# Properties

 Label 1344.2.a.q.1.1 Level $1344$ Weight $2$ Character 1344.1 Self dual yes Analytic conductor $10.732$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1344 = 2^{6} \cdot 3 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1344.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$10.7318940317$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 42) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 1344.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000 q^{3} +2.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{3} +2.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} +4.00000 q^{11} -6.00000 q^{13} +2.00000 q^{15} +2.00000 q^{17} +4.00000 q^{19} -1.00000 q^{21} +8.00000 q^{23} -1.00000 q^{25} +1.00000 q^{27} +2.00000 q^{29} +4.00000 q^{33} -2.00000 q^{35} +10.0000 q^{37} -6.00000 q^{39} -6.00000 q^{41} +4.00000 q^{43} +2.00000 q^{45} +1.00000 q^{49} +2.00000 q^{51} -6.00000 q^{53} +8.00000 q^{55} +4.00000 q^{57} -4.00000 q^{59} -6.00000 q^{61} -1.00000 q^{63} -12.0000 q^{65} -4.00000 q^{67} +8.00000 q^{69} +8.00000 q^{71} +10.0000 q^{73} -1.00000 q^{75} -4.00000 q^{77} +1.00000 q^{81} +4.00000 q^{83} +4.00000 q^{85} +2.00000 q^{87} -6.00000 q^{89} +6.00000 q^{91} +8.00000 q^{95} -14.0000 q^{97} +4.00000 q^{99} +O(q^{100})$$

## 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
$$3$$ 1.00000 0.577350
$$4$$ 0 0
$$5$$ 2.00000 0.894427 0.447214 0.894427i $$-0.352416\pi$$
0.447214 + 0.894427i $$0.352416\pi$$
$$6$$ 0 0
$$7$$ −1.00000 −0.377964
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ 4.00000 1.20605 0.603023 0.797724i $$-0.293963\pi$$
0.603023 + 0.797724i $$0.293963\pi$$
$$12$$ 0 0
$$13$$ −6.00000 −1.66410 −0.832050 0.554700i $$-0.812833\pi$$
−0.832050 + 0.554700i $$0.812833\pi$$
$$14$$ 0 0
$$15$$ 2.00000 0.516398
$$16$$ 0 0
$$17$$ 2.00000 0.485071 0.242536 0.970143i $$-0.422021\pi$$
0.242536 + 0.970143i $$0.422021\pi$$
$$18$$ 0 0
$$19$$ 4.00000 0.917663 0.458831 0.888523i $$-0.348268\pi$$
0.458831 + 0.888523i $$0.348268\pi$$
$$20$$ 0 0
$$21$$ −1.00000 −0.218218
$$22$$ 0 0
$$23$$ 8.00000 1.66812 0.834058 0.551677i $$-0.186012\pi$$
0.834058 + 0.551677i $$0.186012\pi$$
$$24$$ 0 0
$$25$$ −1.00000 −0.200000
$$26$$ 0 0
$$27$$ 1.00000 0.192450
$$28$$ 0 0
$$29$$ 2.00000 0.371391 0.185695 0.982607i $$-0.440546\pi$$
0.185695 + 0.982607i $$0.440546\pi$$
$$30$$ 0 0
$$31$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$32$$ 0 0
$$33$$ 4.00000 0.696311
$$34$$ 0 0
$$35$$ −2.00000 −0.338062
$$36$$ 0 0
$$37$$ 10.0000 1.64399 0.821995 0.569495i $$-0.192861\pi$$
0.821995 + 0.569495i $$0.192861\pi$$
$$38$$ 0 0
$$39$$ −6.00000 −0.960769
$$40$$ 0 0
$$41$$ −6.00000 −0.937043 −0.468521 0.883452i $$-0.655213\pi$$
−0.468521 + 0.883452i $$0.655213\pi$$
$$42$$ 0 0
$$43$$ 4.00000 0.609994 0.304997 0.952353i $$-0.401344\pi$$
0.304997 + 0.952353i $$0.401344\pi$$
$$44$$ 0 0
$$45$$ 2.00000 0.298142
$$46$$ 0 0
$$47$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$48$$ 0 0
$$49$$ 1.00000 0.142857
$$50$$ 0 0
$$51$$ 2.00000 0.280056
$$52$$ 0 0
$$53$$ −6.00000 −0.824163 −0.412082 0.911147i $$-0.635198\pi$$
−0.412082 + 0.911147i $$0.635198\pi$$
$$54$$ 0 0
$$55$$ 8.00000 1.07872
$$56$$ 0 0
$$57$$ 4.00000 0.529813
$$58$$ 0 0
$$59$$ −4.00000 −0.520756 −0.260378 0.965507i $$-0.583847\pi$$
−0.260378 + 0.965507i $$0.583847\pi$$
$$60$$ 0 0
$$61$$ −6.00000 −0.768221 −0.384111 0.923287i $$-0.625492\pi$$
−0.384111 + 0.923287i $$0.625492\pi$$
$$62$$ 0 0
$$63$$ −1.00000 −0.125988
$$64$$ 0 0
$$65$$ −12.0000 −1.48842
$$66$$ 0 0
$$67$$ −4.00000 −0.488678 −0.244339 0.969690i $$-0.578571\pi$$
−0.244339 + 0.969690i $$0.578571\pi$$
$$68$$ 0 0
$$69$$ 8.00000 0.963087
$$70$$ 0 0
$$71$$ 8.00000 0.949425 0.474713 0.880141i $$-0.342552\pi$$
0.474713 + 0.880141i $$0.342552\pi$$
$$72$$ 0 0
$$73$$ 10.0000 1.17041 0.585206 0.810885i $$-0.301014\pi$$
0.585206 + 0.810885i $$0.301014\pi$$
$$74$$ 0 0
$$75$$ −1.00000 −0.115470
$$76$$ 0 0
$$77$$ −4.00000 −0.455842
$$78$$ 0 0
$$79$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ 4.00000 0.439057 0.219529 0.975606i $$-0.429548\pi$$
0.219529 + 0.975606i $$0.429548\pi$$
$$84$$ 0 0
$$85$$ 4.00000 0.433861
$$86$$ 0 0
$$87$$ 2.00000 0.214423
$$88$$ 0 0
$$89$$ −6.00000 −0.635999 −0.317999 0.948091i $$-0.603011\pi$$
−0.317999 + 0.948091i $$0.603011\pi$$
$$90$$ 0 0
$$91$$ 6.00000 0.628971
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 8.00000 0.820783
$$96$$ 0 0
$$97$$ −14.0000 −1.42148 −0.710742 0.703452i $$-0.751641\pi$$
−0.710742 + 0.703452i $$0.751641\pi$$
$$98$$ 0 0
$$99$$ 4.00000 0.402015
$$100$$ 0 0
$$101$$ 2.00000 0.199007 0.0995037 0.995037i $$-0.468274\pi$$
0.0995037 + 0.995037i $$0.468274\pi$$
$$102$$ 0 0
$$103$$ 8.00000 0.788263 0.394132 0.919054i $$-0.371045\pi$$
0.394132 + 0.919054i $$0.371045\pi$$
$$104$$ 0 0
$$105$$ −2.00000 −0.195180
$$106$$ 0 0
$$107$$ −12.0000 −1.16008 −0.580042 0.814587i $$-0.696964\pi$$
−0.580042 + 0.814587i $$0.696964\pi$$
$$108$$ 0 0
$$109$$ 2.00000 0.191565 0.0957826 0.995402i $$-0.469465\pi$$
0.0957826 + 0.995402i $$0.469465\pi$$
$$110$$ 0 0
$$111$$ 10.0000 0.949158
$$112$$ 0 0
$$113$$ −14.0000 −1.31701 −0.658505 0.752577i $$-0.728811\pi$$
−0.658505 + 0.752577i $$0.728811\pi$$
$$114$$ 0 0
$$115$$ 16.0000 1.49201
$$116$$ 0 0
$$117$$ −6.00000 −0.554700
$$118$$ 0 0
$$119$$ −2.00000 −0.183340
$$120$$ 0 0
$$121$$ 5.00000 0.454545
$$122$$ 0 0
$$123$$ −6.00000 −0.541002
$$124$$ 0 0
$$125$$ −12.0000 −1.07331
$$126$$ 0 0
$$127$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$128$$ 0 0
$$129$$ 4.00000 0.352180
$$130$$ 0 0
$$131$$ 20.0000 1.74741 0.873704 0.486458i $$-0.161711\pi$$
0.873704 + 0.486458i $$0.161711\pi$$
$$132$$ 0 0
$$133$$ −4.00000 −0.346844
$$134$$ 0 0
$$135$$ 2.00000 0.172133
$$136$$ 0 0
$$137$$ 10.0000 0.854358 0.427179 0.904167i $$-0.359507\pi$$
0.427179 + 0.904167i $$0.359507\pi$$
$$138$$ 0 0
$$139$$ −4.00000 −0.339276 −0.169638 0.985506i $$-0.554260\pi$$
−0.169638 + 0.985506i $$0.554260\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ −24.0000 −2.00698
$$144$$ 0 0
$$145$$ 4.00000 0.332182
$$146$$ 0 0
$$147$$ 1.00000 0.0824786
$$148$$ 0 0
$$149$$ −6.00000 −0.491539 −0.245770 0.969328i $$-0.579041\pi$$
−0.245770 + 0.969328i $$0.579041\pi$$
$$150$$ 0 0
$$151$$ −8.00000 −0.651031 −0.325515 0.945537i $$-0.605538\pi$$
−0.325515 + 0.945537i $$0.605538\pi$$
$$152$$ 0 0
$$153$$ 2.00000 0.161690
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 10.0000 0.798087 0.399043 0.916932i $$-0.369342\pi$$
0.399043 + 0.916932i $$0.369342\pi$$
$$158$$ 0 0
$$159$$ −6.00000 −0.475831
$$160$$ 0 0
$$161$$ −8.00000 −0.630488
$$162$$ 0 0
$$163$$ −20.0000 −1.56652 −0.783260 0.621694i $$-0.786445\pi$$
−0.783260 + 0.621694i $$0.786445\pi$$
$$164$$ 0 0
$$165$$ 8.00000 0.622799
$$166$$ 0 0
$$167$$ −8.00000 −0.619059 −0.309529 0.950890i $$-0.600171\pi$$
−0.309529 + 0.950890i $$0.600171\pi$$
$$168$$ 0 0
$$169$$ 23.0000 1.76923
$$170$$ 0 0
$$171$$ 4.00000 0.305888
$$172$$ 0 0
$$173$$ −22.0000 −1.67263 −0.836315 0.548250i $$-0.815294\pi$$
−0.836315 + 0.548250i $$0.815294\pi$$
$$174$$ 0 0
$$175$$ 1.00000 0.0755929
$$176$$ 0 0
$$177$$ −4.00000 −0.300658
$$178$$ 0 0
$$179$$ 12.0000 0.896922 0.448461 0.893802i $$-0.351972\pi$$
0.448461 + 0.893802i $$0.351972\pi$$
$$180$$ 0 0
$$181$$ 18.0000 1.33793 0.668965 0.743294i $$-0.266738\pi$$
0.668965 + 0.743294i $$0.266738\pi$$
$$182$$ 0 0
$$183$$ −6.00000 −0.443533
$$184$$ 0 0
$$185$$ 20.0000 1.47043
$$186$$ 0 0
$$187$$ 8.00000 0.585018
$$188$$ 0 0
$$189$$ −1.00000 −0.0727393
$$190$$ 0 0
$$191$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$192$$ 0 0
$$193$$ 2.00000 0.143963 0.0719816 0.997406i $$-0.477068\pi$$
0.0719816 + 0.997406i $$0.477068\pi$$
$$194$$ 0 0
$$195$$ −12.0000 −0.859338
$$196$$ 0 0
$$197$$ 10.0000 0.712470 0.356235 0.934396i $$-0.384060\pi$$
0.356235 + 0.934396i $$0.384060\pi$$
$$198$$ 0 0
$$199$$ 8.00000 0.567105 0.283552 0.958957i $$-0.408487\pi$$
0.283552 + 0.958957i $$0.408487\pi$$
$$200$$ 0 0
$$201$$ −4.00000 −0.282138
$$202$$ 0 0
$$203$$ −2.00000 −0.140372
$$204$$ 0 0
$$205$$ −12.0000 −0.838116
$$206$$ 0 0
$$207$$ 8.00000 0.556038
$$208$$ 0 0
$$209$$ 16.0000 1.10674
$$210$$ 0 0
$$211$$ −20.0000 −1.37686 −0.688428 0.725304i $$-0.741699\pi$$
−0.688428 + 0.725304i $$0.741699\pi$$
$$212$$ 0 0
$$213$$ 8.00000 0.548151
$$214$$ 0 0
$$215$$ 8.00000 0.545595
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ 10.0000 0.675737
$$220$$ 0 0
$$221$$ −12.0000 −0.807207
$$222$$ 0 0
$$223$$ −16.0000 −1.07144 −0.535720 0.844396i $$-0.679960\pi$$
−0.535720 + 0.844396i $$0.679960\pi$$
$$224$$ 0 0
$$225$$ −1.00000 −0.0666667
$$226$$ 0 0
$$227$$ −12.0000 −0.796468 −0.398234 0.917284i $$-0.630377\pi$$
−0.398234 + 0.917284i $$0.630377\pi$$
$$228$$ 0 0
$$229$$ 2.00000 0.132164 0.0660819 0.997814i $$-0.478950\pi$$
0.0660819 + 0.997814i $$0.478950\pi$$
$$230$$ 0 0
$$231$$ −4.00000 −0.263181
$$232$$ 0 0
$$233$$ −22.0000 −1.44127 −0.720634 0.693316i $$-0.756149\pi$$
−0.720634 + 0.693316i $$0.756149\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$240$$ 0 0
$$241$$ 2.00000 0.128831 0.0644157 0.997923i $$-0.479482\pi$$
0.0644157 + 0.997923i $$0.479482\pi$$
$$242$$ 0 0
$$243$$ 1.00000 0.0641500
$$244$$ 0 0
$$245$$ 2.00000 0.127775
$$246$$ 0 0
$$247$$ −24.0000 −1.52708
$$248$$ 0 0
$$249$$ 4.00000 0.253490
$$250$$ 0 0
$$251$$ 12.0000 0.757433 0.378717 0.925513i $$-0.376365\pi$$
0.378717 + 0.925513i $$0.376365\pi$$
$$252$$ 0 0
$$253$$ 32.0000 2.01182
$$254$$ 0 0
$$255$$ 4.00000 0.250490
$$256$$ 0 0
$$257$$ −30.0000 −1.87135 −0.935674 0.352865i $$-0.885208\pi$$
−0.935674 + 0.352865i $$0.885208\pi$$
$$258$$ 0 0
$$259$$ −10.0000 −0.621370
$$260$$ 0 0
$$261$$ 2.00000 0.123797
$$262$$ 0 0
$$263$$ −24.0000 −1.47990 −0.739952 0.672660i $$-0.765152\pi$$
−0.739952 + 0.672660i $$0.765152\pi$$
$$264$$ 0 0
$$265$$ −12.0000 −0.737154
$$266$$ 0 0
$$267$$ −6.00000 −0.367194
$$268$$ 0 0
$$269$$ −22.0000 −1.34136 −0.670682 0.741745i $$-0.733998\pi$$
−0.670682 + 0.741745i $$0.733998\pi$$
$$270$$ 0 0
$$271$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$272$$ 0 0
$$273$$ 6.00000 0.363137
$$274$$ 0 0
$$275$$ −4.00000 −0.241209
$$276$$ 0 0
$$277$$ 10.0000 0.600842 0.300421 0.953807i $$-0.402873\pi$$
0.300421 + 0.953807i $$0.402873\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 26.0000 1.55103 0.775515 0.631329i $$-0.217490\pi$$
0.775515 + 0.631329i $$0.217490\pi$$
$$282$$ 0 0
$$283$$ −4.00000 −0.237775 −0.118888 0.992908i $$-0.537933\pi$$
−0.118888 + 0.992908i $$0.537933\pi$$
$$284$$ 0 0
$$285$$ 8.00000 0.473879
$$286$$ 0 0
$$287$$ 6.00000 0.354169
$$288$$ 0 0
$$289$$ −13.0000 −0.764706
$$290$$ 0 0
$$291$$ −14.0000 −0.820695
$$292$$ 0 0
$$293$$ −30.0000 −1.75262 −0.876309 0.481749i $$-0.840002\pi$$
−0.876309 + 0.481749i $$0.840002\pi$$
$$294$$ 0 0
$$295$$ −8.00000 −0.465778
$$296$$ 0 0
$$297$$ 4.00000 0.232104
$$298$$ 0 0
$$299$$ −48.0000 −2.77591
$$300$$ 0 0
$$301$$ −4.00000 −0.230556
$$302$$ 0 0
$$303$$ 2.00000 0.114897
$$304$$ 0 0
$$305$$ −12.0000 −0.687118
$$306$$ 0 0
$$307$$ −28.0000 −1.59804 −0.799022 0.601302i $$-0.794649\pi$$
−0.799022 + 0.601302i $$0.794649\pi$$
$$308$$ 0 0
$$309$$ 8.00000 0.455104
$$310$$ 0 0
$$311$$ −8.00000 −0.453638 −0.226819 0.973937i $$-0.572833\pi$$
−0.226819 + 0.973937i $$0.572833\pi$$
$$312$$ 0 0
$$313$$ 10.0000 0.565233 0.282617 0.959233i $$-0.408798\pi$$
0.282617 + 0.959233i $$0.408798\pi$$
$$314$$ 0 0
$$315$$ −2.00000 −0.112687
$$316$$ 0 0
$$317$$ 18.0000 1.01098 0.505490 0.862832i $$-0.331312\pi$$
0.505490 + 0.862832i $$0.331312\pi$$
$$318$$ 0 0
$$319$$ 8.00000 0.447914
$$320$$ 0 0
$$321$$ −12.0000 −0.669775
$$322$$ 0 0
$$323$$ 8.00000 0.445132
$$324$$ 0 0
$$325$$ 6.00000 0.332820
$$326$$ 0 0
$$327$$ 2.00000 0.110600
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 4.00000 0.219860 0.109930 0.993939i $$-0.464937\pi$$
0.109930 + 0.993939i $$0.464937\pi$$
$$332$$ 0 0
$$333$$ 10.0000 0.547997
$$334$$ 0 0
$$335$$ −8.00000 −0.437087
$$336$$ 0 0
$$337$$ 18.0000 0.980522 0.490261 0.871576i $$-0.336901\pi$$
0.490261 + 0.871576i $$0.336901\pi$$
$$338$$ 0 0
$$339$$ −14.0000 −0.760376
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ −1.00000 −0.0539949
$$344$$ 0 0
$$345$$ 16.0000 0.861411
$$346$$ 0 0
$$347$$ −12.0000 −0.644194 −0.322097 0.946707i $$-0.604388\pi$$
−0.322097 + 0.946707i $$0.604388\pi$$
$$348$$ 0 0
$$349$$ −22.0000 −1.17763 −0.588817 0.808267i $$-0.700406\pi$$
−0.588817 + 0.808267i $$0.700406\pi$$
$$350$$ 0 0
$$351$$ −6.00000 −0.320256
$$352$$ 0 0
$$353$$ −30.0000 −1.59674 −0.798369 0.602168i $$-0.794304\pi$$
−0.798369 + 0.602168i $$0.794304\pi$$
$$354$$ 0 0
$$355$$ 16.0000 0.849192
$$356$$ 0 0
$$357$$ −2.00000 −0.105851
$$358$$ 0 0
$$359$$ −8.00000 −0.422224 −0.211112 0.977462i $$-0.567708\pi$$
−0.211112 + 0.977462i $$0.567708\pi$$
$$360$$ 0 0
$$361$$ −3.00000 −0.157895
$$362$$ 0 0
$$363$$ 5.00000 0.262432
$$364$$ 0 0
$$365$$ 20.0000 1.04685
$$366$$ 0 0
$$367$$ 32.0000 1.67039 0.835193 0.549957i $$-0.185356\pi$$
0.835193 + 0.549957i $$0.185356\pi$$
$$368$$ 0 0
$$369$$ −6.00000 −0.312348
$$370$$ 0 0
$$371$$ 6.00000 0.311504
$$372$$ 0 0
$$373$$ −22.0000 −1.13912 −0.569558 0.821951i $$-0.692886\pi$$
−0.569558 + 0.821951i $$0.692886\pi$$
$$374$$ 0 0
$$375$$ −12.0000 −0.619677
$$376$$ 0 0
$$377$$ −12.0000 −0.618031
$$378$$ 0 0
$$379$$ 20.0000 1.02733 0.513665 0.857991i $$-0.328287\pi$$
0.513665 + 0.857991i $$0.328287\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ −16.0000 −0.817562 −0.408781 0.912633i $$-0.634046\pi$$
−0.408781 + 0.912633i $$0.634046\pi$$
$$384$$ 0 0
$$385$$ −8.00000 −0.407718
$$386$$ 0 0
$$387$$ 4.00000 0.203331
$$388$$ 0 0
$$389$$ 26.0000 1.31825 0.659126 0.752032i $$-0.270926\pi$$
0.659126 + 0.752032i $$0.270926\pi$$
$$390$$ 0 0
$$391$$ 16.0000 0.809155
$$392$$ 0 0
$$393$$ 20.0000 1.00887
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ −6.00000 −0.301131 −0.150566 0.988600i $$-0.548110\pi$$
−0.150566 + 0.988600i $$0.548110\pi$$
$$398$$ 0 0
$$399$$ −4.00000 −0.200250
$$400$$ 0 0
$$401$$ 18.0000 0.898877 0.449439 0.893311i $$-0.351624\pi$$
0.449439 + 0.893311i $$0.351624\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 0 0
$$405$$ 2.00000 0.0993808
$$406$$ 0 0
$$407$$ 40.0000 1.98273
$$408$$ 0 0
$$409$$ −22.0000 −1.08783 −0.543915 0.839140i $$-0.683059\pi$$
−0.543915 + 0.839140i $$0.683059\pi$$
$$410$$ 0 0
$$411$$ 10.0000 0.493264
$$412$$ 0 0
$$413$$ 4.00000 0.196827
$$414$$ 0 0
$$415$$ 8.00000 0.392705
$$416$$ 0 0
$$417$$ −4.00000 −0.195881
$$418$$ 0 0
$$419$$ 36.0000 1.75872 0.879358 0.476162i $$-0.157972\pi$$
0.879358 + 0.476162i $$0.157972\pi$$
$$420$$ 0 0
$$421$$ −6.00000 −0.292422 −0.146211 0.989253i $$-0.546708\pi$$
−0.146211 + 0.989253i $$0.546708\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ −2.00000 −0.0970143
$$426$$ 0 0
$$427$$ 6.00000 0.290360
$$428$$ 0 0
$$429$$ −24.0000 −1.15873
$$430$$ 0 0
$$431$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$432$$ 0 0
$$433$$ 2.00000 0.0961139 0.0480569 0.998845i $$-0.484697\pi$$
0.0480569 + 0.998845i $$0.484697\pi$$
$$434$$ 0 0
$$435$$ 4.00000 0.191785
$$436$$ 0 0
$$437$$ 32.0000 1.53077
$$438$$ 0 0
$$439$$ −24.0000 −1.14546 −0.572729 0.819745i $$-0.694115\pi$$
−0.572729 + 0.819745i $$0.694115\pi$$
$$440$$ 0 0
$$441$$ 1.00000 0.0476190
$$442$$ 0 0
$$443$$ 4.00000 0.190046 0.0950229 0.995475i $$-0.469708\pi$$
0.0950229 + 0.995475i $$0.469708\pi$$
$$444$$ 0 0
$$445$$ −12.0000 −0.568855
$$446$$ 0 0
$$447$$ −6.00000 −0.283790
$$448$$ 0 0
$$449$$ 34.0000 1.60456 0.802280 0.596948i $$-0.203620\pi$$
0.802280 + 0.596948i $$0.203620\pi$$
$$450$$ 0 0
$$451$$ −24.0000 −1.13012
$$452$$ 0 0
$$453$$ −8.00000 −0.375873
$$454$$ 0 0
$$455$$ 12.0000 0.562569
$$456$$ 0 0
$$457$$ 10.0000 0.467780 0.233890 0.972263i $$-0.424854\pi$$
0.233890 + 0.972263i $$0.424854\pi$$
$$458$$ 0 0
$$459$$ 2.00000 0.0933520
$$460$$ 0 0
$$461$$ −22.0000 −1.02464 −0.512321 0.858794i $$-0.671214\pi$$
−0.512321 + 0.858794i $$0.671214\pi$$
$$462$$ 0 0
$$463$$ −32.0000 −1.48717 −0.743583 0.668644i $$-0.766875\pi$$
−0.743583 + 0.668644i $$0.766875\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −28.0000 −1.29569 −0.647843 0.761774i $$-0.724329\pi$$
−0.647843 + 0.761774i $$0.724329\pi$$
$$468$$ 0 0
$$469$$ 4.00000 0.184703
$$470$$ 0 0
$$471$$ 10.0000 0.460776
$$472$$ 0 0
$$473$$ 16.0000 0.735681
$$474$$ 0 0
$$475$$ −4.00000 −0.183533
$$476$$ 0 0
$$477$$ −6.00000 −0.274721
$$478$$ 0 0
$$479$$ −16.0000 −0.731059 −0.365529 0.930800i $$-0.619112\pi$$
−0.365529 + 0.930800i $$0.619112\pi$$
$$480$$ 0 0
$$481$$ −60.0000 −2.73576
$$482$$ 0 0
$$483$$ −8.00000 −0.364013
$$484$$ 0 0
$$485$$ −28.0000 −1.27141
$$486$$ 0 0
$$487$$ 8.00000 0.362515 0.181257 0.983436i $$-0.441983\pi$$
0.181257 + 0.983436i $$0.441983\pi$$
$$488$$ 0 0
$$489$$ −20.0000 −0.904431
$$490$$ 0 0
$$491$$ −12.0000 −0.541552 −0.270776 0.962642i $$-0.587280\pi$$
−0.270776 + 0.962642i $$0.587280\pi$$
$$492$$ 0 0
$$493$$ 4.00000 0.180151
$$494$$ 0 0
$$495$$ 8.00000 0.359573
$$496$$ 0 0
$$497$$ −8.00000 −0.358849
$$498$$ 0 0
$$499$$ 44.0000 1.96971 0.984855 0.173379i $$-0.0554684\pi$$
0.984855 + 0.173379i $$0.0554684\pi$$
$$500$$ 0 0
$$501$$ −8.00000 −0.357414
$$502$$ 0 0
$$503$$ 24.0000 1.07011 0.535054 0.844818i $$-0.320291\pi$$
0.535054 + 0.844818i $$0.320291\pi$$
$$504$$ 0 0
$$505$$ 4.00000 0.177998
$$506$$ 0 0
$$507$$ 23.0000 1.02147
$$508$$ 0 0
$$509$$ −6.00000 −0.265945 −0.132973 0.991120i $$-0.542452\pi$$
−0.132973 + 0.991120i $$0.542452\pi$$
$$510$$ 0 0
$$511$$ −10.0000 −0.442374
$$512$$ 0 0
$$513$$ 4.00000 0.176604
$$514$$ 0 0
$$515$$ 16.0000 0.705044
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 0 0
$$519$$ −22.0000 −0.965693
$$520$$ 0 0
$$521$$ −6.00000 −0.262865 −0.131432 0.991325i $$-0.541958\pi$$
−0.131432 + 0.991325i $$0.541958\pi$$
$$522$$ 0 0
$$523$$ −20.0000 −0.874539 −0.437269 0.899331i $$-0.644054\pi$$
−0.437269 + 0.899331i $$0.644054\pi$$
$$524$$ 0 0
$$525$$ 1.00000 0.0436436
$$526$$ 0 0
$$527$$ 0 0
$$528$$ 0 0
$$529$$ 41.0000 1.78261
$$530$$ 0 0
$$531$$ −4.00000 −0.173585
$$532$$ 0 0
$$533$$ 36.0000 1.55933
$$534$$ 0 0
$$535$$ −24.0000 −1.03761
$$536$$ 0 0
$$537$$ 12.0000 0.517838
$$538$$ 0 0
$$539$$ 4.00000 0.172292
$$540$$ 0 0
$$541$$ −30.0000 −1.28980 −0.644900 0.764267i $$-0.723101\pi$$
−0.644900 + 0.764267i $$0.723101\pi$$
$$542$$ 0 0
$$543$$ 18.0000 0.772454
$$544$$ 0 0
$$545$$ 4.00000 0.171341
$$546$$ 0 0
$$547$$ 12.0000 0.513083 0.256541 0.966533i $$-0.417417\pi$$
0.256541 + 0.966533i $$0.417417\pi$$
$$548$$ 0 0
$$549$$ −6.00000 −0.256074
$$550$$ 0 0
$$551$$ 8.00000 0.340811
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ 20.0000 0.848953
$$556$$ 0 0
$$557$$ 2.00000 0.0847427 0.0423714 0.999102i $$-0.486509\pi$$
0.0423714 + 0.999102i $$0.486509\pi$$
$$558$$ 0 0
$$559$$ −24.0000 −1.01509
$$560$$ 0 0
$$561$$ 8.00000 0.337760
$$562$$ 0 0
$$563$$ −44.0000 −1.85438 −0.927189 0.374593i $$-0.877783\pi$$
−0.927189 + 0.374593i $$0.877783\pi$$
$$564$$ 0 0
$$565$$ −28.0000 −1.17797
$$566$$ 0 0
$$567$$ −1.00000 −0.0419961
$$568$$ 0 0
$$569$$ −6.00000 −0.251533 −0.125767 0.992060i $$-0.540139\pi$$
−0.125767 + 0.992060i $$0.540139\pi$$
$$570$$ 0 0
$$571$$ −12.0000 −0.502184 −0.251092 0.967963i $$-0.580790\pi$$
−0.251092 + 0.967963i $$0.580790\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ −8.00000 −0.333623
$$576$$ 0 0
$$577$$ 34.0000 1.41544 0.707719 0.706494i $$-0.249724\pi$$
0.707719 + 0.706494i $$0.249724\pi$$
$$578$$ 0 0
$$579$$ 2.00000 0.0831172
$$580$$ 0 0
$$581$$ −4.00000 −0.165948
$$582$$ 0 0
$$583$$ −24.0000 −0.993978
$$584$$ 0 0
$$585$$ −12.0000 −0.496139
$$586$$ 0 0
$$587$$ 28.0000 1.15568 0.577842 0.816149i $$-0.303895\pi$$
0.577842 + 0.816149i $$0.303895\pi$$
$$588$$ 0 0
$$589$$ 0 0
$$590$$ 0 0
$$591$$ 10.0000 0.411345
$$592$$ 0 0
$$593$$ 18.0000 0.739171 0.369586 0.929197i $$-0.379500\pi$$
0.369586 + 0.929197i $$0.379500\pi$$
$$594$$ 0 0
$$595$$ −4.00000 −0.163984
$$596$$ 0 0
$$597$$ 8.00000 0.327418
$$598$$ 0 0
$$599$$ 24.0000 0.980613 0.490307 0.871550i $$-0.336885\pi$$
0.490307 + 0.871550i $$0.336885\pi$$
$$600$$ 0 0
$$601$$ 26.0000 1.06056 0.530281 0.847822i $$-0.322086\pi$$
0.530281 + 0.847822i $$0.322086\pi$$
$$602$$ 0 0
$$603$$ −4.00000 −0.162893
$$604$$ 0 0
$$605$$ 10.0000 0.406558
$$606$$ 0 0
$$607$$ 48.0000 1.94826 0.974130 0.225989i $$-0.0725612\pi$$
0.974130 + 0.225989i $$0.0725612\pi$$
$$608$$ 0 0
$$609$$ −2.00000 −0.0810441
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 0 0
$$613$$ 42.0000 1.69636 0.848182 0.529705i $$-0.177697\pi$$
0.848182 + 0.529705i $$0.177697\pi$$
$$614$$ 0 0
$$615$$ −12.0000 −0.483887
$$616$$ 0 0
$$617$$ −22.0000 −0.885687 −0.442843 0.896599i $$-0.646030\pi$$
−0.442843 + 0.896599i $$0.646030\pi$$
$$618$$ 0 0
$$619$$ 44.0000 1.76851 0.884255 0.467005i $$-0.154667\pi$$
0.884255 + 0.467005i $$0.154667\pi$$
$$620$$ 0 0
$$621$$ 8.00000 0.321029
$$622$$ 0 0
$$623$$ 6.00000 0.240385
$$624$$ 0 0
$$625$$ −19.0000 −0.760000
$$626$$ 0 0
$$627$$ 16.0000 0.638978
$$628$$ 0 0
$$629$$ 20.0000 0.797452
$$630$$ 0 0
$$631$$ 8.00000 0.318475 0.159237 0.987240i $$-0.449096\pi$$
0.159237 + 0.987240i $$0.449096\pi$$
$$632$$ 0 0
$$633$$ −20.0000 −0.794929
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ −6.00000 −0.237729
$$638$$ 0 0
$$639$$ 8.00000 0.316475
$$640$$ 0 0
$$641$$ 2.00000 0.0789953 0.0394976 0.999220i $$-0.487424\pi$$
0.0394976 + 0.999220i $$0.487424\pi$$
$$642$$ 0 0
$$643$$ 4.00000 0.157745 0.0788723 0.996885i $$-0.474868\pi$$
0.0788723 + 0.996885i $$0.474868\pi$$
$$644$$ 0 0
$$645$$ 8.00000 0.315000
$$646$$ 0 0
$$647$$ 24.0000 0.943537 0.471769 0.881722i $$-0.343616\pi$$
0.471769 + 0.881722i $$0.343616\pi$$
$$648$$ 0 0
$$649$$ −16.0000 −0.628055
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 18.0000 0.704394 0.352197 0.935926i $$-0.385435\pi$$
0.352197 + 0.935926i $$0.385435\pi$$
$$654$$ 0 0
$$655$$ 40.0000 1.56293
$$656$$ 0 0
$$657$$ 10.0000 0.390137
$$658$$ 0 0
$$659$$ 28.0000 1.09073 0.545363 0.838200i $$-0.316392\pi$$
0.545363 + 0.838200i $$0.316392\pi$$
$$660$$ 0 0
$$661$$ 2.00000 0.0777910 0.0388955 0.999243i $$-0.487616\pi$$
0.0388955 + 0.999243i $$0.487616\pi$$
$$662$$ 0 0
$$663$$ −12.0000 −0.466041
$$664$$ 0 0
$$665$$ −8.00000 −0.310227
$$666$$ 0 0
$$667$$ 16.0000 0.619522
$$668$$ 0 0
$$669$$ −16.0000 −0.618596
$$670$$ 0 0
$$671$$ −24.0000 −0.926510
$$672$$ 0 0
$$673$$ 2.00000 0.0770943 0.0385472 0.999257i $$-0.487727\pi$$
0.0385472 + 0.999257i $$0.487727\pi$$
$$674$$ 0 0
$$675$$ −1.00000 −0.0384900
$$676$$ 0 0
$$677$$ 18.0000 0.691796 0.345898 0.938272i $$-0.387574\pi$$
0.345898 + 0.938272i $$0.387574\pi$$
$$678$$ 0 0
$$679$$ 14.0000 0.537271
$$680$$ 0 0
$$681$$ −12.0000 −0.459841
$$682$$ 0 0
$$683$$ −12.0000 −0.459167 −0.229584 0.973289i $$-0.573736\pi$$
−0.229584 + 0.973289i $$0.573736\pi$$
$$684$$ 0 0
$$685$$ 20.0000 0.764161
$$686$$ 0 0
$$687$$ 2.00000 0.0763048
$$688$$ 0 0
$$689$$ 36.0000 1.37149
$$690$$ 0 0
$$691$$ 4.00000 0.152167 0.0760836 0.997101i $$-0.475758\pi$$
0.0760836 + 0.997101i $$0.475758\pi$$
$$692$$ 0 0
$$693$$ −4.00000 −0.151947
$$694$$ 0 0
$$695$$ −8.00000 −0.303457
$$696$$ 0 0
$$697$$ −12.0000 −0.454532
$$698$$ 0 0
$$699$$ −22.0000 −0.832116
$$700$$ 0 0
$$701$$ 2.00000 0.0755390 0.0377695 0.999286i $$-0.487975\pi$$
0.0377695 + 0.999286i $$0.487975\pi$$
$$702$$ 0 0
$$703$$ 40.0000 1.50863
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ −2.00000 −0.0752177
$$708$$ 0 0
$$709$$ 10.0000 0.375558 0.187779 0.982211i $$-0.439871\pi$$
0.187779 + 0.982211i $$0.439871\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 0 0
$$714$$ 0 0
$$715$$ −48.0000 −1.79510
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$720$$ 0 0
$$721$$ −8.00000 −0.297936
$$722$$ 0 0
$$723$$ 2.00000 0.0743808
$$724$$ 0 0
$$725$$ −2.00000 −0.0742781
$$726$$ 0 0
$$727$$ −8.00000 −0.296704 −0.148352 0.988935i $$-0.547397\pi$$
−0.148352 + 0.988935i $$0.547397\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ 8.00000 0.295891
$$732$$ 0 0
$$733$$ −6.00000 −0.221615 −0.110808 0.993842i $$-0.535344\pi$$
−0.110808 + 0.993842i $$0.535344\pi$$
$$734$$ 0 0
$$735$$ 2.00000 0.0737711
$$736$$ 0 0
$$737$$ −16.0000 −0.589368
$$738$$ 0 0
$$739$$ 12.0000 0.441427 0.220714 0.975339i $$-0.429161\pi$$
0.220714 + 0.975339i $$0.429161\pi$$
$$740$$ 0 0
$$741$$ −24.0000 −0.881662
$$742$$ 0 0
$$743$$ 24.0000 0.880475 0.440237 0.897881i $$-0.354894\pi$$
0.440237 + 0.897881i $$0.354894\pi$$
$$744$$ 0 0
$$745$$ −12.0000 −0.439646
$$746$$ 0 0
$$747$$ 4.00000 0.146352
$$748$$ 0 0
$$749$$ 12.0000 0.438470
$$750$$ 0 0
$$751$$ 48.0000 1.75154 0.875772 0.482724i $$-0.160353\pi$$
0.875772 + 0.482724i $$0.160353\pi$$
$$752$$ 0 0
$$753$$ 12.0000 0.437304
$$754$$ 0 0
$$755$$ −16.0000 −0.582300
$$756$$ 0 0
$$757$$ −6.00000 −0.218074 −0.109037 0.994038i $$-0.534777\pi$$
−0.109037 + 0.994038i $$0.534777\pi$$
$$758$$ 0 0
$$759$$ 32.0000 1.16153
$$760$$ 0 0
$$761$$ −22.0000 −0.797499 −0.398750 0.917060i $$-0.630556\pi$$
−0.398750 + 0.917060i $$0.630556\pi$$
$$762$$ 0 0
$$763$$ −2.00000 −0.0724049
$$764$$ 0 0
$$765$$ 4.00000 0.144620
$$766$$ 0 0
$$767$$ 24.0000 0.866590
$$768$$ 0 0
$$769$$ −14.0000 −0.504853 −0.252426 0.967616i $$-0.581229\pi$$
−0.252426 + 0.967616i $$0.581229\pi$$
$$770$$ 0 0
$$771$$ −30.0000 −1.08042
$$772$$ 0 0
$$773$$ 2.00000 0.0719350 0.0359675 0.999353i $$-0.488549\pi$$
0.0359675 + 0.999353i $$0.488549\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ −10.0000 −0.358748
$$778$$ 0 0
$$779$$ −24.0000 −0.859889
$$780$$ 0 0
$$781$$ 32.0000 1.14505
$$782$$ 0 0
$$783$$ 2.00000 0.0714742
$$784$$ 0 0
$$785$$ 20.0000 0.713831
$$786$$ 0 0
$$787$$ 36.0000 1.28326 0.641631 0.767014i $$-0.278258\pi$$
0.641631 + 0.767014i $$0.278258\pi$$
$$788$$ 0 0
$$789$$ −24.0000 −0.854423
$$790$$ 0 0
$$791$$ 14.0000 0.497783
$$792$$ 0 0
$$793$$ 36.0000 1.27840
$$794$$ 0 0
$$795$$ −12.0000 −0.425596
$$796$$ 0 0
$$797$$ −6.00000 −0.212531 −0.106265 0.994338i $$-0.533889\pi$$
−0.106265 + 0.994338i $$0.533889\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ 0 0
$$801$$ −6.00000 −0.212000
$$802$$ 0 0
$$803$$ 40.0000 1.41157
$$804$$ 0 0
$$805$$ −16.0000 −0.563926
$$806$$ 0 0
$$807$$ −22.0000 −0.774437
$$808$$ 0 0
$$809$$ 10.0000 0.351581 0.175791 0.984428i $$-0.443752\pi$$
0.175791 + 0.984428i $$0.443752\pi$$
$$810$$ 0 0
$$811$$ 44.0000 1.54505 0.772524 0.634985i $$-0.218994\pi$$
0.772524 + 0.634985i $$0.218994\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ −40.0000 −1.40114
$$816$$ 0 0
$$817$$ 16.0000 0.559769
$$818$$ 0 0
$$819$$ 6.00000 0.209657
$$820$$ 0 0
$$821$$ −38.0000 −1.32621 −0.663105 0.748527i $$-0.730762\pi$$
−0.663105 + 0.748527i $$0.730762\pi$$
$$822$$ 0 0
$$823$$ −56.0000 −1.95204 −0.976019 0.217687i $$-0.930149\pi$$
−0.976019 + 0.217687i $$0.930149\pi$$
$$824$$ 0 0
$$825$$ −4.00000 −0.139262
$$826$$ 0 0
$$827$$ 36.0000 1.25184 0.625921 0.779886i $$-0.284723\pi$$
0.625921 + 0.779886i $$0.284723\pi$$
$$828$$ 0 0
$$829$$ 26.0000 0.903017 0.451509 0.892267i $$-0.350886\pi$$
0.451509 + 0.892267i $$0.350886\pi$$
$$830$$ 0 0
$$831$$ 10.0000 0.346896
$$832$$ 0 0
$$833$$ 2.00000 0.0692959
$$834$$ 0 0
$$835$$ −16.0000 −0.553703
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ 56.0000 1.93333 0.966667 0.256036i $$-0.0824164\pi$$
0.966667 + 0.256036i $$0.0824164\pi$$
$$840$$ 0 0
$$841$$ −25.0000 −0.862069
$$842$$ 0 0
$$843$$ 26.0000 0.895488
$$844$$ 0 0
$$845$$ 46.0000 1.58245
$$846$$ 0 0
$$847$$ −5.00000 −0.171802
$$848$$ 0 0
$$849$$ −4.00000 −0.137280
$$850$$ 0 0
$$851$$ 80.0000 2.74236
$$852$$ 0 0
$$853$$ −14.0000 −0.479351 −0.239675 0.970853i $$-0.577041\pi$$
−0.239675 + 0.970853i $$0.577041\pi$$
$$854$$ 0 0
$$855$$ 8.00000 0.273594
$$856$$ 0 0
$$857$$ 42.0000 1.43469 0.717346 0.696717i $$-0.245357\pi$$
0.717346 + 0.696717i $$0.245357\pi$$
$$858$$ 0 0
$$859$$ −20.0000 −0.682391 −0.341196 0.939992i $$-0.610832\pi$$
−0.341196 + 0.939992i $$0.610832\pi$$
$$860$$ 0 0
$$861$$ 6.00000 0.204479
$$862$$ 0 0
$$863$$ −32.0000 −1.08929 −0.544646 0.838666i $$-0.683336\pi$$
−0.544646 + 0.838666i $$0.683336\pi$$
$$864$$ 0 0
$$865$$ −44.0000 −1.49604
$$866$$ 0 0
$$867$$ −13.0000 −0.441503
$$868$$ 0 0
$$869$$ 0 0
$$870$$ 0 0
$$871$$ 24.0000 0.813209
$$872$$ 0 0
$$873$$ −14.0000 −0.473828
$$874$$ 0 0
$$875$$ 12.0000 0.405674
$$876$$ 0 0
$$877$$ 2.00000 0.0675352 0.0337676 0.999430i $$-0.489249\pi$$
0.0337676 + 0.999430i $$0.489249\pi$$
$$878$$ 0 0
$$879$$ −30.0000 −1.01187
$$880$$ 0 0
$$881$$ 18.0000 0.606435 0.303218 0.952921i $$-0.401939\pi$$
0.303218 + 0.952921i $$0.401939\pi$$
$$882$$ 0 0
$$883$$ −20.0000 −0.673054 −0.336527 0.941674i $$-0.609252\pi$$
−0.336527 + 0.941674i $$0.609252\pi$$
$$884$$ 0 0
$$885$$ −8.00000 −0.268917
$$886$$ 0 0
$$887$$ −24.0000 −0.805841 −0.402921 0.915235i $$-0.632005\pi$$
−0.402921 + 0.915235i $$0.632005\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ 4.00000 0.134005
$$892$$ 0 0
$$893$$ 0 0
$$894$$ 0 0
$$895$$ 24.0000 0.802232
$$896$$ 0 0
$$897$$ −48.0000 −1.60267
$$898$$ 0 0
$$899$$ 0 0
$$900$$ 0 0
$$901$$ −12.0000 −0.399778
$$902$$ 0 0
$$903$$ −4.00000 −0.133112
$$904$$ 0 0
$$905$$ 36.0000 1.19668
$$906$$ 0 0
$$907$$ −12.0000 −0.398453 −0.199227 0.979953i $$-0.563843\pi$$
−0.199227 + 0.979953i $$0.563843\pi$$
$$908$$ 0 0
$$909$$ 2.00000 0.0663358
$$910$$ 0 0
$$911$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$912$$ 0 0
$$913$$ 16.0000 0.529523
$$914$$ 0 0
$$915$$ −12.0000 −0.396708
$$916$$ 0 0
$$917$$ −20.0000 −0.660458
$$918$$ 0 0
$$919$$ 40.0000 1.31948 0.659739 0.751495i $$-0.270667\pi$$
0.659739 + 0.751495i $$0.270667\pi$$
$$920$$ 0 0
$$921$$ −28.0000 −0.922631
$$922$$ 0 0
$$923$$ −48.0000 −1.57994
$$924$$ 0 0
$$925$$ −10.0000 −0.328798
$$926$$ 0 0
$$927$$ 8.00000 0.262754
$$928$$ 0 0
$$929$$ 18.0000 0.590561 0.295280 0.955411i $$-0.404587\pi$$
0.295280 + 0.955411i $$0.404587\pi$$
$$930$$ 0 0
$$931$$ 4.00000 0.131095
$$932$$ 0 0
$$933$$ −8.00000 −0.261908
$$934$$ 0 0
$$935$$ 16.0000 0.523256
$$936$$ 0 0
$$937$$ −22.0000 −0.718709 −0.359354 0.933201i $$-0.617003\pi$$
−0.359354 + 0.933201i $$0.617003\pi$$
$$938$$ 0 0
$$939$$ 10.0000 0.326338
$$940$$ 0 0
$$941$$ 26.0000 0.847576 0.423788 0.905761i $$-0.360700\pi$$
0.423788 + 0.905761i $$0.360700\pi$$
$$942$$ 0 0
$$943$$ −48.0000 −1.56310
$$944$$ 0 0
$$945$$ −2.00000 −0.0650600
$$946$$ 0 0
$$947$$ −4.00000 −0.129983 −0.0649913 0.997886i $$-0.520702\pi$$
−0.0649913 + 0.997886i $$0.520702\pi$$
$$948$$ 0 0
$$949$$ −60.0000 −1.94768
$$950$$ 0 0
$$951$$ 18.0000 0.583690
$$952$$ 0 0
$$953$$ 26.0000 0.842223 0.421111 0.907009i $$-0.361640\pi$$
0.421111 + 0.907009i $$0.361640\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 8.00000 0.258603
$$958$$ 0 0
$$959$$ −10.0000 −0.322917
$$960$$ 0 0
$$961$$ −31.0000 −1.00000
$$962$$ 0 0
$$963$$ −12.0000 −0.386695
$$964$$ 0 0
$$965$$ 4.00000 0.128765
$$966$$ 0 0
$$967$$ 8.00000 0.257263 0.128631 0.991692i $$-0.458942\pi$$
0.128631 + 0.991692i $$0.458942\pi$$
$$968$$ 0 0
$$969$$ 8.00000 0.256997
$$970$$ 0 0
$$971$$ 12.0000 0.385098 0.192549 0.981287i $$-0.438325\pi$$
0.192549 + 0.981287i $$0.438325\pi$$
$$972$$ 0 0
$$973$$ 4.00000 0.128234
$$974$$ 0 0
$$975$$ 6.00000 0.192154
$$976$$ 0 0
$$977$$ 18.0000 0.575871 0.287936 0.957650i $$-0.407031\pi$$
0.287936 + 0.957650i $$0.407031\pi$$
$$978$$ 0 0
$$979$$ −24.0000 −0.767043
$$980$$ 0 0
$$981$$ 2.00000 0.0638551
$$982$$ 0 0
$$983$$ −24.0000 −0.765481 −0.382741 0.923856i $$-0.625020\pi$$
−0.382741 + 0.923856i $$0.625020\pi$$
$$984$$ 0 0
$$985$$ 20.0000 0.637253
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 32.0000 1.01754
$$990$$ 0 0
$$991$$ −16.0000 −0.508257 −0.254128 0.967170i $$-0.581789\pi$$
−0.254128 + 0.967170i $$0.581789\pi$$
$$992$$ 0 0
$$993$$ 4.00000 0.126936
$$994$$ 0 0
$$995$$ 16.0000 0.507234
$$996$$ 0 0
$$997$$ −14.0000 −0.443384 −0.221692 0.975117i $$-0.571158\pi$$
−0.221692 + 0.975117i $$0.571158\pi$$
$$998$$ 0 0
$$999$$ 10.0000 0.316386
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 1344.2.a.q.1.1 1
3.2 odd 2 4032.2.a.e.1.1 1
4.3 odd 2 1344.2.a.i.1.1 1
7.6 odd 2 9408.2.a.n.1.1 1
8.3 odd 2 336.2.a.d.1.1 1
8.5 even 2 42.2.a.a.1.1 1
12.11 even 2 4032.2.a.m.1.1 1
16.3 odd 4 5376.2.c.e.2689.1 2
16.5 even 4 5376.2.c.bc.2689.1 2
16.11 odd 4 5376.2.c.e.2689.2 2
16.13 even 4 5376.2.c.bc.2689.2 2
24.5 odd 2 126.2.a.a.1.1 1
24.11 even 2 1008.2.a.j.1.1 1
28.27 even 2 9408.2.a.bw.1.1 1
40.13 odd 4 1050.2.g.a.799.1 2
40.19 odd 2 8400.2.a.k.1.1 1
40.29 even 2 1050.2.a.i.1.1 1
40.37 odd 4 1050.2.g.a.799.2 2
56.3 even 6 2352.2.q.n.961.1 2
56.5 odd 6 294.2.e.a.67.1 2
56.11 odd 6 2352.2.q.i.961.1 2
56.13 odd 2 294.2.a.g.1.1 1
56.19 even 6 2352.2.q.n.1537.1 2
56.27 even 2 2352.2.a.l.1.1 1
56.37 even 6 294.2.e.c.67.1 2
56.45 odd 6 294.2.e.a.79.1 2
56.51 odd 6 2352.2.q.i.1537.1 2
56.53 even 6 294.2.e.c.79.1 2
72.5 odd 6 1134.2.f.j.379.1 2
72.13 even 6 1134.2.f.g.379.1 2
72.29 odd 6 1134.2.f.j.757.1 2
72.61 even 6 1134.2.f.g.757.1 2
88.21 odd 2 5082.2.a.d.1.1 1
104.77 even 2 7098.2.a.f.1.1 1
120.29 odd 2 3150.2.a.bo.1.1 1
120.53 even 4 3150.2.g.r.2899.2 2
120.77 even 4 3150.2.g.r.2899.1 2
168.5 even 6 882.2.g.j.361.1 2
168.53 odd 6 882.2.g.h.667.1 2
168.83 odd 2 7056.2.a.k.1.1 1
168.101 even 6 882.2.g.j.667.1 2
168.125 even 2 882.2.a.b.1.1 1
168.149 odd 6 882.2.g.h.361.1 2
280.69 odd 2 7350.2.a.f.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
42.2.a.a.1.1 1 8.5 even 2
126.2.a.a.1.1 1 24.5 odd 2
294.2.a.g.1.1 1 56.13 odd 2
294.2.e.a.67.1 2 56.5 odd 6
294.2.e.a.79.1 2 56.45 odd 6
294.2.e.c.67.1 2 56.37 even 6
294.2.e.c.79.1 2 56.53 even 6
336.2.a.d.1.1 1 8.3 odd 2
882.2.a.b.1.1 1 168.125 even 2
882.2.g.h.361.1 2 168.149 odd 6
882.2.g.h.667.1 2 168.53 odd 6
882.2.g.j.361.1 2 168.5 even 6
882.2.g.j.667.1 2 168.101 even 6
1008.2.a.j.1.1 1 24.11 even 2
1050.2.a.i.1.1 1 40.29 even 2
1050.2.g.a.799.1 2 40.13 odd 4
1050.2.g.a.799.2 2 40.37 odd 4
1134.2.f.g.379.1 2 72.13 even 6
1134.2.f.g.757.1 2 72.61 even 6
1134.2.f.j.379.1 2 72.5 odd 6
1134.2.f.j.757.1 2 72.29 odd 6
1344.2.a.i.1.1 1 4.3 odd 2
1344.2.a.q.1.1 1 1.1 even 1 trivial
2352.2.a.l.1.1 1 56.27 even 2
2352.2.q.i.961.1 2 56.11 odd 6
2352.2.q.i.1537.1 2 56.51 odd 6
2352.2.q.n.961.1 2 56.3 even 6
2352.2.q.n.1537.1 2 56.19 even 6
3150.2.a.bo.1.1 1 120.29 odd 2
3150.2.g.r.2899.1 2 120.77 even 4
3150.2.g.r.2899.2 2 120.53 even 4
4032.2.a.e.1.1 1 3.2 odd 2
4032.2.a.m.1.1 1 12.11 even 2
5082.2.a.d.1.1 1 88.21 odd 2
5376.2.c.e.2689.1 2 16.3 odd 4
5376.2.c.e.2689.2 2 16.11 odd 4
5376.2.c.bc.2689.1 2 16.5 even 4
5376.2.c.bc.2689.2 2 16.13 even 4
7056.2.a.k.1.1 1 168.83 odd 2
7098.2.a.f.1.1 1 104.77 even 2
7350.2.a.f.1.1 1 280.69 odd 2
8400.2.a.k.1.1 1 40.19 odd 2
9408.2.a.n.1.1 1 7.6 odd 2
9408.2.a.bw.1.1 1 28.27 even 2