# Properties

 Label 378.2.a.c.1.1 Level $378$ Weight $2$ Character 378.1 Self dual yes Analytic conductor $3.018$ Analytic rank $1$ 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: $$1$$ 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 $$-0.571783\pi$$
−0.223607 + 0.974679i $$0.571783\pi$$
$$6$$ 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 $$-0.771771\pi$$
−0.753778 + 0.657129i $$0.771771\pi$$
$$12$$ 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 $$-0.577979\pi$$
−0.242536 + 0.970143i $$0.577979\pi$$
$$18$$ 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 $$-0.466753\pi$$
0.104257 + 0.994550i $$0.466753\pi$$
$$24$$ 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 $$-0.621119\pi$$
−0.371391 + 0.928477i $$0.621119\pi$$
$$30$$ 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 $$-0.251941\pi$$
0.702782 + 0.711405i $$0.251941\pi$$
$$42$$ 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 $$-0.644170\pi$$
−0.437595 + 0.899172i $$0.644170\pi$$
$$48$$ 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 $$-0.808354\pi$$
−0.824163 + 0.566352i $$0.808354\pi$$
$$54$$ 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 $$-0.135063\pi$$
0.911322 + 0.411693i $$0.135063\pi$$
$$60$$ 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 $$-0.219553\pi$$
0.771408 + 0.636341i $$0.219553\pi$$
$$72$$ 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 $$-0.429548\pi$$
0.219529 + 0.975606i $$0.429548\pi$$
$$84$$ 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 $$-0.341725\pi$$
0.476999 + 0.878904i $$0.341725\pi$$
$$90$$ 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 $$-0.254728\pi$$
0.696526 + 0.717532i $$0.254728\pi$$
$$102$$ 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 $$-0.696964\pi$$
−0.580042 + 0.814587i $$0.696964\pi$$
$$108$$ 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 $$-0.470012\pi$$
0.0940721 + 0.995565i $$0.470012\pi$$
$$114$$ 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 $$-0.910895\pi$$
−0.961074 + 0.276289i $$0.910895\pi$$
$$132$$ 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 $$-0.739537\pi$$
−0.683486 + 0.729964i $$0.739537\pi$$
$$138$$ 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 $$-0.579041\pi$$
−0.245770 + 0.969328i $$0.579041\pi$$
$$150$$ 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 $$-0.626458\pi$$
−0.386912 + 0.922117i $$0.626458\pi$$
$$168$$ 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 $$-0.585735\pi$$
−0.266100 + 0.963945i $$0.585735\pi$$
$$174$$ 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 $$-0.854202\pi$$
−0.896922 + 0.442189i $$0.854202\pi$$
$$180$$ 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 $$-0.465384\pi$$
0.108536 + 0.994092i $$0.465384\pi$$
$$192$$ 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 $$-0.384060\pi$$
0.356235 + 0.934396i $$0.384060\pi$$
$$198$$ 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 $$-0.563807\pi$$
−0.199117 + 0.979976i $$0.563807\pi$$
$$228$$ 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 $$-0.651644\pi$$
−0.458585 + 0.888650i $$0.651644\pi$$
$$234$$ 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 $$-0.782863\pi$$
−0.776215 + 0.630468i $$0.782863\pi$$
$$240$$ 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 $$-0.226453\pi$$
0.757433 + 0.652913i $$0.226453\pi$$
$$252$$ 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 $$-0.818685\pi$$
−0.842107 + 0.539311i $$0.818685\pi$$
$$258$$ 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 $$-0.275832\pi$$
0.647458 + 0.762101i $$0.275832\pi$$
$$264$$ 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 $$-0.629710\pi$$
−0.396312 + 0.918116i $$0.629710\pi$$
$$270$$ 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 $$-0.403589\pi$$
0.298275 + 0.954480i $$0.403589\pi$$
$$282$$ 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 $$-0.676229\pi$$
−0.525786 + 0.850617i $$0.676229\pi$$
$$294$$ 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 $$-0.427167\pi$$
0.226819 + 0.973937i $$0.427167\pi$$
$$312$$ 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 $$-0.331312\pi$$
0.505490 + 0.862832i $$0.331312\pi$$
$$318$$ 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 $$-0.474341\pi$$
0.0805242 + 0.996753i $$0.474341\pi$$
$$348$$ 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 $$-0.474560\pi$$
0.0798369 + 0.996808i $$0.474560\pi$$
$$354$$ 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 $$-0.533662\pi$$
−0.105556 + 0.994413i $$0.533662\pi$$
$$360$$ 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 $$-0.417764\pi$$
0.255488 + 0.966812i $$0.417764\pi$$
$$384$$ 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 $$-0.330747\pi$$
0.507020 + 0.861934i $$0.330747\pi$$
$$390$$ 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 $$-0.403138\pi$$
0.299626 + 0.954057i $$0.403138\pi$$
$$402$$ 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 $$-0.546820\pi$$
−0.146560 + 0.989202i $$0.546820\pi$$
$$420$$ 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 $$-0.382346\pi$$
0.361262 + 0.932464i $$0.382346\pi$$
$$432$$ 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 $$-0.584155\pi$$
−0.261313 + 0.965254i $$0.584155\pi$$
$$444$$ 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 $$-0.575825\pi$$
−0.235965 + 0.971762i $$0.575825\pi$$
$$450$$ 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 $$-0.756734\pi$$
−0.721907 + 0.691990i $$0.756734\pi$$
$$462$$ 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 $$-0.294543\pi$$
0.601568 + 0.798821i $$0.294543\pi$$
$$468$$ 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 $$-0.760973\pi$$
−0.731059 + 0.682315i $$0.760973\pi$$
$$480$$ 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 $$-0.565096\pi$$
−0.203082 + 0.979162i $$0.565096\pi$$
$$492$$ 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 $$-0.796540\pi$$
−0.802580 + 0.596544i $$0.796540\pi$$
$$504$$ 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 $$-0.630614\pi$$
−0.398918 + 0.916987i $$0.630614\pi$$
$$510$$ 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 $$-0.393430\pi$$
0.328581 + 0.944476i $$0.393430\pi$$
$$522$$ 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 $$-0.654336\pi$$
−0.466085 + 0.884740i $$0.654336\pi$$
$$558$$ 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 $$-0.473138\pi$$
0.0842900 + 0.996441i $$0.473138\pi$$
$$564$$ 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 $$-0.716467\pi$$
−0.628833 + 0.777541i $$0.716467\pi$$
$$570$$ 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 $$-0.593296\pi$$
−0.288921 + 0.957353i $$0.593296\pi$$
$$588$$ 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 $$-0.559161\pi$$
−0.184793 + 0.982777i $$0.559161\pi$$
$$594$$ 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 $$-0.314019\pi$$
0.551595 + 0.834112i $$0.314019\pi$$
$$600$$ 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 $$-0.590935\pi$$
−0.281809 + 0.959470i $$0.590935\pi$$
$$618$$ 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 $$-0.356934\pi$$
0.434474 + 0.900684i $$0.356934\pi$$
$$642$$ 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 $$-0.809160\pi$$
−0.825595 + 0.564263i $$0.809160\pi$$
$$648$$ 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 $$-0.300311\pi$$
0.586995 + 0.809590i $$0.300311\pi$$
$$654$$ 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 $$-0.607424\pi$$
−0.331113 + 0.943591i $$0.607424\pi$$
$$660$$ 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 $$-0.673639\pi$$
−0.518847 + 0.854867i $$0.673639\pi$$
$$678$$ 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 $$-0.444916\pi$$
0.172188 + 0.985064i $$0.444916\pi$$
$$684$$ 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 $$-0.376717\pi$$
0.377695 + 0.925930i $$0.376717\pi$$
$$702$$ 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 $$-0.311029\pi$$
0.559406 + 0.828894i $$0.311029\pi$$
$$720$$ 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 $$-0.374130\pi$$
0.385208 + 0.922830i $$0.374130\pi$$
$$744$$ 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 $$-0.369444\pi$$
0.398750 + 0.917060i $$0.369444\pi$$
$$762$$ 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 $$-0.611000\pi$$
−0.341691 + 0.939812i $$0.611000\pi$$
$$774$$ 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 $$-0.301306\pi$$
0.584460 + 0.811423i $$0.301306\pi$$
$$798$$ 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 $$-0.385644\pi$$
0.351581 + 0.936157i $$0.385644\pi$$
$$810$$ 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 $$-0.444169\pi$$
0.174501 + 0.984657i $$0.444169\pi$$
$$822$$ 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 $$-0.305484\pi$$
0.573761 + 0.819023i $$0.305484\pi$$
$$828$$ 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 $$-0.478004\pi$$
0.0690477 + 0.997613i $$0.478004\pi$$
$$840$$ 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 $$-0.516317\pi$$
−0.0512390 + 0.998686i $$0.516317\pi$$
$$858$$ 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 $$-0.587792\pi$$
−0.272323 + 0.962206i $$0.587792\pi$$
$$864$$ 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 $$-0.418688\pi$$
0.252681 + 0.967550i $$0.418688\pi$$
$$882$$ 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 $$-0.706579\pi$$
−0.604381 + 0.796696i $$0.706579\pi$$
$$888$$ 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 $$-0.369851\pi$$
0.397578 + 0.917568i $$0.369851\pi$$
$$912$$ 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 $$-0.258055\pi$$
0.688988 + 0.724773i $$0.258055\pi$$
$$930$$ 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 $$-0.567963\pi$$
−0.211894 + 0.977293i $$0.567963\pi$$
$$942$$ 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 $$-0.410921\pi$$
0.276213 + 0.961096i $$0.410921\pi$$
$$948$$ 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 $$-0.752505\pi$$
−0.712650 + 0.701520i $$0.752505\pi$$
$$954$$ 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 $$-0.696029\pi$$
−0.577647 + 0.816286i $$0.696029\pi$$
$$972$$ 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 $$-0.909421\pi$$
−0.959785 + 0.280736i $$0.909421\pi$$
$$978$$ 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 $$-0.341208\pi$$
0.478426 + 0.878128i $$0.341208\pi$$
$$984$$ 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.c.1.1 1
3.2 odd 2 378.2.a.f.1.1 yes 1
4.3 odd 2 3024.2.a.m.1.1 1
5.4 even 2 9450.2.a.dc.1.1 1
7.6 odd 2 2646.2.a.i.1.1 1
9.2 odd 6 1134.2.f.c.757.1 2
9.4 even 3 1134.2.f.n.379.1 2
9.5 odd 6 1134.2.f.c.379.1 2
9.7 even 3 1134.2.f.n.757.1 2
12.11 even 2 3024.2.a.t.1.1 1
15.14 odd 2 9450.2.a.bx.1.1 1
21.20 even 2 2646.2.a.v.1.1 1

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