
$$643$$ 32.3746 1.27673 0.638365 0.769734i $$-0.279611\pi$$
0.638365 + 0.769734i $$0.279611\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −34.0000 −1.33668 −0.668339 0.743857i $$-0.732994\pi$$
−0.668339 + 0.743857i $$0.732994\pi$$
$$648$$ 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
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 18.1993 0.708946 0.354473 0.935066i $$-0.384660\pi$$
0.354473 + 0.935066i $$0.384660\pi$$
$$660$$ 0 0
$$661$$ 29.8248 1.16005 0.580024 0.814599i $$-0.303043\pi$$
0.580024 + 0.814599i $$0.303043\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 9.09967 0.352341
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 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
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 5.72508 0.220033 0.110016 0.993930i $$-0.464910\pi$$
0.110016 + 0.993930i $$0.464910\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 34.8248 1.33253 0.666266 0.745714i $$-0.267892\pi$$
0.666266 + 0.745714i $$0.267892\pi$$
$$684$$ 0 0
$$685$$ −70.7492 −2.70319
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ 2.19934 0.0837881
$$690$$ 0 0
$$691$$ −11.8248 −0.449835 −0.224917 0.974378i $$-0.572211\pi$$
−0.224917 + 0.974378i $$0.572211\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ −3.09967 −0.117577
$$696$$ 0 0
$$697$$ −42.1993 −1.59841
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 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
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 32.1993 1.20927 0.604636 0.796502i $$-0.293319\pi$$
0.604636 + 0.796502i $$0.293319\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 4.00000 0.149801
$$714$$ 0 0
$$715$$ 23.2990 0.871333
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ −32.1993 −1.20083 −0.600416 0.799688i $$-0.704998\pi$$
−0.600416 + 0.799688i $$0.704998\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 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
$$729$$ 0 0
$$730$$ 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
$$741$$ 0 0
$$742$$ 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 0
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 25.5498 0.932327 0.466163 0.884699i $$-0.345636\pi$$
0.466163 + 0.884699i $$0.345636\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 65.7251 2.39198
$$756$$ 0 0
$$757$$ −22.0000 −0.799604 −0.399802 0.916602i $$-0.630921\pi$$
−0.399802 + 0.916602i $$0.630921\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 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
$$765$$ 0 0
$$766$$ 0 0
$$767$$ −8.00000 −0.288863
$$768$$ 0 0
$$769$$ 12.6495 0.456153 0.228076 0.973643i $$-0.426756\pi$$
0.228076 + 0.973643i $$0.426756\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ 27.0997 0.974707 0.487354 0.873205i $$-0.337962\pi$$
0.487354 + 0.873205i $$0.337962\pi$$
$$774$$ 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
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 62.1993 2.21999
$$786$$ 0 0
$$787$$ 12.5498 0.447353 0.223677 0.974663i $$-0.428194\pi$$
0.223677 + 0.974663i $$0.428194\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ −12.7492 −0.452736
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ −36.4743 −1.29198 −0.645992 0.763344i $$-0.723556\pi$$
−0.645992 + 0.763344i $$0.723556\pi$$
$$798$$ 0 0
$$799$$ −24.0000 −0.849059
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ −14.0000 −0.494049
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ −15.6495 −0.550207 −0.275104 0.961415i $$-0.588712\pi$$
−0.275104 + 0.961415i $$0.588712\pi$$
$$810$$ 0 0
$$811$$ 27.4502 0.963906 0.481953 0.876197i $$-0.339928\pi$$
0.481953 + 0.876197i $$0.339928\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 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 $$-0.650942\pi$$
−0.456626 + 0.889659i $$0.650942\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 39.0241 1.35700 0.678500 0.734600i $$-0.262630\pi$$
0.678500 + 0.734600i $$0.262630\pi$$
$$828$$ 0 0
$$829$$ 22.7251 0.789275 0.394637 0.918837i $$-0.370870\pi$$
0.394637 + 0.918837i $$0.370870\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 0 0
$$835$$ −25.6495 −0.887638
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ 30.1993 1.04260 0.521298 0.853374i $$-0.325448\pi$$
0.521298 + 0.853374i $$0.325448\pi$$
$$840$$ 0 0
$$841$$ −23.8248 −0.821543
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ −48.6254 −1.67277
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 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
$$855$$ 0 0
$$856$$ 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 $$-0.443497\pi$$
0.176577 + 0.984287i $$0.443497\pi$$
$$860$$ 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
$$867$$ 0 0
$$868$$ 0 0
$$869$$ −15.1752 −0.514785
$$870$$ 0 0
$$871$$ 9.27492 0.314269
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −28.7492 −0.970791 −0.485395 0.874295i $$-0.661324\pi$$
−0.485395 + 0.874295i $$0.661324\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 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
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −11.6495 −0.391152 −0.195576 0.980689i $$-0.562658\pi$$
−0.195576 + 0.980689i $$0.562658\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ 7.64950 0.255981
$$894$$ 0 0
$$895$$ −6.19934 −0.207221
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 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
$$909$$ 0 0
$$910$$ 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
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ −19.8248 −0.653958 −0.326979 0.945032i $$-0.606031\pi$$
−0.326979 + 0.945032i $$0.606031\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ −2.54983 −0.0839288
$$924$$ 0 0
$$925$$ 70.0241 2.30238
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 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
$$933$$ 0 0
$$934$$ 0 0
$$935$$ −73.0997 −2.39061
$$936$$ 0 0
$$937$$ 24.0997 0.787302 0.393651 0.919260i $$-0.371212\pi$$
0.393651 + 0.919260i $$0.371212\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −29.1752 −0.951086 −0.475543 0.879693i $$-0.657748\pi$$
−0.475543 + 0.879693i $$0.657748\pi$$
$$942$$ 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 $$-0.689721\pi$$
−0.561359 + 0.827572i $$0.689721\pi$$
$$948$$ 0 0
$$949$$ −4.17525 −0.135534
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ −10.3505 −0.335285 −0.167643 0.985848i $$-0.553616\pi$$
−0.167643 + 0.985848i $$0.553616\pi$$
$$954$$ 0 0
$$955$$ −45.0997 −1.45939
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −30.0000 −0.967742
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ 66.4743 2.13988
$$966$$ 0 0
$$967$$ 38.4502 1.23647 0.618237 0.785992i $$-0.287847\pi$$
0.618237 + 0.785992i $$0.287847\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 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
$$975$$ 0 0
$$976$$ 0 0
$$977$$ −39.0997 −1.25091 −0.625455 0.780261i $$-0.715086\pi$$
−0.625455 + 0.780261i $$0.715086\pi$$
$$978$$ 0 0
$$979$$ 19.4502 0.621630
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ −51.2990 −1.63618 −0.818092 0.575087i $$-0.804968\pi$$
−0.818092 + 0.575087i $$0.804968\pi$$
$$984$$ 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 $$-0.582310\pi$$
−0.255711 + 0.966753i $$0.582310\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 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
$$999$$ 0 0
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 7056.2.a.cu.1.2 2
3.2 odd 2 2352.2.a.bf.1.1 2
4.3 odd 2 3528.2.a.bk.1.2 2
7.2 even 3 1008.2.s.r.865.1 4
7.4 even 3 1008.2.s.r.289.1 4
7.6 odd 2 7056.2.a.ch.1.1 2
12.11 even 2 1176.2.a.k.1.1 2
21.2 odd 6 336.2.q.g.193.2 4
21.5 even 6 2352.2.q.bf.1537.1 4
21.11 odd 6 336.2.q.g.289.2 4
21.17 even 6 2352.2.q.bf.961.1 4
21.20 even 2 2352.2.a.ba.1.2 2
24.5 odd 2 9408.2.a.dp.1.2 2
24.11 even 2 9408.2.a.ec.1.2 2
28.3 even 6 3528.2.s.bk.3313.2 4
28.11 odd 6 504.2.s.i.289.1 4
28.19 even 6 3528.2.s.bk.361.2 4
28.23 odd 6 504.2.s.i.361.1 4
28.27 even 2 3528.2.a.bd.1.1 2
84.11 even 6 168.2.q.c.121.2 yes 4
84.23 even 6 168.2.q.c.25.2 4
84.47 odd 6 1176.2.q.l.361.1 4
84.59 odd 6 1176.2.q.l.961.1 4
84.83 odd 2 1176.2.a.n.1.2 2
168.11 even 6 1344.2.q.w.961.1 4
168.53 odd 6 1344.2.q.x.961.1 4
168.83 odd 2 9408.2.a.dj.1.1 2
168.107 even 6 1344.2.q.w.193.1 4
168.125 even 2 9408.2.a.dw.1.1 2
168.149 odd 6 1344.2.q.x.193.1 4

By twisted newform
Twist Min Dim Char Parity Ord Type
168.2.q.c.25.2 4 84.23 even 6
168.2.q.c.121.2 yes 4 84.11 even 6
336.2.q.g.193.2 4 21.2 odd 6
336.2.q.g.289.2 4 21.11 odd 6
504.2.s.i.289.1 4 28.11 odd 6
504.2.s.i.361.1 4 28.23 odd 6
1008.2.s.r.289.1 4 7.4 even 3
1008.2.s.r.865.1 4 7.2 even 3
1176.2.a.k.1.1 2 12.11 even 2
1176.2.a.n.1.2 2 84.83 odd 2
1176.2.q.l.361.1 4 84.47 odd 6
1176.2.q.l.961.1 4 84.59 odd 6
1344.2.q.w.193.1 4 168.107 even 6
1344.2.q.w.961.1 4 168.11 even 6
1344.2.q.x.193.1 4 168.149 odd 6
1344.2.q.x.961.1 4 168.53 odd 6
2352.2.a.ba.1.2 2 21.20 even 2
2352.2.a.bf.1.1 2 3.2 odd 2
2352.2.q.bf.961.1 4 21.17 even 6
2352.2.q.bf.1537.1 4 21.5 even 6
3528.2.a.bd.1.1 2 28.27 even 2
3528.2.a.bk.1.2 2 4.3 odd 2
3528.2.s.bk.361.2 4 28.19 even 6
3528.2.s.bk.3313.2 4 28.3 even 6
7056.2.a.ch.1.1 2 7.6 odd 2
7056.2.a.cu.1.2 2 1.1 even 1 trivial
9408.2.a.dj.1.1 2 168.83 odd 2
9408.2.a.dp.1.2 2 24.5 odd 2
9408.2.a.dw.1.1 2 168.125 even 2
9408.2.a.ec.1.2 2 24.11 even 2