# Properties

 Label 378.2.a.f.1.1 Level $378$ Weight $2$ Character 378.1 Self dual yes Analytic conductor $3.018$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$378 = 2 \cdot 3^{3} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 378.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$3.01834519640$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 378.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{8} +O(q^{10})$$ $$q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{10} +5.00000 q^{11} -1.00000 q^{14} +1.00000 q^{16} +2.00000 q^{17} -1.00000 q^{19} +1.00000 q^{20} +5.00000 q^{22} -1.00000 q^{23} -4.00000 q^{25} -1.00000 q^{28} +4.00000 q^{29} -9.00000 q^{31} +1.00000 q^{32} +2.00000 q^{34} -1.00000 q^{35} +5.00000 q^{37} -1.00000 q^{38} +1.00000 q^{40} -9.00000 q^{41} -10.0000 q^{43} +5.00000 q^{44} -1.00000 q^{46} +6.00000 q^{47} +1.00000 q^{49} -4.00000 q^{50} +12.0000 q^{53} +5.00000 q^{55} -1.00000 q^{56} +4.00000 q^{58} -14.0000 q^{59} -9.00000 q^{62} +1.00000 q^{64} -8.00000 q^{67} +2.00000 q^{68} -1.00000 q^{70} -13.0000 q^{71} -2.00000 q^{73} +5.00000 q^{74} -1.00000 q^{76} -5.00000 q^{77} +6.00000 q^{79} +1.00000 q^{80} -9.00000 q^{82} -4.00000 q^{83} +2.00000 q^{85} -10.0000 q^{86} +5.00000 q^{88} -9.00000 q^{89} -1.00000 q^{92} +6.00000 q^{94} -1.00000 q^{95} +16.0000 q^{97} +1.00000 q^{98} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.00000 0.707107
$$3$$ 0 0
$$4$$ 1.00000 0.500000
$$5$$ 1.00000 0.447214 0.223607 0.974679i $$-0.428217\pi$$
0.223607 + 0.974679i $$0.428217\pi$$
$$6$$ 0 0
$$7$$ −1.00000 −0.377964
$$8$$ 1.00000 0.353553
$$9$$ 0 0
$$10$$ 1.00000 0.316228
$$11$$ 5.00000 1.50756 0.753778 0.657129i $$-0.228229\pi$$
0.753778 + 0.657129i $$0.228229\pi$$
$$12$$ 0 0
$$13$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$14$$ −1.00000 −0.267261
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 2.00000 0.485071 0.242536 0.970143i $$-0.422021\pi$$
0.242536 + 0.970143i $$0.422021\pi$$
$$18$$ 0 0
$$19$$ −1.00000 −0.229416 −0.114708 0.993399i $$-0.536593\pi$$
−0.114708 + 0.993399i $$0.536593\pi$$
$$20$$ 1.00000 0.223607
$$21$$ 0 0
$$22$$ 5.00000 1.06600
$$23$$ −1.00000 −0.208514 −0.104257 0.994550i $$-0.533247\pi$$
−0.104257 + 0.994550i $$0.533247\pi$$
$$24$$ 0 0
$$25$$ −4.00000 −0.800000
$$26$$ 0 0
$$27$$ 0 0
$$28$$ −1.00000 −0.188982
$$29$$ 4.00000 0.742781 0.371391 0.928477i $$-0.378881\pi$$
0.371391 + 0.928477i $$0.378881\pi$$
$$30$$ 0 0
$$31$$ −9.00000 −1.61645 −0.808224 0.588875i $$-0.799571\pi$$
−0.808224 + 0.588875i $$0.799571\pi$$
$$32$$ 1.00000 0.176777
$$33$$ 0 0
$$34$$ 2.00000 0.342997
$$35$$ −1.00000 −0.169031
$$36$$ 0 0
$$37$$ 5.00000 0.821995 0.410997 0.911636i $$-0.365181\pi$$
0.410997 + 0.911636i $$0.365181\pi$$
$$38$$ −1.00000 −0.162221
$$39$$ 0 0
$$40$$ 1.00000 0.158114
$$41$$ −9.00000 −1.40556 −0.702782 0.711405i $$-0.748059\pi$$
−0.702782 + 0.711405i $$0.748059\pi$$
$$42$$ 0 0
$$43$$ −10.0000 −1.52499 −0.762493 0.646997i $$-0.776025\pi$$
−0.762493 + 0.646997i $$0.776025\pi$$
$$44$$ 5.00000 0.753778
$$45$$ 0 0
$$46$$ −1.00000 −0.147442
$$47$$ 6.00000 0.875190 0.437595 0.899172i $$-0.355830\pi$$
0.437595 + 0.899172i $$0.355830\pi$$
$$48$$ 0 0
$$49$$ 1.00000 0.142857
$$50$$ −4.00000 −0.565685
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 12.0000 1.64833 0.824163 0.566352i $$-0.191646\pi$$
0.824163 + 0.566352i $$0.191646\pi$$
$$54$$ 0 0
$$55$$ 5.00000 0.674200
$$56$$ −1.00000 −0.133631
$$57$$ 0 0
$$58$$ 4.00000 0.525226
$$59$$ −14.0000 −1.82264 −0.911322 0.411693i $$-0.864937\pi$$
−0.911322 + 0.411693i $$0.864937\pi$$
$$60$$ 0 0
$$61$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$62$$ −9.00000 −1.14300
$$63$$ 0 0
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −8.00000 −0.977356 −0.488678 0.872464i $$-0.662521\pi$$
−0.488678 + 0.872464i $$0.662521\pi$$
$$68$$ 2.00000 0.242536
$$69$$ 0 0
$$70$$ −1.00000 −0.119523
$$71$$ −13.0000 −1.54282 −0.771408 0.636341i $$-0.780447\pi$$
−0.771408 + 0.636341i $$0.780447\pi$$
$$72$$ 0 0
$$73$$ −2.00000 −0.234082 −0.117041 0.993127i $$-0.537341\pi$$
−0.117041 + 0.993127i $$0.537341\pi$$
$$74$$ 5.00000 0.581238
$$75$$ 0 0
$$76$$ −1.00000 −0.114708
$$77$$ −5.00000 −0.569803
$$78$$ 0 0
$$79$$ 6.00000 0.675053 0.337526 0.941316i $$-0.390410\pi$$
0.337526 + 0.941316i $$0.390410\pi$$
$$80$$ 1.00000 0.111803
$$81$$ 0 0
$$82$$ −9.00000 −0.993884
$$83$$ −4.00000 −0.439057 −0.219529 0.975606i $$-0.570452\pi$$
−0.219529 + 0.975606i $$0.570452\pi$$
$$84$$ 0 0
$$85$$ 2.00000 0.216930
$$86$$ −10.0000 −1.07833
$$87$$ 0 0
$$88$$ 5.00000 0.533002
$$89$$ −9.00000 −0.953998 −0.476999 0.878904i $$-0.658275\pi$$
−0.476999 + 0.878904i $$0.658275\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ −1.00000 −0.104257
$$93$$ 0 0
$$94$$ 6.00000 0.618853
$$95$$ −1.00000 −0.102598
$$96$$ 0 0
$$97$$ 16.0000 1.62455 0.812277 0.583272i $$-0.198228\pi$$
0.812277 + 0.583272i $$0.198228\pi$$
$$98$$ 1.00000 0.101015
$$99$$ 0 0
$$100$$ −4.00000 −0.400000
$$101$$ −14.0000 −1.39305 −0.696526 0.717532i $$-0.745272\pi$$
−0.696526 + 0.717532i $$0.745272\pi$$
$$102$$ 0 0
$$103$$ −1.00000 −0.0985329 −0.0492665 0.998786i $$-0.515688\pi$$
−0.0492665 + 0.998786i $$0.515688\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 12.0000 1.16554
$$107$$ 12.0000 1.16008 0.580042 0.814587i $$-0.303036\pi$$
0.580042 + 0.814587i $$0.303036\pi$$
$$108$$ 0 0
$$109$$ 7.00000 0.670478 0.335239 0.942133i $$-0.391183\pi$$
0.335239 + 0.942133i $$0.391183\pi$$
$$110$$ 5.00000 0.476731
$$111$$ 0 0
$$112$$ −1.00000 −0.0944911
$$113$$ −2.00000 −0.188144 −0.0940721 0.995565i $$-0.529988\pi$$
−0.0940721 + 0.995565i $$0.529988\pi$$
$$114$$ 0 0
$$115$$ −1.00000 −0.0932505
$$116$$ 4.00000 0.371391
$$117$$ 0 0
$$118$$ −14.0000 −1.28880
$$119$$ −2.00000 −0.183340
$$120$$ 0 0
$$121$$ 14.0000 1.27273
$$122$$ 0 0
$$123$$ 0 0
$$124$$ −9.00000 −0.808224
$$125$$ −9.00000 −0.804984
$$126$$ 0 0
$$127$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 22.0000 1.92215 0.961074 0.276289i $$-0.0891049\pi$$
0.961074 + 0.276289i $$0.0891049\pi$$
$$132$$ 0 0
$$133$$ 1.00000 0.0867110
$$134$$ −8.00000 −0.691095
$$135$$ 0 0
$$136$$ 2.00000 0.171499
$$137$$ 16.0000 1.36697 0.683486 0.729964i $$-0.260463\pi$$
0.683486 + 0.729964i $$0.260463\pi$$
$$138$$ 0 0
$$139$$ −20.0000 −1.69638 −0.848189 0.529694i $$-0.822307\pi$$
−0.848189 + 0.529694i $$0.822307\pi$$
$$140$$ −1.00000 −0.0845154
$$141$$ 0 0
$$142$$ −13.0000 −1.09094
$$143$$ 0 0
$$144$$ 0 0
$$145$$ 4.00000 0.332182
$$146$$ −2.00000 −0.165521
$$147$$ 0 0
$$148$$ 5.00000 0.410997
$$149$$ 6.00000 0.491539 0.245770 0.969328i $$-0.420959\pi$$
0.245770 + 0.969328i $$0.420959\pi$$
$$150$$ 0 0
$$151$$ 10.0000 0.813788 0.406894 0.913475i $$-0.366612\pi$$
0.406894 + 0.913475i $$0.366612\pi$$
$$152$$ −1.00000 −0.0811107
$$153$$ 0 0
$$154$$ −5.00000 −0.402911
$$155$$ −9.00000 −0.722897
$$156$$ 0 0
$$157$$ 8.00000 0.638470 0.319235 0.947676i $$-0.396574\pi$$
0.319235 + 0.947676i $$0.396574\pi$$
$$158$$ 6.00000 0.477334
$$159$$ 0 0
$$160$$ 1.00000 0.0790569
$$161$$ 1.00000 0.0788110
$$162$$ 0 0
$$163$$ −4.00000 −0.313304 −0.156652 0.987654i $$-0.550070\pi$$
−0.156652 + 0.987654i $$0.550070\pi$$
$$164$$ −9.00000 −0.702782
$$165$$ 0 0
$$166$$ −4.00000 −0.310460
$$167$$ 10.0000 0.773823 0.386912 0.922117i $$-0.373542\pi$$
0.386912 + 0.922117i $$0.373542\pi$$
$$168$$ 0 0
$$169$$ −13.0000 −1.00000
$$170$$ 2.00000 0.153393
$$171$$ 0 0
$$172$$ −10.0000 −0.762493
$$173$$ 7.00000 0.532200 0.266100 0.963945i $$-0.414265\pi$$
0.266100 + 0.963945i $$0.414265\pi$$
$$174$$ 0 0
$$175$$ 4.00000 0.302372
$$176$$ 5.00000 0.376889
$$177$$ 0 0
$$178$$ −9.00000 −0.674579
$$179$$ 24.0000 1.79384 0.896922 0.442189i $$-0.145798\pi$$
0.896922 + 0.442189i $$0.145798\pi$$
$$180$$ 0 0
$$181$$ 18.0000 1.33793 0.668965 0.743294i $$-0.266738\pi$$
0.668965 + 0.743294i $$0.266738\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ −1.00000 −0.0737210
$$185$$ 5.00000 0.367607
$$186$$ 0 0
$$187$$ 10.0000 0.731272
$$188$$ 6.00000 0.437595
$$189$$ 0 0
$$190$$ −1.00000 −0.0725476
$$191$$ −3.00000 −0.217072 −0.108536 0.994092i $$-0.534616\pi$$
−0.108536 + 0.994092i $$0.534616\pi$$
$$192$$ 0 0
$$193$$ −10.0000 −0.719816 −0.359908 0.932988i $$-0.617192\pi$$
−0.359908 + 0.932988i $$0.617192\pi$$
$$194$$ 16.0000 1.14873
$$195$$ 0 0
$$196$$ 1.00000 0.0714286
$$197$$ −10.0000 −0.712470 −0.356235 0.934396i $$-0.615940\pi$$
−0.356235 + 0.934396i $$0.615940\pi$$
$$198$$ 0 0
$$199$$ −13.0000 −0.921546 −0.460773 0.887518i $$-0.652428\pi$$
−0.460773 + 0.887518i $$0.652428\pi$$
$$200$$ −4.00000 −0.282843
$$201$$ 0 0
$$202$$ −14.0000 −0.985037
$$203$$ −4.00000 −0.280745
$$204$$ 0 0
$$205$$ −9.00000 −0.628587
$$206$$ −1.00000 −0.0696733
$$207$$ 0 0
$$208$$ 0 0
$$209$$ −5.00000 −0.345857
$$210$$ 0 0
$$211$$ −22.0000 −1.51454 −0.757271 0.653101i $$-0.773468\pi$$
−0.757271 + 0.653101i $$0.773468\pi$$
$$212$$ 12.0000 0.824163
$$213$$ 0 0
$$214$$ 12.0000 0.820303
$$215$$ −10.0000 −0.681994
$$216$$ 0 0
$$217$$ 9.00000 0.610960
$$218$$ 7.00000 0.474100
$$219$$ 0 0
$$220$$ 5.00000 0.337100
$$221$$ 0 0
$$222$$ 0 0
$$223$$ 5.00000 0.334825 0.167412 0.985887i $$-0.446459\pi$$
0.167412 + 0.985887i $$0.446459\pi$$
$$224$$ −1.00000 −0.0668153
$$225$$ 0 0
$$226$$ −2.00000 −0.133038
$$227$$ 6.00000 0.398234 0.199117 0.979976i $$-0.436193\pi$$
0.199117 + 0.979976i $$0.436193\pi$$
$$228$$ 0 0
$$229$$ 28.0000 1.85029 0.925146 0.379611i $$-0.123942\pi$$
0.925146 + 0.379611i $$0.123942\pi$$
$$230$$ −1.00000 −0.0659380
$$231$$ 0 0
$$232$$ 4.00000 0.262613
$$233$$ 14.0000 0.917170 0.458585 0.888650i $$-0.348356\pi$$
0.458585 + 0.888650i $$0.348356\pi$$
$$234$$ 0 0
$$235$$ 6.00000 0.391397
$$236$$ −14.0000 −0.911322
$$237$$ 0 0
$$238$$ −2.00000 −0.129641
$$239$$ 24.0000 1.55243 0.776215 0.630468i $$-0.217137\pi$$
0.776215 + 0.630468i $$0.217137\pi$$
$$240$$ 0 0
$$241$$ 14.0000 0.901819 0.450910 0.892570i $$-0.351100\pi$$
0.450910 + 0.892570i $$0.351100\pi$$
$$242$$ 14.0000 0.899954
$$243$$ 0 0
$$244$$ 0 0
$$245$$ 1.00000 0.0638877
$$246$$ 0 0
$$247$$ 0 0
$$248$$ −9.00000 −0.571501
$$249$$ 0 0
$$250$$ −9.00000 −0.569210
$$251$$ −24.0000 −1.51487 −0.757433 0.652913i $$-0.773547\pi$$
−0.757433 + 0.652913i $$0.773547\pi$$
$$252$$ 0 0
$$253$$ −5.00000 −0.314347
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 27.0000 1.68421 0.842107 0.539311i $$-0.181315\pi$$
0.842107 + 0.539311i $$0.181315\pi$$
$$258$$ 0 0
$$259$$ −5.00000 −0.310685
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 22.0000 1.35916
$$263$$ −21.0000 −1.29492 −0.647458 0.762101i $$-0.724168\pi$$
−0.647458 + 0.762101i $$0.724168\pi$$
$$264$$ 0 0
$$265$$ 12.0000 0.737154
$$266$$ 1.00000 0.0613139
$$267$$ 0 0
$$268$$ −8.00000 −0.488678
$$269$$ 13.0000 0.792624 0.396312 0.918116i $$-0.370290\pi$$
0.396312 + 0.918116i $$0.370290\pi$$
$$270$$ 0 0
$$271$$ 24.0000 1.45790 0.728948 0.684569i $$-0.240010\pi$$
0.728948 + 0.684569i $$0.240010\pi$$
$$272$$ 2.00000 0.121268
$$273$$ 0 0
$$274$$ 16.0000 0.966595
$$275$$ −20.0000 −1.20605
$$276$$ 0 0
$$277$$ −19.0000 −1.14160 −0.570800 0.821089i $$-0.693367\pi$$
−0.570800 + 0.821089i $$0.693367\pi$$
$$278$$ −20.0000 −1.19952
$$279$$ 0 0
$$280$$ −1.00000 −0.0597614
$$281$$ −10.0000 −0.596550 −0.298275 0.954480i $$-0.596411\pi$$
−0.298275 + 0.954480i $$0.596411\pi$$
$$282$$ 0 0
$$283$$ −20.0000 −1.18888 −0.594438 0.804141i $$-0.702626\pi$$
−0.594438 + 0.804141i $$0.702626\pi$$
$$284$$ −13.0000 −0.771408
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 9.00000 0.531253
$$288$$ 0 0
$$289$$ −13.0000 −0.764706
$$290$$ 4.00000 0.234888
$$291$$ 0 0
$$292$$ −2.00000 −0.117041
$$293$$ 18.0000 1.05157 0.525786 0.850617i $$-0.323771\pi$$
0.525786 + 0.850617i $$0.323771\pi$$
$$294$$ 0 0
$$295$$ −14.0000 −0.815112
$$296$$ 5.00000 0.290619
$$297$$ 0 0
$$298$$ 6.00000 0.347571
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 10.0000 0.576390
$$302$$ 10.0000 0.575435
$$303$$ 0 0
$$304$$ −1.00000 −0.0573539
$$305$$ 0 0
$$306$$ 0 0
$$307$$ −5.00000 −0.285365 −0.142683 0.989769i $$-0.545573\pi$$
−0.142683 + 0.989769i $$0.545573\pi$$
$$308$$ −5.00000 −0.284901
$$309$$ 0 0
$$310$$ −9.00000 −0.511166
$$311$$ −8.00000 −0.453638 −0.226819 0.973937i $$-0.572833\pi$$
−0.226819 + 0.973937i $$0.572833\pi$$
$$312$$ 0 0
$$313$$ −8.00000 −0.452187 −0.226093 0.974106i $$-0.572595\pi$$
−0.226093 + 0.974106i $$0.572595\pi$$
$$314$$ 8.00000 0.451466
$$315$$ 0 0
$$316$$ 6.00000 0.337526
$$317$$ −18.0000 −1.01098 −0.505490 0.862832i $$-0.668688\pi$$
−0.505490 + 0.862832i $$0.668688\pi$$
$$318$$ 0 0
$$319$$ 20.0000 1.11979
$$320$$ 1.00000 0.0559017
$$321$$ 0 0
$$322$$ 1.00000 0.0557278
$$323$$ −2.00000 −0.111283
$$324$$ 0 0
$$325$$ 0 0
$$326$$ −4.00000 −0.221540
$$327$$ 0 0
$$328$$ −9.00000 −0.496942
$$329$$ −6.00000 −0.330791
$$330$$ 0 0
$$331$$ −4.00000 −0.219860 −0.109930 0.993939i $$-0.535063\pi$$
−0.109930 + 0.993939i $$0.535063\pi$$
$$332$$ −4.00000 −0.219529
$$333$$ 0 0
$$334$$ 10.0000 0.547176
$$335$$ −8.00000 −0.437087
$$336$$ 0 0
$$337$$ 27.0000 1.47078 0.735392 0.677642i $$-0.236998\pi$$
0.735392 + 0.677642i $$0.236998\pi$$
$$338$$ −13.0000 −0.707107
$$339$$ 0 0
$$340$$ 2.00000 0.108465
$$341$$ −45.0000 −2.43689
$$342$$ 0 0
$$343$$ −1.00000 −0.0539949
$$344$$ −10.0000 −0.539164
$$345$$ 0 0
$$346$$ 7.00000 0.376322
$$347$$ −3.00000 −0.161048 −0.0805242 0.996753i $$-0.525659\pi$$
−0.0805242 + 0.996753i $$0.525659\pi$$
$$348$$ 0 0
$$349$$ −26.0000 −1.39175 −0.695874 0.718164i $$-0.744983\pi$$
−0.695874 + 0.718164i $$0.744983\pi$$
$$350$$ 4.00000 0.213809
$$351$$ 0 0
$$352$$ 5.00000 0.266501
$$353$$ −3.00000 −0.159674 −0.0798369 0.996808i $$-0.525440\pi$$
−0.0798369 + 0.996808i $$0.525440\pi$$
$$354$$ 0 0
$$355$$ −13.0000 −0.689968
$$356$$ −9.00000 −0.476999
$$357$$ 0 0
$$358$$ 24.0000 1.26844
$$359$$ 4.00000 0.211112 0.105556 0.994413i $$-0.466338\pi$$
0.105556 + 0.994413i $$0.466338\pi$$
$$360$$ 0 0
$$361$$ −18.0000 −0.947368
$$362$$ 18.0000 0.946059
$$363$$ 0 0
$$364$$ 0 0
$$365$$ −2.00000 −0.104685
$$366$$ 0 0
$$367$$ 35.0000 1.82699 0.913493 0.406855i $$-0.133375\pi$$
0.913493 + 0.406855i $$0.133375\pi$$
$$368$$ −1.00000 −0.0521286
$$369$$ 0 0
$$370$$ 5.00000 0.259938
$$371$$ −12.0000 −0.623009
$$372$$ 0 0
$$373$$ −17.0000 −0.880227 −0.440113 0.897942i $$-0.645062\pi$$
−0.440113 + 0.897942i $$0.645062\pi$$
$$374$$ 10.0000 0.517088
$$375$$ 0 0
$$376$$ 6.00000 0.309426
$$377$$ 0 0
$$378$$ 0 0
$$379$$ −14.0000 −0.719132 −0.359566 0.933120i $$-0.617075\pi$$
−0.359566 + 0.933120i $$0.617075\pi$$
$$380$$ −1.00000 −0.0512989
$$381$$ 0 0
$$382$$ −3.00000 −0.153493
$$383$$ −10.0000 −0.510976 −0.255488 0.966812i $$-0.582236\pi$$
−0.255488 + 0.966812i $$0.582236\pi$$
$$384$$ 0 0
$$385$$ −5.00000 −0.254824
$$386$$ −10.0000 −0.508987
$$387$$ 0 0
$$388$$ 16.0000 0.812277
$$389$$ −20.0000 −1.01404 −0.507020 0.861934i $$-0.669253\pi$$
−0.507020 + 0.861934i $$0.669253\pi$$
$$390$$ 0 0
$$391$$ −2.00000 −0.101144
$$392$$ 1.00000 0.0505076
$$393$$ 0 0
$$394$$ −10.0000 −0.503793
$$395$$ 6.00000 0.301893
$$396$$ 0 0
$$397$$ −30.0000 −1.50566 −0.752828 0.658217i $$-0.771311\pi$$
−0.752828 + 0.658217i $$0.771311\pi$$
$$398$$ −13.0000 −0.651631
$$399$$ 0 0
$$400$$ −4.00000 −0.200000
$$401$$ −12.0000 −0.599251 −0.299626 0.954057i $$-0.596862\pi$$
−0.299626 + 0.954057i $$0.596862\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ −14.0000 −0.696526
$$405$$ 0 0
$$406$$ −4.00000 −0.198517
$$407$$ 25.0000 1.23920
$$408$$ 0 0
$$409$$ −10.0000 −0.494468 −0.247234 0.968956i $$-0.579522\pi$$
−0.247234 + 0.968956i $$0.579522\pi$$
$$410$$ −9.00000 −0.444478
$$411$$ 0 0
$$412$$ −1.00000 −0.0492665
$$413$$ 14.0000 0.688895
$$414$$ 0 0
$$415$$ −4.00000 −0.196352
$$416$$ 0 0
$$417$$ 0 0
$$418$$ −5.00000 −0.244558
$$419$$ 6.00000 0.293119 0.146560 0.989202i $$-0.453180\pi$$
0.146560 + 0.989202i $$0.453180\pi$$
$$420$$ 0 0
$$421$$ −27.0000 −1.31590 −0.657950 0.753062i $$-0.728576\pi$$
−0.657950 + 0.753062i $$0.728576\pi$$
$$422$$ −22.0000 −1.07094
$$423$$ 0 0
$$424$$ 12.0000 0.582772
$$425$$ −8.00000 −0.388057
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 12.0000 0.580042
$$429$$ 0 0
$$430$$ −10.0000 −0.482243
$$431$$ −15.0000 −0.722525 −0.361262 0.932464i $$-0.617654\pi$$
−0.361262 + 0.932464i $$0.617654\pi$$
$$432$$ 0 0
$$433$$ −4.00000 −0.192228 −0.0961139 0.995370i $$-0.530641\pi$$
−0.0961139 + 0.995370i $$0.530641\pi$$
$$434$$ 9.00000 0.432014
$$435$$ 0 0
$$436$$ 7.00000 0.335239
$$437$$ 1.00000 0.0478365
$$438$$ 0 0
$$439$$ 24.0000 1.14546 0.572729 0.819745i $$-0.305885\pi$$
0.572729 + 0.819745i $$0.305885\pi$$
$$440$$ 5.00000 0.238366
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 11.0000 0.522626 0.261313 0.965254i $$-0.415845\pi$$
0.261313 + 0.965254i $$0.415845\pi$$
$$444$$ 0 0
$$445$$ −9.00000 −0.426641
$$446$$ 5.00000 0.236757
$$447$$ 0 0
$$448$$ −1.00000 −0.0472456
$$449$$ 10.0000 0.471929 0.235965 0.971762i $$-0.424175\pi$$
0.235965 + 0.971762i $$0.424175\pi$$
$$450$$ 0 0
$$451$$ −45.0000 −2.11897
$$452$$ −2.00000 −0.0940721
$$453$$ 0 0
$$454$$ 6.00000 0.281594
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 13.0000 0.608114 0.304057 0.952654i $$-0.401659\pi$$
0.304057 + 0.952654i $$0.401659\pi$$
$$458$$ 28.0000 1.30835
$$459$$ 0 0
$$460$$ −1.00000 −0.0466252
$$461$$ 31.0000 1.44381 0.721907 0.691990i $$-0.243266\pi$$
0.721907 + 0.691990i $$0.243266\pi$$
$$462$$ 0 0
$$463$$ −14.0000 −0.650635 −0.325318 0.945605i $$-0.605471\pi$$
−0.325318 + 0.945605i $$0.605471\pi$$
$$464$$ 4.00000 0.185695
$$465$$ 0 0
$$466$$ 14.0000 0.648537
$$467$$ −26.0000 −1.20314 −0.601568 0.798821i $$-0.705457\pi$$
−0.601568 + 0.798821i $$0.705457\pi$$
$$468$$ 0 0
$$469$$ 8.00000 0.369406
$$470$$ 6.00000 0.276759
$$471$$ 0 0
$$472$$ −14.0000 −0.644402
$$473$$ −50.0000 −2.29900
$$474$$ 0 0
$$475$$ 4.00000 0.183533
$$476$$ −2.00000 −0.0916698
$$477$$ 0 0
$$478$$ 24.0000 1.09773
$$479$$ 32.0000 1.46212 0.731059 0.682315i $$-0.239027\pi$$
0.731059 + 0.682315i $$0.239027\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 14.0000 0.637683
$$483$$ 0 0
$$484$$ 14.0000 0.636364
$$485$$ 16.0000 0.726523
$$486$$ 0 0
$$487$$ 26.0000 1.17817 0.589086 0.808070i $$-0.299488\pi$$
0.589086 + 0.808070i $$0.299488\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 1.00000 0.0451754
$$491$$ 9.00000 0.406164 0.203082 0.979162i $$-0.434904\pi$$
0.203082 + 0.979162i $$0.434904\pi$$
$$492$$ 0 0
$$493$$ 8.00000 0.360302
$$494$$ 0 0
$$495$$ 0 0
$$496$$ −9.00000 −0.404112
$$497$$ 13.0000 0.583130
$$498$$ 0 0
$$499$$ 10.0000 0.447661 0.223831 0.974628i $$-0.428144\pi$$
0.223831 + 0.974628i $$0.428144\pi$$
$$500$$ −9.00000 −0.402492
$$501$$ 0 0
$$502$$ −24.0000 −1.07117
$$503$$ 36.0000 1.60516 0.802580 0.596544i $$-0.203460\pi$$
0.802580 + 0.596544i $$0.203460\pi$$
$$504$$ 0 0
$$505$$ −14.0000 −0.622992
$$506$$ −5.00000 −0.222277
$$507$$ 0 0
$$508$$ 0 0
$$509$$ 18.0000 0.797836 0.398918 0.916987i $$-0.369386\pi$$
0.398918 + 0.916987i $$0.369386\pi$$
$$510$$ 0 0
$$511$$ 2.00000 0.0884748
$$512$$ 1.00000 0.0441942
$$513$$ 0 0
$$514$$ 27.0000 1.19092
$$515$$ −1.00000 −0.0440653
$$516$$ 0 0
$$517$$ 30.0000 1.31940
$$518$$ −5.00000 −0.219687
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −15.0000 −0.657162 −0.328581 0.944476i $$-0.606570\pi$$
−0.328581 + 0.944476i $$0.606570\pi$$
$$522$$ 0 0
$$523$$ 11.0000 0.480996 0.240498 0.970650i $$-0.422689\pi$$
0.240498 + 0.970650i $$0.422689\pi$$
$$524$$ 22.0000 0.961074
$$525$$ 0 0
$$526$$ −21.0000 −0.915644
$$527$$ −18.0000 −0.784092
$$528$$ 0 0
$$529$$ −22.0000 −0.956522
$$530$$ 12.0000 0.521247
$$531$$ 0 0
$$532$$ 1.00000 0.0433555
$$533$$ 0 0
$$534$$ 0 0
$$535$$ 12.0000 0.518805
$$536$$ −8.00000 −0.345547
$$537$$ 0 0
$$538$$ 13.0000 0.560470
$$539$$ 5.00000 0.215365
$$540$$ 0 0
$$541$$ −3.00000 −0.128980 −0.0644900 0.997918i $$-0.520542\pi$$
−0.0644900 + 0.997918i $$0.520542\pi$$
$$542$$ 24.0000 1.03089
$$543$$ 0 0
$$544$$ 2.00000 0.0857493
$$545$$ 7.00000 0.299847
$$546$$ 0 0
$$547$$ 12.0000 0.513083 0.256541 0.966533i $$-0.417417\pi$$
0.256541 + 0.966533i $$0.417417\pi$$
$$548$$ 16.0000 0.683486
$$549$$ 0 0
$$550$$ −20.0000 −0.852803
$$551$$ −4.00000 −0.170406
$$552$$ 0 0
$$553$$ −6.00000 −0.255146
$$554$$ −19.0000 −0.807233
$$555$$ 0 0
$$556$$ −20.0000 −0.848189
$$557$$ 22.0000 0.932170 0.466085 0.884740i $$-0.345664\pi$$
0.466085 + 0.884740i $$0.345664\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ −1.00000 −0.0422577
$$561$$ 0 0
$$562$$ −10.0000 −0.421825
$$563$$ −4.00000 −0.168580 −0.0842900 0.996441i $$-0.526862\pi$$
−0.0842900 + 0.996441i $$0.526862\pi$$
$$564$$ 0 0
$$565$$ −2.00000 −0.0841406
$$566$$ −20.0000 −0.840663
$$567$$ 0 0
$$568$$ −13.0000 −0.545468
$$569$$ 30.0000 1.25767 0.628833 0.777541i $$-0.283533\pi$$
0.628833 + 0.777541i $$0.283533\pi$$
$$570$$ 0 0
$$571$$ 18.0000 0.753277 0.376638 0.926360i $$-0.377080\pi$$
0.376638 + 0.926360i $$0.377080\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 9.00000 0.375653
$$575$$ 4.00000 0.166812
$$576$$ 0 0
$$577$$ −14.0000 −0.582828 −0.291414 0.956597i $$-0.594126\pi$$
−0.291414 + 0.956597i $$0.594126\pi$$
$$578$$ −13.0000 −0.540729
$$579$$ 0 0
$$580$$ 4.00000 0.166091
$$581$$ 4.00000 0.165948
$$582$$ 0 0
$$583$$ 60.0000 2.48495
$$584$$ −2.00000 −0.0827606
$$585$$ 0 0
$$586$$ 18.0000 0.743573
$$587$$ 14.0000 0.577842 0.288921 0.957353i $$-0.406704\pi$$
0.288921 + 0.957353i $$0.406704\pi$$
$$588$$ 0 0
$$589$$ 9.00000 0.370839
$$590$$ −14.0000 −0.576371
$$591$$ 0 0
$$592$$ 5.00000 0.205499
$$593$$ 9.00000 0.369586 0.184793 0.982777i $$-0.440839\pi$$
0.184793 + 0.982777i $$0.440839\pi$$
$$594$$ 0 0
$$595$$ −2.00000 −0.0819920
$$596$$ 6.00000 0.245770
$$597$$ 0 0
$$598$$ 0 0
$$599$$ −27.0000 −1.10319 −0.551595 0.834112i $$-0.685981\pi$$
−0.551595 + 0.834112i $$0.685981\pi$$
$$600$$ 0 0
$$601$$ 8.00000 0.326327 0.163163 0.986599i $$-0.447830\pi$$
0.163163 + 0.986599i $$0.447830\pi$$
$$602$$ 10.0000 0.407570
$$603$$ 0 0
$$604$$ 10.0000 0.406894
$$605$$ 14.0000 0.569181
$$606$$ 0 0
$$607$$ 12.0000 0.487065 0.243532 0.969893i $$-0.421694\pi$$
0.243532 + 0.969893i $$0.421694\pi$$
$$608$$ −1.00000 −0.0405554
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 0 0
$$613$$ −15.0000 −0.605844 −0.302922 0.953015i $$-0.597962\pi$$
−0.302922 + 0.953015i $$0.597962\pi$$
$$614$$ −5.00000 −0.201784
$$615$$ 0 0
$$616$$ −5.00000 −0.201456
$$617$$ 14.0000 0.563619 0.281809 0.959470i $$-0.409065\pi$$
0.281809 + 0.959470i $$0.409065\pi$$
$$618$$ 0 0
$$619$$ −11.0000 −0.442127 −0.221064 0.975259i $$-0.570953\pi$$
−0.221064 + 0.975259i $$0.570953\pi$$
$$620$$ −9.00000 −0.361449
$$621$$ 0 0
$$622$$ −8.00000 −0.320771
$$623$$ 9.00000 0.360577
$$624$$ 0 0
$$625$$ 11.0000 0.440000
$$626$$ −8.00000 −0.319744
$$627$$ 0 0
$$628$$ 8.00000 0.319235
$$629$$ 10.0000 0.398726
$$630$$ 0 0
$$631$$ 38.0000 1.51276 0.756378 0.654135i $$-0.226967\pi$$
0.756378 + 0.654135i $$0.226967\pi$$
$$632$$ 6.00000 0.238667
$$633$$ 0 0
$$634$$ −18.0000 −0.714871
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 20.0000 0.791808
$$639$$ 0 0
$$640$$ 1.00000 0.0395285
$$641$$ −22.0000 −0.868948 −0.434474 0.900684i $$-0.643066\pi$$
−0.434474 + 0.900684i $$0.643066\pi$$
$$642$$ 0 0
$$643$$ −13.0000 −0.512670 −0.256335 0.966588i $$-0.582515\pi$$
−0.256335 + 0.966588i $$0.582515\pi$$
$$644$$ 1.00000 0.0394055
$$645$$ 0 0
$$646$$ −2.00000 −0.0786889
$$647$$ 42.0000 1.65119 0.825595 0.564263i $$-0.190840\pi$$
0.825595 + 0.564263i $$0.190840\pi$$
$$648$$ 0 0
$$649$$ −70.0000 −2.74774
$$650$$ 0 0
$$651$$ 0 0
$$652$$ −4.00000 −0.156652
$$653$$ −30.0000 −1.17399 −0.586995 0.809590i $$-0.699689\pi$$
−0.586995 + 0.809590i $$0.699689\pi$$
$$654$$ 0 0
$$655$$ 22.0000 0.859611
$$656$$ −9.00000 −0.351391
$$657$$ 0 0
$$658$$ −6.00000 −0.233904
$$659$$ 17.0000 0.662226 0.331113 0.943591i $$-0.392576\pi$$
0.331113 + 0.943591i $$0.392576\pi$$
$$660$$ 0 0
$$661$$ 28.0000 1.08907 0.544537 0.838737i $$-0.316705\pi$$
0.544537 + 0.838737i $$0.316705\pi$$
$$662$$ −4.00000 −0.155464
$$663$$ 0 0
$$664$$ −4.00000 −0.155230
$$665$$ 1.00000 0.0387783
$$666$$ 0 0
$$667$$ −4.00000 −0.154881
$$668$$ 10.0000 0.386912
$$669$$ 0 0
$$670$$ −8.00000 −0.309067
$$671$$ 0 0
$$672$$ 0 0
$$673$$ 26.0000 1.00223 0.501113 0.865382i $$-0.332924\pi$$
0.501113 + 0.865382i $$0.332924\pi$$
$$674$$ 27.0000 1.04000
$$675$$ 0 0
$$676$$ −13.0000 −0.500000
$$677$$ 27.0000 1.03769 0.518847 0.854867i $$-0.326361\pi$$
0.518847 + 0.854867i $$0.326361\pi$$
$$678$$ 0 0
$$679$$ −16.0000 −0.614024
$$680$$ 2.00000 0.0766965
$$681$$ 0 0
$$682$$ −45.0000 −1.72314
$$683$$ −9.00000 −0.344375 −0.172188 0.985064i $$-0.555084\pi$$
−0.172188 + 0.985064i $$0.555084\pi$$
$$684$$ 0 0
$$685$$ 16.0000 0.611329
$$686$$ −1.00000 −0.0381802
$$687$$ 0 0
$$688$$ −10.0000 −0.381246
$$689$$ 0 0
$$690$$ 0 0
$$691$$ −4.00000 −0.152167 −0.0760836 0.997101i $$-0.524242\pi$$
−0.0760836 + 0.997101i $$0.524242\pi$$
$$692$$ 7.00000 0.266100
$$693$$ 0 0
$$694$$ −3.00000 −0.113878
$$695$$ −20.0000 −0.758643
$$696$$ 0 0
$$697$$ −18.0000 −0.681799
$$698$$ −26.0000 −0.984115
$$699$$ 0 0
$$700$$ 4.00000 0.151186
$$701$$ −20.0000 −0.755390 −0.377695 0.925930i $$-0.623283\pi$$
−0.377695 + 0.925930i $$0.623283\pi$$
$$702$$ 0 0
$$703$$ −5.00000 −0.188579
$$704$$ 5.00000 0.188445
$$705$$ 0 0
$$706$$ −3.00000 −0.112906
$$707$$ 14.0000 0.526524
$$708$$ 0 0
$$709$$ −25.0000 −0.938895 −0.469447 0.882960i $$-0.655547\pi$$
−0.469447 + 0.882960i $$0.655547\pi$$
$$710$$ −13.0000 −0.487881
$$711$$ 0 0
$$712$$ −9.00000 −0.337289
$$713$$ 9.00000 0.337053
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 24.0000 0.896922
$$717$$ 0 0
$$718$$ 4.00000 0.149279
$$719$$ −30.0000 −1.11881 −0.559406 0.828894i $$-0.688971\pi$$
−0.559406 + 0.828894i $$0.688971\pi$$
$$720$$ 0 0
$$721$$ 1.00000 0.0372419
$$722$$ −18.0000 −0.669891
$$723$$ 0 0
$$724$$ 18.0000 0.668965
$$725$$ −16.0000 −0.594225
$$726$$ 0 0
$$727$$ −32.0000 −1.18681 −0.593407 0.804902i $$-0.702218\pi$$
−0.593407 + 0.804902i $$0.702218\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ −2.00000 −0.0740233
$$731$$ −20.0000 −0.739727
$$732$$ 0 0
$$733$$ −18.0000 −0.664845 −0.332423 0.943131i $$-0.607866\pi$$
−0.332423 + 0.943131i $$0.607866\pi$$
$$734$$ 35.0000 1.29187
$$735$$ 0 0
$$736$$ −1.00000 −0.0368605
$$737$$ −40.0000 −1.47342
$$738$$ 0 0
$$739$$ 18.0000 0.662141 0.331070 0.943606i $$-0.392590\pi$$
0.331070 + 0.943606i $$0.392590\pi$$
$$740$$ 5.00000 0.183804
$$741$$ 0 0
$$742$$ −12.0000 −0.440534
$$743$$ −21.0000 −0.770415 −0.385208 0.922830i $$-0.625870\pi$$
−0.385208 + 0.922830i $$0.625870\pi$$
$$744$$ 0 0
$$745$$ 6.00000 0.219823
$$746$$ −17.0000 −0.622414
$$747$$ 0 0
$$748$$ 10.0000 0.365636
$$749$$ −12.0000 −0.438470
$$750$$ 0 0
$$751$$ 18.0000 0.656829 0.328415 0.944534i $$-0.393486\pi$$
0.328415 + 0.944534i $$0.393486\pi$$
$$752$$ 6.00000 0.218797
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 10.0000 0.363937
$$756$$ 0 0
$$757$$ 42.0000 1.52652 0.763258 0.646094i $$-0.223599\pi$$
0.763258 + 0.646094i $$0.223599\pi$$
$$758$$ −14.0000 −0.508503
$$759$$ 0 0
$$760$$ −1.00000 −0.0362738
$$761$$ −22.0000 −0.797499 −0.398750 0.917060i $$-0.630556\pi$$
−0.398750 + 0.917060i $$0.630556\pi$$
$$762$$ 0 0
$$763$$ −7.00000 −0.253417
$$764$$ −3.00000 −0.108536
$$765$$ 0 0
$$766$$ −10.0000 −0.361315
$$767$$ 0 0
$$768$$ 0 0
$$769$$ 40.0000 1.44244 0.721218 0.692708i $$-0.243582\pi$$
0.721218 + 0.692708i $$0.243582\pi$$
$$770$$ −5.00000 −0.180187
$$771$$ 0 0
$$772$$ −10.0000 −0.359908
$$773$$ 19.0000 0.683383 0.341691 0.939812i $$-0.389000\pi$$
0.341691 + 0.939812i $$0.389000\pi$$
$$774$$ 0 0
$$775$$ 36.0000 1.29316
$$776$$ 16.0000 0.574367
$$777$$ 0 0
$$778$$ −20.0000 −0.717035
$$779$$ 9.00000 0.322458
$$780$$ 0 0
$$781$$ −65.0000 −2.32588
$$782$$ −2.00000 −0.0715199
$$783$$ 0 0
$$784$$ 1.00000 0.0357143
$$785$$ 8.00000 0.285532
$$786$$ 0 0
$$787$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$788$$ −10.0000 −0.356235
$$789$$ 0 0
$$790$$ 6.00000 0.213470
$$791$$ 2.00000 0.0711118
$$792$$ 0 0
$$793$$ 0 0
$$794$$ −30.0000 −1.06466
$$795$$ 0 0
$$796$$ −13.0000 −0.460773
$$797$$ −33.0000 −1.16892 −0.584460 0.811423i $$-0.698694\pi$$
−0.584460 + 0.811423i $$0.698694\pi$$
$$798$$ 0 0
$$799$$ 12.0000 0.424529
$$800$$ −4.00000 −0.141421
$$801$$ 0 0
$$802$$ −12.0000 −0.423735
$$803$$ −10.0000 −0.352892
$$804$$ 0 0
$$805$$ 1.00000 0.0352454
$$806$$ 0 0
$$807$$ 0 0
$$808$$ −14.0000 −0.492518
$$809$$ −20.0000 −0.703163 −0.351581 0.936157i $$-0.614356\pi$$
−0.351581 + 0.936157i $$0.614356\pi$$
$$810$$ 0 0
$$811$$ 1.00000 0.0351147 0.0175574 0.999846i $$-0.494411\pi$$
0.0175574 + 0.999846i $$0.494411\pi$$
$$812$$ −4.00000 −0.140372
$$813$$ 0 0
$$814$$ 25.0000 0.876250
$$815$$ −4.00000 −0.140114
$$816$$ 0 0
$$817$$ 10.0000 0.349856
$$818$$ −10.0000 −0.349642
$$819$$ 0 0
$$820$$ −9.00000 −0.314294
$$821$$ −10.0000 −0.349002 −0.174501 0.984657i $$-0.555831\pi$$
−0.174501 + 0.984657i $$0.555831\pi$$
$$822$$ 0 0
$$823$$ 34.0000 1.18517 0.592583 0.805510i $$-0.298108\pi$$
0.592583 + 0.805510i $$0.298108\pi$$
$$824$$ −1.00000 −0.0348367
$$825$$ 0 0
$$826$$ 14.0000 0.487122
$$827$$ −33.0000 −1.14752 −0.573761 0.819023i $$-0.694516\pi$$
−0.573761 + 0.819023i $$0.694516\pi$$
$$828$$ 0 0
$$829$$ 10.0000 0.347314 0.173657 0.984806i $$-0.444442\pi$$
0.173657 + 0.984806i $$0.444442\pi$$
$$830$$ −4.00000 −0.138842
$$831$$ 0 0
$$832$$ 0 0
$$833$$ 2.00000 0.0692959
$$834$$ 0 0
$$835$$ 10.0000 0.346064
$$836$$ −5.00000 −0.172929
$$837$$ 0 0
$$838$$ 6.00000 0.207267
$$839$$ −4.00000 −0.138095 −0.0690477 0.997613i $$-0.521996\pi$$
−0.0690477 + 0.997613i $$0.521996\pi$$
$$840$$ 0 0
$$841$$ −13.0000 −0.448276
$$842$$ −27.0000 −0.930481
$$843$$ 0 0
$$844$$ −22.0000 −0.757271
$$845$$ −13.0000 −0.447214
$$846$$ 0 0
$$847$$ −14.0000 −0.481046
$$848$$ 12.0000 0.412082
$$849$$ 0 0
$$850$$ −8.00000 −0.274398
$$851$$ −5.00000 −0.171398
$$852$$ 0 0
$$853$$ −16.0000 −0.547830 −0.273915 0.961754i $$-0.588319\pi$$
−0.273915 + 0.961754i $$0.588319\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 12.0000 0.410152
$$857$$ 3.00000 0.102478 0.0512390 0.998686i $$-0.483683\pi$$
0.0512390 + 0.998686i $$0.483683\pi$$
$$858$$ 0 0
$$859$$ −25.0000 −0.852989 −0.426494 0.904490i $$-0.640252\pi$$
−0.426494 + 0.904490i $$0.640252\pi$$
$$860$$ −10.0000 −0.340997
$$861$$ 0 0
$$862$$ −15.0000 −0.510902
$$863$$ 16.0000 0.544646 0.272323 0.962206i $$-0.412208\pi$$
0.272323 + 0.962206i $$0.412208\pi$$
$$864$$ 0 0
$$865$$ 7.00000 0.238007
$$866$$ −4.00000 −0.135926
$$867$$ 0 0
$$868$$ 9.00000 0.305480
$$869$$ 30.0000 1.01768
$$870$$ 0 0
$$871$$ 0 0
$$872$$ 7.00000 0.237050
$$873$$ 0 0
$$874$$ 1.00000 0.0338255
$$875$$ 9.00000 0.304256
$$876$$ 0 0
$$877$$ −50.0000 −1.68838 −0.844190 0.536044i $$-0.819918\pi$$
−0.844190 + 0.536044i $$0.819918\pi$$
$$878$$ 24.0000 0.809961
$$879$$ 0 0
$$880$$ 5.00000 0.168550
$$881$$ −15.0000 −0.505363 −0.252681 0.967550i $$-0.581312\pi$$
−0.252681 + 0.967550i $$0.581312\pi$$
$$882$$ 0 0
$$883$$ 32.0000 1.07689 0.538443 0.842662i $$-0.319013\pi$$
0.538443 + 0.842662i $$0.319013\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 11.0000 0.369552
$$887$$ 36.0000 1.20876 0.604381 0.796696i $$-0.293421\pi$$
0.604381 + 0.796696i $$0.293421\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ −9.00000 −0.301681
$$891$$ 0 0
$$892$$ 5.00000 0.167412
$$893$$ −6.00000 −0.200782
$$894$$ 0 0
$$895$$ 24.0000 0.802232
$$896$$ −1.00000 −0.0334077
$$897$$ 0 0
$$898$$ 10.0000 0.333704
$$899$$ −36.0000 −1.20067
$$900$$ 0 0
$$901$$ 24.0000 0.799556
$$902$$ −45.0000 −1.49834
$$903$$ 0 0
$$904$$ −2.00000 −0.0665190
$$905$$ 18.0000 0.598340
$$906$$ 0 0
$$907$$ 30.0000 0.996134 0.498067 0.867139i $$-0.334043\pi$$
0.498067 + 0.867139i $$0.334043\pi$$
$$908$$ 6.00000 0.199117
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −24.0000 −0.795155 −0.397578 0.917568i $$-0.630149\pi$$
−0.397578 + 0.917568i $$0.630149\pi$$
$$912$$ 0 0
$$913$$ −20.0000 −0.661903
$$914$$ 13.0000 0.430002
$$915$$ 0 0
$$916$$ 28.0000 0.925146
$$917$$ −22.0000 −0.726504
$$918$$ 0 0
$$919$$ 4.00000 0.131948 0.0659739 0.997821i $$-0.478985\pi$$
0.0659739 + 0.997821i $$0.478985\pi$$
$$920$$ −1.00000 −0.0329690
$$921$$ 0 0
$$922$$ 31.0000 1.02093
$$923$$ 0 0
$$924$$ 0 0
$$925$$ −20.0000 −0.657596
$$926$$ −14.0000 −0.460069
$$927$$ 0 0
$$928$$ 4.00000 0.131306
$$929$$ −42.0000 −1.37798 −0.688988 0.724773i $$-0.741945\pi$$
−0.688988 + 0.724773i $$0.741945\pi$$
$$930$$ 0 0
$$931$$ −1.00000 −0.0327737
$$932$$ 14.0000 0.458585
$$933$$ 0 0
$$934$$ −26.0000 −0.850746
$$935$$ 10.0000 0.327035
$$936$$ 0 0
$$937$$ −28.0000 −0.914720 −0.457360 0.889282i $$-0.651205\pi$$
−0.457360 + 0.889282i $$0.651205\pi$$
$$938$$ 8.00000 0.261209
$$939$$ 0 0
$$940$$ 6.00000 0.195698
$$941$$ 13.0000 0.423788 0.211894 0.977293i $$-0.432037\pi$$
0.211894 + 0.977293i $$0.432037\pi$$
$$942$$ 0 0
$$943$$ 9.00000 0.293080
$$944$$ −14.0000 −0.455661
$$945$$ 0 0
$$946$$ −50.0000 −1.62564
$$947$$ −17.0000 −0.552426 −0.276213 0.961096i $$-0.589079\pi$$
−0.276213 + 0.961096i $$0.589079\pi$$
$$948$$ 0 0
$$949$$ 0 0
$$950$$ 4.00000 0.129777
$$951$$ 0 0
$$952$$ −2.00000 −0.0648204
$$953$$ 44.0000 1.42530 0.712650 0.701520i $$-0.247495\pi$$
0.712650 + 0.701520i $$0.247495\pi$$
$$954$$ 0 0
$$955$$ −3.00000 −0.0970777
$$956$$ 24.0000 0.776215
$$957$$ 0 0
$$958$$ 32.0000 1.03387
$$959$$ −16.0000 −0.516667
$$960$$ 0 0
$$961$$ 50.0000 1.61290
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 14.0000 0.450910
$$965$$ −10.0000 −0.321911
$$966$$ 0 0
$$967$$ 2.00000 0.0643157 0.0321578 0.999483i $$-0.489762\pi$$
0.0321578 + 0.999483i $$0.489762\pi$$
$$968$$ 14.0000 0.449977
$$969$$ 0 0
$$970$$ 16.0000 0.513729
$$971$$ 36.0000 1.15529 0.577647 0.816286i $$-0.303971\pi$$
0.577647 + 0.816286i $$0.303971\pi$$
$$972$$ 0 0
$$973$$ 20.0000 0.641171
$$974$$ 26.0000 0.833094
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 60.0000 1.91957 0.959785 0.280736i $$-0.0905785\pi$$
0.959785 + 0.280736i $$0.0905785\pi$$
$$978$$ 0 0
$$979$$ −45.0000 −1.43821
$$980$$ 1.00000 0.0319438
$$981$$ 0 0
$$982$$ 9.00000 0.287202
$$983$$ −30.0000 −0.956851 −0.478426 0.878128i $$-0.658792\pi$$
−0.478426 + 0.878128i $$0.658792\pi$$
$$984$$ 0 0
$$985$$ −10.0000 −0.318626
$$986$$ 8.00000 0.254772
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 10.0000 0.317982
$$990$$ 0 0
$$991$$ −4.00000 −0.127064 −0.0635321 0.997980i $$-0.520237\pi$$
−0.0635321 + 0.997980i $$0.520237\pi$$
$$992$$ −9.00000 −0.285750
$$993$$ 0 0
$$994$$ 13.0000 0.412335
$$995$$ −13.0000 −0.412128
$$996$$ 0 0
$$997$$ 38.0000 1.20347 0.601736 0.798695i $$-0.294476\pi$$
0.601736 + 0.798695i $$0.294476\pi$$
$$998$$ 10.0000 0.316544
$$999$$ 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 378.2.a.f.1.1 yes 1
3.2 odd 2 378.2.a.c.1.1 1
4.3 odd 2 3024.2.a.t.1.1 1
5.4 even 2 9450.2.a.bx.1.1 1
7.6 odd 2 2646.2.a.v.1.1 1
9.2 odd 6 1134.2.f.n.757.1 2
9.4 even 3 1134.2.f.c.379.1 2
9.5 odd 6 1134.2.f.n.379.1 2
9.7 even 3 1134.2.f.c.757.1 2
12.11 even 2 3024.2.a.m.1.1 1
15.14 odd 2 9450.2.a.dc.1.1 1
21.20 even 2 2646.2.a.i.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
378.2.a.c.1.1 1 3.2 odd 2
378.2.a.f.1.1 yes 1 1.1 even 1 trivial
1134.2.f.c.379.1 2 9.4 even 3
1134.2.f.c.757.1 2 9.7 even 3
1134.2.f.n.379.1 2 9.5 odd 6
1134.2.f.n.757.1 2 9.2 odd 6
2646.2.a.i.1.1 1 21.20 even 2
2646.2.a.v.1.1 1 7.6 odd 2
3024.2.a.m.1.1 1 12.11 even 2
3024.2.a.t.1.1 1 4.3 odd 2
9450.2.a.bx.1.1 1 5.4 even 2
9450.2.a.dc.1.1 1 15.14 odd 2