# Properties

 Label 1950.2.a.ba.1.1 Level $1950$ Weight $2$ Character 1950.1 Self dual yes Analytic conductor $15.571$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Learn more

## Newspace parameters

 Level: $$N$$ $$=$$ $$1950 = 2 \cdot 3 \cdot 5^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1950.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$15.5708283941$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 390) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 1950.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +2.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +2.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +4.00000 q^{11} +1.00000 q^{12} +1.00000 q^{13} +2.00000 q^{14} +1.00000 q^{16} -4.00000 q^{17} +1.00000 q^{18} -2.00000 q^{19} +2.00000 q^{21} +4.00000 q^{22} -2.00000 q^{23} +1.00000 q^{24} +1.00000 q^{26} +1.00000 q^{27} +2.00000 q^{28} +8.00000 q^{29} +4.00000 q^{31} +1.00000 q^{32} +4.00000 q^{33} -4.00000 q^{34} +1.00000 q^{36} -6.00000 q^{37} -2.00000 q^{38} +1.00000 q^{39} +10.0000 q^{41} +2.00000 q^{42} -4.00000 q^{43} +4.00000 q^{44} -2.00000 q^{46} +1.00000 q^{48} -3.00000 q^{49} -4.00000 q^{51} +1.00000 q^{52} -6.00000 q^{53} +1.00000 q^{54} +2.00000 q^{56} -2.00000 q^{57} +8.00000 q^{58} -12.0000 q^{59} -2.00000 q^{61} +4.00000 q^{62} +2.00000 q^{63} +1.00000 q^{64} +4.00000 q^{66} +8.00000 q^{67} -4.00000 q^{68} -2.00000 q^{69} +1.00000 q^{72} -6.00000 q^{74} -2.00000 q^{76} +8.00000 q^{77} +1.00000 q^{78} -8.00000 q^{79} +1.00000 q^{81} +10.0000 q^{82} +12.0000 q^{83} +2.00000 q^{84} -4.00000 q^{86} +8.00000 q^{87} +4.00000 q^{88} -10.0000 q^{89} +2.00000 q^{91} -2.00000 q^{92} +4.00000 q^{93} +1.00000 q^{96} +8.00000 q^{97} -3.00000 q^{98} +4.00000 q^{99} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.00000 0.707107
$$3$$ 1.00000 0.577350
$$4$$ 1.00000 0.500000
$$5$$ 0 0
$$6$$ 1.00000 0.408248
$$7$$ 2.00000 0.755929 0.377964 0.925820i $$-0.376624\pi$$
0.377964 + 0.925820i $$0.376624\pi$$
$$8$$ 1.00000 0.353553
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ 4.00000 1.20605 0.603023 0.797724i $$-0.293963\pi$$
0.603023 + 0.797724i $$0.293963\pi$$
$$12$$ 1.00000 0.288675
$$13$$ 1.00000 0.277350
$$14$$ 2.00000 0.534522
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ −4.00000 −0.970143 −0.485071 0.874475i $$-0.661206\pi$$
−0.485071 + 0.874475i $$0.661206\pi$$
$$18$$ 1.00000 0.235702
$$19$$ −2.00000 −0.458831 −0.229416 0.973329i $$-0.573682\pi$$
−0.229416 + 0.973329i $$0.573682\pi$$
$$20$$ 0 0
$$21$$ 2.00000 0.436436
$$22$$ 4.00000 0.852803
$$23$$ −2.00000 −0.417029 −0.208514 0.978019i $$-0.566863\pi$$
−0.208514 + 0.978019i $$0.566863\pi$$
$$24$$ 1.00000 0.204124
$$25$$ 0 0
$$26$$ 1.00000 0.196116
$$27$$ 1.00000 0.192450
$$28$$ 2.00000 0.377964
$$29$$ 8.00000 1.48556 0.742781 0.669534i $$-0.233506\pi$$
0.742781 + 0.669534i $$0.233506\pi$$
$$30$$ 0 0
$$31$$ 4.00000 0.718421 0.359211 0.933257i $$-0.383046\pi$$
0.359211 + 0.933257i $$0.383046\pi$$
$$32$$ 1.00000 0.176777
$$33$$ 4.00000 0.696311
$$34$$ −4.00000 −0.685994
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ −6.00000 −0.986394 −0.493197 0.869918i $$-0.664172\pi$$
−0.493197 + 0.869918i $$0.664172\pi$$
$$38$$ −2.00000 −0.324443
$$39$$ 1.00000 0.160128
$$40$$ 0 0
$$41$$ 10.0000 1.56174 0.780869 0.624695i $$-0.214777\pi$$
0.780869 + 0.624695i $$0.214777\pi$$
$$42$$ 2.00000 0.308607
$$43$$ −4.00000 −0.609994 −0.304997 0.952353i $$-0.598656\pi$$
−0.304997 + 0.952353i $$0.598656\pi$$
$$44$$ 4.00000 0.603023
$$45$$ 0 0
$$46$$ −2.00000 −0.294884
$$47$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$48$$ 1.00000 0.144338
$$49$$ −3.00000 −0.428571
$$50$$ 0 0
$$51$$ −4.00000 −0.560112
$$52$$ 1.00000 0.138675
$$53$$ −6.00000 −0.824163 −0.412082 0.911147i $$-0.635198\pi$$
−0.412082 + 0.911147i $$0.635198\pi$$
$$54$$ 1.00000 0.136083
$$55$$ 0 0
$$56$$ 2.00000 0.267261
$$57$$ −2.00000 −0.264906
$$58$$ 8.00000 1.05045
$$59$$ −12.0000 −1.56227 −0.781133 0.624364i $$-0.785358\pi$$
−0.781133 + 0.624364i $$0.785358\pi$$
$$60$$ 0 0
$$61$$ −2.00000 −0.256074 −0.128037 0.991769i $$-0.540868\pi$$
−0.128037 + 0.991769i $$0.540868\pi$$
$$62$$ 4.00000 0.508001
$$63$$ 2.00000 0.251976
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ 4.00000 0.492366
$$67$$ 8.00000 0.977356 0.488678 0.872464i $$-0.337479\pi$$
0.488678 + 0.872464i $$0.337479\pi$$
$$68$$ −4.00000 −0.485071
$$69$$ −2.00000 −0.240772
$$70$$ 0 0
$$71$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$72$$ 1.00000 0.117851
$$73$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$74$$ −6.00000 −0.697486
$$75$$ 0 0
$$76$$ −2.00000 −0.229416
$$77$$ 8.00000 0.911685
$$78$$ 1.00000 0.113228
$$79$$ −8.00000 −0.900070 −0.450035 0.893011i $$-0.648589\pi$$
−0.450035 + 0.893011i $$0.648589\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 10.0000 1.10432
$$83$$ 12.0000 1.31717 0.658586 0.752506i $$-0.271155\pi$$
0.658586 + 0.752506i $$0.271155\pi$$
$$84$$ 2.00000 0.218218
$$85$$ 0 0
$$86$$ −4.00000 −0.431331
$$87$$ 8.00000 0.857690
$$88$$ 4.00000 0.426401
$$89$$ −10.0000 −1.06000 −0.529999 0.847998i $$-0.677808\pi$$
−0.529999 + 0.847998i $$0.677808\pi$$
$$90$$ 0 0
$$91$$ 2.00000 0.209657
$$92$$ −2.00000 −0.208514
$$93$$ 4.00000 0.414781
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 1.00000 0.102062
$$97$$ 8.00000 0.812277 0.406138 0.913812i $$-0.366875\pi$$
0.406138 + 0.913812i $$0.366875\pi$$
$$98$$ −3.00000 −0.303046
$$99$$ 4.00000 0.402015
$$100$$ 0 0
$$101$$ 20.0000 1.99007 0.995037 0.0995037i $$-0.0317255\pi$$
0.995037 + 0.0995037i $$0.0317255\pi$$
$$102$$ −4.00000 −0.396059
$$103$$ −4.00000 −0.394132 −0.197066 0.980390i $$-0.563141\pi$$
−0.197066 + 0.980390i $$0.563141\pi$$
$$104$$ 1.00000 0.0980581
$$105$$ 0 0
$$106$$ −6.00000 −0.582772
$$107$$ −4.00000 −0.386695 −0.193347 0.981130i $$-0.561934\pi$$
−0.193347 + 0.981130i $$0.561934\pi$$
$$108$$ 1.00000 0.0962250
$$109$$ 4.00000 0.383131 0.191565 0.981480i $$-0.438644\pi$$
0.191565 + 0.981480i $$0.438644\pi$$
$$110$$ 0 0
$$111$$ −6.00000 −0.569495
$$112$$ 2.00000 0.188982
$$113$$ 16.0000 1.50515 0.752577 0.658505i $$-0.228811\pi$$
0.752577 + 0.658505i $$0.228811\pi$$
$$114$$ −2.00000 −0.187317
$$115$$ 0 0
$$116$$ 8.00000 0.742781
$$117$$ 1.00000 0.0924500
$$118$$ −12.0000 −1.10469
$$119$$ −8.00000 −0.733359
$$120$$ 0 0
$$121$$ 5.00000 0.454545
$$122$$ −2.00000 −0.181071
$$123$$ 10.0000 0.901670
$$124$$ 4.00000 0.359211
$$125$$ 0 0
$$126$$ 2.00000 0.178174
$$127$$ −20.0000 −1.77471 −0.887357 0.461084i $$-0.847461\pi$$
−0.887357 + 0.461084i $$0.847461\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ −4.00000 −0.352180
$$130$$ 0 0
$$131$$ 14.0000 1.22319 0.611593 0.791173i $$-0.290529\pi$$
0.611593 + 0.791173i $$0.290529\pi$$
$$132$$ 4.00000 0.348155
$$133$$ −4.00000 −0.346844
$$134$$ 8.00000 0.691095
$$135$$ 0 0
$$136$$ −4.00000 −0.342997
$$137$$ 2.00000 0.170872 0.0854358 0.996344i $$-0.472772\pi$$
0.0854358 + 0.996344i $$0.472772\pi$$
$$138$$ −2.00000 −0.170251
$$139$$ −16.0000 −1.35710 −0.678551 0.734553i $$-0.737392\pi$$
−0.678551 + 0.734553i $$0.737392\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 4.00000 0.334497
$$144$$ 1.00000 0.0833333
$$145$$ 0 0
$$146$$ 0 0
$$147$$ −3.00000 −0.247436
$$148$$ −6.00000 −0.493197
$$149$$ −22.0000 −1.80231 −0.901155 0.433497i $$-0.857280\pi$$
−0.901155 + 0.433497i $$0.857280\pi$$
$$150$$ 0 0
$$151$$ −4.00000 −0.325515 −0.162758 0.986666i $$-0.552039\pi$$
−0.162758 + 0.986666i $$0.552039\pi$$
$$152$$ −2.00000 −0.162221
$$153$$ −4.00000 −0.323381
$$154$$ 8.00000 0.644658
$$155$$ 0 0
$$156$$ 1.00000 0.0800641
$$157$$ 10.0000 0.798087 0.399043 0.916932i $$-0.369342\pi$$
0.399043 + 0.916932i $$0.369342\pi$$
$$158$$ −8.00000 −0.636446
$$159$$ −6.00000 −0.475831
$$160$$ 0 0
$$161$$ −4.00000 −0.315244
$$162$$ 1.00000 0.0785674
$$163$$ −12.0000 −0.939913 −0.469956 0.882690i $$-0.655730\pi$$
−0.469956 + 0.882690i $$0.655730\pi$$
$$164$$ 10.0000 0.780869
$$165$$ 0 0
$$166$$ 12.0000 0.931381
$$167$$ 12.0000 0.928588 0.464294 0.885681i $$-0.346308\pi$$
0.464294 + 0.885681i $$0.346308\pi$$
$$168$$ 2.00000 0.154303
$$169$$ 1.00000 0.0769231
$$170$$ 0 0
$$171$$ −2.00000 −0.152944
$$172$$ −4.00000 −0.304997
$$173$$ −14.0000 −1.06440 −0.532200 0.846619i $$-0.678635\pi$$
−0.532200 + 0.846619i $$0.678635\pi$$
$$174$$ 8.00000 0.606478
$$175$$ 0 0
$$176$$ 4.00000 0.301511
$$177$$ −12.0000 −0.901975
$$178$$ −10.0000 −0.749532
$$179$$ 2.00000 0.149487 0.0747435 0.997203i $$-0.476186\pi$$
0.0747435 + 0.997203i $$0.476186\pi$$
$$180$$ 0 0
$$181$$ −22.0000 −1.63525 −0.817624 0.575753i $$-0.804709\pi$$
−0.817624 + 0.575753i $$0.804709\pi$$
$$182$$ 2.00000 0.148250
$$183$$ −2.00000 −0.147844
$$184$$ −2.00000 −0.147442
$$185$$ 0 0
$$186$$ 4.00000 0.293294
$$187$$ −16.0000 −1.17004
$$188$$ 0 0
$$189$$ 2.00000 0.145479
$$190$$ 0 0
$$191$$ −8.00000 −0.578860 −0.289430 0.957199i $$-0.593466\pi$$
−0.289430 + 0.957199i $$0.593466\pi$$
$$192$$ 1.00000 0.0721688
$$193$$ −12.0000 −0.863779 −0.431889 0.901927i $$-0.642153\pi$$
−0.431889 + 0.901927i $$0.642153\pi$$
$$194$$ 8.00000 0.574367
$$195$$ 0 0
$$196$$ −3.00000 −0.214286
$$197$$ −10.0000 −0.712470 −0.356235 0.934396i $$-0.615940\pi$$
−0.356235 + 0.934396i $$0.615940\pi$$
$$198$$ 4.00000 0.284268
$$199$$ 24.0000 1.70131 0.850657 0.525720i $$-0.176204\pi$$
0.850657 + 0.525720i $$0.176204\pi$$
$$200$$ 0 0
$$201$$ 8.00000 0.564276
$$202$$ 20.0000 1.40720
$$203$$ 16.0000 1.12298
$$204$$ −4.00000 −0.280056
$$205$$ 0 0
$$206$$ −4.00000 −0.278693
$$207$$ −2.00000 −0.139010
$$208$$ 1.00000 0.0693375
$$209$$ −8.00000 −0.553372
$$210$$ 0 0
$$211$$ −4.00000 −0.275371 −0.137686 0.990476i $$-0.543966\pi$$
−0.137686 + 0.990476i $$0.543966\pi$$
$$212$$ −6.00000 −0.412082
$$213$$ 0 0
$$214$$ −4.00000 −0.273434
$$215$$ 0 0
$$216$$ 1.00000 0.0680414
$$217$$ 8.00000 0.543075
$$218$$ 4.00000 0.270914
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −4.00000 −0.269069
$$222$$ −6.00000 −0.402694
$$223$$ 22.0000 1.47323 0.736614 0.676313i $$-0.236423\pi$$
0.736614 + 0.676313i $$0.236423\pi$$
$$224$$ 2.00000 0.133631
$$225$$ 0 0
$$226$$ 16.0000 1.06430
$$227$$ −12.0000 −0.796468 −0.398234 0.917284i $$-0.630377\pi$$
−0.398234 + 0.917284i $$0.630377\pi$$
$$228$$ −2.00000 −0.132453
$$229$$ 28.0000 1.85029 0.925146 0.379611i $$-0.123942\pi$$
0.925146 + 0.379611i $$0.123942\pi$$
$$230$$ 0 0
$$231$$ 8.00000 0.526361
$$232$$ 8.00000 0.525226
$$233$$ −28.0000 −1.83434 −0.917170 0.398495i $$-0.869533\pi$$
−0.917170 + 0.398495i $$0.869533\pi$$
$$234$$ 1.00000 0.0653720
$$235$$ 0 0
$$236$$ −12.0000 −0.781133
$$237$$ −8.00000 −0.519656
$$238$$ −8.00000 −0.518563
$$239$$ −8.00000 −0.517477 −0.258738 0.965947i $$-0.583307\pi$$
−0.258738 + 0.965947i $$0.583307\pi$$
$$240$$ 0 0
$$241$$ −22.0000 −1.41714 −0.708572 0.705638i $$-0.750660\pi$$
−0.708572 + 0.705638i $$0.750660\pi$$
$$242$$ 5.00000 0.321412
$$243$$ 1.00000 0.0641500
$$244$$ −2.00000 −0.128037
$$245$$ 0 0
$$246$$ 10.0000 0.637577
$$247$$ −2.00000 −0.127257
$$248$$ 4.00000 0.254000
$$249$$ 12.0000 0.760469
$$250$$ 0 0
$$251$$ 10.0000 0.631194 0.315597 0.948893i $$-0.397795\pi$$
0.315597 + 0.948893i $$0.397795\pi$$
$$252$$ 2.00000 0.125988
$$253$$ −8.00000 −0.502956
$$254$$ −20.0000 −1.25491
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 8.00000 0.499026 0.249513 0.968371i $$-0.419729\pi$$
0.249513 + 0.968371i $$0.419729\pi$$
$$258$$ −4.00000 −0.249029
$$259$$ −12.0000 −0.745644
$$260$$ 0 0
$$261$$ 8.00000 0.495188
$$262$$ 14.0000 0.864923
$$263$$ 18.0000 1.10993 0.554964 0.831875i $$-0.312732\pi$$
0.554964 + 0.831875i $$0.312732\pi$$
$$264$$ 4.00000 0.246183
$$265$$ 0 0
$$266$$ −4.00000 −0.245256
$$267$$ −10.0000 −0.611990
$$268$$ 8.00000 0.488678
$$269$$ −4.00000 −0.243884 −0.121942 0.992537i $$-0.538912\pi$$
−0.121942 + 0.992537i $$0.538912\pi$$
$$270$$ 0 0
$$271$$ −28.0000 −1.70088 −0.850439 0.526073i $$-0.823664\pi$$
−0.850439 + 0.526073i $$0.823664\pi$$
$$272$$ −4.00000 −0.242536
$$273$$ 2.00000 0.121046
$$274$$ 2.00000 0.120824
$$275$$ 0 0
$$276$$ −2.00000 −0.120386
$$277$$ −22.0000 −1.32185 −0.660926 0.750451i $$-0.729836\pi$$
−0.660926 + 0.750451i $$0.729836\pi$$
$$278$$ −16.0000 −0.959616
$$279$$ 4.00000 0.239474
$$280$$ 0 0
$$281$$ −10.0000 −0.596550 −0.298275 0.954480i $$-0.596411\pi$$
−0.298275 + 0.954480i $$0.596411\pi$$
$$282$$ 0 0
$$283$$ −28.0000 −1.66443 −0.832214 0.554455i $$-0.812927\pi$$
−0.832214 + 0.554455i $$0.812927\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 4.00000 0.236525
$$287$$ 20.0000 1.18056
$$288$$ 1.00000 0.0589256
$$289$$ −1.00000 −0.0588235
$$290$$ 0 0
$$291$$ 8.00000 0.468968
$$292$$ 0 0
$$293$$ 6.00000 0.350524 0.175262 0.984522i $$-0.443923\pi$$
0.175262 + 0.984522i $$0.443923\pi$$
$$294$$ −3.00000 −0.174964
$$295$$ 0 0
$$296$$ −6.00000 −0.348743
$$297$$ 4.00000 0.232104
$$298$$ −22.0000 −1.27443
$$299$$ −2.00000 −0.115663
$$300$$ 0 0
$$301$$ −8.00000 −0.461112
$$302$$ −4.00000 −0.230174
$$303$$ 20.0000 1.14897
$$304$$ −2.00000 −0.114708
$$305$$ 0 0
$$306$$ −4.00000 −0.228665
$$307$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$308$$ 8.00000 0.455842
$$309$$ −4.00000 −0.227552
$$310$$ 0 0
$$311$$ −16.0000 −0.907277 −0.453638 0.891186i $$-0.649874\pi$$
−0.453638 + 0.891186i $$0.649874\pi$$
$$312$$ 1.00000 0.0566139
$$313$$ 10.0000 0.565233 0.282617 0.959233i $$-0.408798\pi$$
0.282617 + 0.959233i $$0.408798\pi$$
$$314$$ 10.0000 0.564333
$$315$$ 0 0
$$316$$ −8.00000 −0.450035
$$317$$ 2.00000 0.112331 0.0561656 0.998421i $$-0.482113\pi$$
0.0561656 + 0.998421i $$0.482113\pi$$
$$318$$ −6.00000 −0.336463
$$319$$ 32.0000 1.79166
$$320$$ 0 0
$$321$$ −4.00000 −0.223258
$$322$$ −4.00000 −0.222911
$$323$$ 8.00000 0.445132
$$324$$ 1.00000 0.0555556
$$325$$ 0 0
$$326$$ −12.0000 −0.664619
$$327$$ 4.00000 0.221201
$$328$$ 10.0000 0.552158
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −10.0000 −0.549650 −0.274825 0.961494i $$-0.588620\pi$$
−0.274825 + 0.961494i $$0.588620\pi$$
$$332$$ 12.0000 0.658586
$$333$$ −6.00000 −0.328798
$$334$$ 12.0000 0.656611
$$335$$ 0 0
$$336$$ 2.00000 0.109109
$$337$$ 26.0000 1.41631 0.708155 0.706057i $$-0.249528\pi$$
0.708155 + 0.706057i $$0.249528\pi$$
$$338$$ 1.00000 0.0543928
$$339$$ 16.0000 0.869001
$$340$$ 0 0
$$341$$ 16.0000 0.866449
$$342$$ −2.00000 −0.108148
$$343$$ −20.0000 −1.07990
$$344$$ −4.00000 −0.215666
$$345$$ 0 0
$$346$$ −14.0000 −0.752645
$$347$$ −16.0000 −0.858925 −0.429463 0.903085i $$-0.641297\pi$$
−0.429463 + 0.903085i $$0.641297\pi$$
$$348$$ 8.00000 0.428845
$$349$$ −28.0000 −1.49881 −0.749403 0.662114i $$-0.769659\pi$$
−0.749403 + 0.662114i $$0.769659\pi$$
$$350$$ 0 0
$$351$$ 1.00000 0.0533761
$$352$$ 4.00000 0.213201
$$353$$ −6.00000 −0.319348 −0.159674 0.987170i $$-0.551044\pi$$
−0.159674 + 0.987170i $$0.551044\pi$$
$$354$$ −12.0000 −0.637793
$$355$$ 0 0
$$356$$ −10.0000 −0.529999
$$357$$ −8.00000 −0.423405
$$358$$ 2.00000 0.105703
$$359$$ 24.0000 1.26667 0.633336 0.773877i $$-0.281685\pi$$
0.633336 + 0.773877i $$0.281685\pi$$
$$360$$ 0 0
$$361$$ −15.0000 −0.789474
$$362$$ −22.0000 −1.15629
$$363$$ 5.00000 0.262432
$$364$$ 2.00000 0.104828
$$365$$ 0 0
$$366$$ −2.00000 −0.104542
$$367$$ 4.00000 0.208798 0.104399 0.994535i $$-0.466708\pi$$
0.104399 + 0.994535i $$0.466708\pi$$
$$368$$ −2.00000 −0.104257
$$369$$ 10.0000 0.520579
$$370$$ 0 0
$$371$$ −12.0000 −0.623009
$$372$$ 4.00000 0.207390
$$373$$ −14.0000 −0.724893 −0.362446 0.932005i $$-0.618058\pi$$
−0.362446 + 0.932005i $$0.618058\pi$$
$$374$$ −16.0000 −0.827340
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 8.00000 0.412021
$$378$$ 2.00000 0.102869
$$379$$ −14.0000 −0.719132 −0.359566 0.933120i $$-0.617075\pi$$
−0.359566 + 0.933120i $$0.617075\pi$$
$$380$$ 0 0
$$381$$ −20.0000 −1.02463
$$382$$ −8.00000 −0.409316
$$383$$ 20.0000 1.02195 0.510976 0.859595i $$-0.329284\pi$$
0.510976 + 0.859595i $$0.329284\pi$$
$$384$$ 1.00000 0.0510310
$$385$$ 0 0
$$386$$ −12.0000 −0.610784
$$387$$ −4.00000 −0.203331
$$388$$ 8.00000 0.406138
$$389$$ 20.0000 1.01404 0.507020 0.861934i $$-0.330747\pi$$
0.507020 + 0.861934i $$0.330747\pi$$
$$390$$ 0 0
$$391$$ 8.00000 0.404577
$$392$$ −3.00000 −0.151523
$$393$$ 14.0000 0.706207
$$394$$ −10.0000 −0.503793
$$395$$ 0 0
$$396$$ 4.00000 0.201008
$$397$$ −6.00000 −0.301131 −0.150566 0.988600i $$-0.548110\pi$$
−0.150566 + 0.988600i $$0.548110\pi$$
$$398$$ 24.0000 1.20301
$$399$$ −4.00000 −0.200250
$$400$$ 0 0
$$401$$ −30.0000 −1.49813 −0.749064 0.662497i $$-0.769497\pi$$
−0.749064 + 0.662497i $$0.769497\pi$$
$$402$$ 8.00000 0.399004
$$403$$ 4.00000 0.199254
$$404$$ 20.0000 0.995037
$$405$$ 0 0
$$406$$ 16.0000 0.794067
$$407$$ −24.0000 −1.18964
$$408$$ −4.00000 −0.198030
$$409$$ −18.0000 −0.890043 −0.445021 0.895520i $$-0.646804\pi$$
−0.445021 + 0.895520i $$0.646804\pi$$
$$410$$ 0 0
$$411$$ 2.00000 0.0986527
$$412$$ −4.00000 −0.197066
$$413$$ −24.0000 −1.18096
$$414$$ −2.00000 −0.0982946
$$415$$ 0 0
$$416$$ 1.00000 0.0490290
$$417$$ −16.0000 −0.783523
$$418$$ −8.00000 −0.391293
$$419$$ 14.0000 0.683945 0.341972 0.939710i $$-0.388905\pi$$
0.341972 + 0.939710i $$0.388905\pi$$
$$420$$ 0 0
$$421$$ −16.0000 −0.779792 −0.389896 0.920859i $$-0.627489\pi$$
−0.389896 + 0.920859i $$0.627489\pi$$
$$422$$ −4.00000 −0.194717
$$423$$ 0 0
$$424$$ −6.00000 −0.291386
$$425$$ 0 0
$$426$$ 0 0
$$427$$ −4.00000 −0.193574
$$428$$ −4.00000 −0.193347
$$429$$ 4.00000 0.193122
$$430$$ 0 0
$$431$$ 16.0000 0.770693 0.385346 0.922772i $$-0.374082\pi$$
0.385346 + 0.922772i $$0.374082\pi$$
$$432$$ 1.00000 0.0481125
$$433$$ 26.0000 1.24948 0.624740 0.780833i $$-0.285205\pi$$
0.624740 + 0.780833i $$0.285205\pi$$
$$434$$ 8.00000 0.384012
$$435$$ 0 0
$$436$$ 4.00000 0.191565
$$437$$ 4.00000 0.191346
$$438$$ 0 0
$$439$$ 8.00000 0.381819 0.190910 0.981608i $$-0.438856\pi$$
0.190910 + 0.981608i $$0.438856\pi$$
$$440$$ 0 0
$$441$$ −3.00000 −0.142857
$$442$$ −4.00000 −0.190261
$$443$$ 16.0000 0.760183 0.380091 0.924949i $$-0.375893\pi$$
0.380091 + 0.924949i $$0.375893\pi$$
$$444$$ −6.00000 −0.284747
$$445$$ 0 0
$$446$$ 22.0000 1.04173
$$447$$ −22.0000 −1.04056
$$448$$ 2.00000 0.0944911
$$449$$ −18.0000 −0.849473 −0.424736 0.905317i $$-0.639633\pi$$
−0.424736 + 0.905317i $$0.639633\pi$$
$$450$$ 0 0
$$451$$ 40.0000 1.88353
$$452$$ 16.0000 0.752577
$$453$$ −4.00000 −0.187936
$$454$$ −12.0000 −0.563188
$$455$$ 0 0
$$456$$ −2.00000 −0.0936586
$$457$$ 16.0000 0.748448 0.374224 0.927338i $$-0.377909\pi$$
0.374224 + 0.927338i $$0.377909\pi$$
$$458$$ 28.0000 1.30835
$$459$$ −4.00000 −0.186704
$$460$$ 0 0
$$461$$ 30.0000 1.39724 0.698620 0.715493i $$-0.253798\pi$$
0.698620 + 0.715493i $$0.253798\pi$$
$$462$$ 8.00000 0.372194
$$463$$ −2.00000 −0.0929479 −0.0464739 0.998920i $$-0.514798\pi$$
−0.0464739 + 0.998920i $$0.514798\pi$$
$$464$$ 8.00000 0.371391
$$465$$ 0 0
$$466$$ −28.0000 −1.29707
$$467$$ −12.0000 −0.555294 −0.277647 0.960683i $$-0.589555\pi$$
−0.277647 + 0.960683i $$0.589555\pi$$
$$468$$ 1.00000 0.0462250
$$469$$ 16.0000 0.738811
$$470$$ 0 0
$$471$$ 10.0000 0.460776
$$472$$ −12.0000 −0.552345
$$473$$ −16.0000 −0.735681
$$474$$ −8.00000 −0.367452
$$475$$ 0 0
$$476$$ −8.00000 −0.366679
$$477$$ −6.00000 −0.274721
$$478$$ −8.00000 −0.365911
$$479$$ 24.0000 1.09659 0.548294 0.836286i $$-0.315277\pi$$
0.548294 + 0.836286i $$0.315277\pi$$
$$480$$ 0 0
$$481$$ −6.00000 −0.273576
$$482$$ −22.0000 −1.00207
$$483$$ −4.00000 −0.182006
$$484$$ 5.00000 0.227273
$$485$$ 0 0
$$486$$ 1.00000 0.0453609
$$487$$ −38.0000 −1.72194 −0.860972 0.508652i $$-0.830144\pi$$
−0.860972 + 0.508652i $$0.830144\pi$$
$$488$$ −2.00000 −0.0905357
$$489$$ −12.0000 −0.542659
$$490$$ 0 0
$$491$$ 38.0000 1.71492 0.857458 0.514554i $$-0.172042\pi$$
0.857458 + 0.514554i $$0.172042\pi$$
$$492$$ 10.0000 0.450835
$$493$$ −32.0000 −1.44121
$$494$$ −2.00000 −0.0899843
$$495$$ 0 0
$$496$$ 4.00000 0.179605
$$497$$ 0 0
$$498$$ 12.0000 0.537733
$$499$$ −14.0000 −0.626726 −0.313363 0.949633i $$-0.601456\pi$$
−0.313363 + 0.949633i $$0.601456\pi$$
$$500$$ 0 0
$$501$$ 12.0000 0.536120
$$502$$ 10.0000 0.446322
$$503$$ 42.0000 1.87269 0.936344 0.351085i $$-0.114187\pi$$
0.936344 + 0.351085i $$0.114187\pi$$
$$504$$ 2.00000 0.0890871
$$505$$ 0 0
$$506$$ −8.00000 −0.355643
$$507$$ 1.00000 0.0444116
$$508$$ −20.0000 −0.887357
$$509$$ 38.0000 1.68432 0.842160 0.539227i $$-0.181284\pi$$
0.842160 + 0.539227i $$0.181284\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 1.00000 0.0441942
$$513$$ −2.00000 −0.0883022
$$514$$ 8.00000 0.352865
$$515$$ 0 0
$$516$$ −4.00000 −0.176090
$$517$$ 0 0
$$518$$ −12.0000 −0.527250
$$519$$ −14.0000 −0.614532
$$520$$ 0 0
$$521$$ 6.00000 0.262865 0.131432 0.991325i $$-0.458042\pi$$
0.131432 + 0.991325i $$0.458042\pi$$
$$522$$ 8.00000 0.350150
$$523$$ 4.00000 0.174908 0.0874539 0.996169i $$-0.472127\pi$$
0.0874539 + 0.996169i $$0.472127\pi$$
$$524$$ 14.0000 0.611593
$$525$$ 0 0
$$526$$ 18.0000 0.784837
$$527$$ −16.0000 −0.696971
$$528$$ 4.00000 0.174078
$$529$$ −19.0000 −0.826087
$$530$$ 0 0
$$531$$ −12.0000 −0.520756
$$532$$ −4.00000 −0.173422
$$533$$ 10.0000 0.433148
$$534$$ −10.0000 −0.432742
$$535$$ 0 0
$$536$$ 8.00000 0.345547
$$537$$ 2.00000 0.0863064
$$538$$ −4.00000 −0.172452
$$539$$ −12.0000 −0.516877
$$540$$ 0 0
$$541$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$542$$ −28.0000 −1.20270
$$543$$ −22.0000 −0.944110
$$544$$ −4.00000 −0.171499
$$545$$ 0 0
$$546$$ 2.00000 0.0855921
$$547$$ 12.0000 0.513083 0.256541 0.966533i $$-0.417417\pi$$
0.256541 + 0.966533i $$0.417417\pi$$
$$548$$ 2.00000 0.0854358
$$549$$ −2.00000 −0.0853579
$$550$$ 0 0
$$551$$ −16.0000 −0.681623
$$552$$ −2.00000 −0.0851257
$$553$$ −16.0000 −0.680389
$$554$$ −22.0000 −0.934690
$$555$$ 0 0
$$556$$ −16.0000 −0.678551
$$557$$ 10.0000 0.423714 0.211857 0.977301i $$-0.432049\pi$$
0.211857 + 0.977301i $$0.432049\pi$$
$$558$$ 4.00000 0.169334
$$559$$ −4.00000 −0.169182
$$560$$ 0 0
$$561$$ −16.0000 −0.675521
$$562$$ −10.0000 −0.421825
$$563$$ 24.0000 1.01148 0.505740 0.862686i $$-0.331220\pi$$
0.505740 + 0.862686i $$0.331220\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ −28.0000 −1.17693
$$567$$ 2.00000 0.0839921
$$568$$ 0 0
$$569$$ −10.0000 −0.419222 −0.209611 0.977785i $$-0.567220\pi$$
−0.209611 + 0.977785i $$0.567220\pi$$
$$570$$ 0 0
$$571$$ −16.0000 −0.669579 −0.334790 0.942293i $$-0.608665\pi$$
−0.334790 + 0.942293i $$0.608665\pi$$
$$572$$ 4.00000 0.167248
$$573$$ −8.00000 −0.334205
$$574$$ 20.0000 0.834784
$$575$$ 0 0
$$576$$ 1.00000 0.0416667
$$577$$ 4.00000 0.166522 0.0832611 0.996528i $$-0.473466\pi$$
0.0832611 + 0.996528i $$0.473466\pi$$
$$578$$ −1.00000 −0.0415945
$$579$$ −12.0000 −0.498703
$$580$$ 0 0
$$581$$ 24.0000 0.995688
$$582$$ 8.00000 0.331611
$$583$$ −24.0000 −0.993978
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 6.00000 0.247858
$$587$$ −12.0000 −0.495293 −0.247647 0.968850i $$-0.579657\pi$$
−0.247647 + 0.968850i $$0.579657\pi$$
$$588$$ −3.00000 −0.123718
$$589$$ −8.00000 −0.329634
$$590$$ 0 0
$$591$$ −10.0000 −0.411345
$$592$$ −6.00000 −0.246598
$$593$$ −34.0000 −1.39621 −0.698106 0.715994i $$-0.745974\pi$$
−0.698106 + 0.715994i $$0.745974\pi$$
$$594$$ 4.00000 0.164122
$$595$$ 0 0
$$596$$ −22.0000 −0.901155
$$597$$ 24.0000 0.982255
$$598$$ −2.00000 −0.0817861
$$599$$ −24.0000 −0.980613 −0.490307 0.871550i $$-0.663115\pi$$
−0.490307 + 0.871550i $$0.663115\pi$$
$$600$$ 0 0
$$601$$ −42.0000 −1.71322 −0.856608 0.515968i $$-0.827432\pi$$
−0.856608 + 0.515968i $$0.827432\pi$$
$$602$$ −8.00000 −0.326056
$$603$$ 8.00000 0.325785
$$604$$ −4.00000 −0.162758
$$605$$ 0 0
$$606$$ 20.0000 0.812444
$$607$$ 12.0000 0.487065 0.243532 0.969893i $$-0.421694\pi$$
0.243532 + 0.969893i $$0.421694\pi$$
$$608$$ −2.00000 −0.0811107
$$609$$ 16.0000 0.648353
$$610$$ 0 0
$$611$$ 0 0
$$612$$ −4.00000 −0.161690
$$613$$ 2.00000 0.0807792 0.0403896 0.999184i $$-0.487140\pi$$
0.0403896 + 0.999184i $$0.487140\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 8.00000 0.322329
$$617$$ −38.0000 −1.52982 −0.764911 0.644136i $$-0.777217\pi$$
−0.764911 + 0.644136i $$0.777217\pi$$
$$618$$ −4.00000 −0.160904
$$619$$ 26.0000 1.04503 0.522514 0.852631i $$-0.324994\pi$$
0.522514 + 0.852631i $$0.324994\pi$$
$$620$$ 0 0
$$621$$ −2.00000 −0.0802572
$$622$$ −16.0000 −0.641542
$$623$$ −20.0000 −0.801283
$$624$$ 1.00000 0.0400320
$$625$$ 0 0
$$626$$ 10.0000 0.399680
$$627$$ −8.00000 −0.319489
$$628$$ 10.0000 0.399043
$$629$$ 24.0000 0.956943
$$630$$ 0 0
$$631$$ 40.0000 1.59237 0.796187 0.605050i $$-0.206847\pi$$
0.796187 + 0.605050i $$0.206847\pi$$
$$632$$ −8.00000 −0.318223
$$633$$ −4.00000 −0.158986
$$634$$ 2.00000 0.0794301
$$635$$ 0 0
$$636$$ −6.00000 −0.237915
$$637$$ −3.00000 −0.118864
$$638$$ 32.0000 1.26689
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 18.0000 0.710957 0.355479 0.934684i $$-0.384318\pi$$
0.355479 + 0.934684i $$0.384318\pi$$
$$642$$ −4.00000 −0.157867
$$643$$ 40.0000 1.57745 0.788723 0.614749i $$-0.210743\pi$$
0.788723 + 0.614749i $$0.210743\pi$$
$$644$$ −4.00000 −0.157622
$$645$$ 0 0
$$646$$ 8.00000 0.314756
$$647$$ 18.0000 0.707653 0.353827 0.935311i $$-0.384880\pi$$
0.353827 + 0.935311i $$0.384880\pi$$
$$648$$ 1.00000 0.0392837
$$649$$ −48.0000 −1.88416
$$650$$ 0 0
$$651$$ 8.00000 0.313545
$$652$$ −12.0000 −0.469956
$$653$$ 38.0000 1.48705 0.743527 0.668705i $$-0.233151\pi$$
0.743527 + 0.668705i $$0.233151\pi$$
$$654$$ 4.00000 0.156412
$$655$$ 0 0
$$656$$ 10.0000 0.390434
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 30.0000 1.16863 0.584317 0.811525i $$-0.301362\pi$$
0.584317 + 0.811525i $$0.301362\pi$$
$$660$$ 0 0
$$661$$ −16.0000 −0.622328 −0.311164 0.950356i $$-0.600719\pi$$
−0.311164 + 0.950356i $$0.600719\pi$$
$$662$$ −10.0000 −0.388661
$$663$$ −4.00000 −0.155347
$$664$$ 12.0000 0.465690
$$665$$ 0 0
$$666$$ −6.00000 −0.232495
$$667$$ −16.0000 −0.619522
$$668$$ 12.0000 0.464294
$$669$$ 22.0000 0.850569
$$670$$ 0 0
$$671$$ −8.00000 −0.308837
$$672$$ 2.00000 0.0771517
$$673$$ 46.0000 1.77317 0.886585 0.462566i $$-0.153071\pi$$
0.886585 + 0.462566i $$0.153071\pi$$
$$674$$ 26.0000 1.00148
$$675$$ 0 0
$$676$$ 1.00000 0.0384615
$$677$$ 22.0000 0.845529 0.422764 0.906240i $$-0.361060\pi$$
0.422764 + 0.906240i $$0.361060\pi$$
$$678$$ 16.0000 0.614476
$$679$$ 16.0000 0.614024
$$680$$ 0 0
$$681$$ −12.0000 −0.459841
$$682$$ 16.0000 0.612672
$$683$$ −36.0000 −1.37750 −0.688751 0.724998i $$-0.741841\pi$$
−0.688751 + 0.724998i $$0.741841\pi$$
$$684$$ −2.00000 −0.0764719
$$685$$ 0 0
$$686$$ −20.0000 −0.763604
$$687$$ 28.0000 1.06827
$$688$$ −4.00000 −0.152499
$$689$$ −6.00000 −0.228582
$$690$$ 0 0
$$691$$ −18.0000 −0.684752 −0.342376 0.939563i $$-0.611232\pi$$
−0.342376 + 0.939563i $$0.611232\pi$$
$$692$$ −14.0000 −0.532200
$$693$$ 8.00000 0.303895
$$694$$ −16.0000 −0.607352
$$695$$ 0 0
$$696$$ 8.00000 0.303239
$$697$$ −40.0000 −1.51511
$$698$$ −28.0000 −1.05982
$$699$$ −28.0000 −1.05906
$$700$$ 0 0
$$701$$ −20.0000 −0.755390 −0.377695 0.925930i $$-0.623283\pi$$
−0.377695 + 0.925930i $$0.623283\pi$$
$$702$$ 1.00000 0.0377426
$$703$$ 12.0000 0.452589
$$704$$ 4.00000 0.150756
$$705$$ 0 0
$$706$$ −6.00000 −0.225813
$$707$$ 40.0000 1.50435
$$708$$ −12.0000 −0.450988
$$709$$ 4.00000 0.150223 0.0751116 0.997175i $$-0.476069\pi$$
0.0751116 + 0.997175i $$0.476069\pi$$
$$710$$ 0 0
$$711$$ −8.00000 −0.300023
$$712$$ −10.0000 −0.374766
$$713$$ −8.00000 −0.299602
$$714$$ −8.00000 −0.299392
$$715$$ 0 0
$$716$$ 2.00000 0.0747435
$$717$$ −8.00000 −0.298765
$$718$$ 24.0000 0.895672
$$719$$ 36.0000 1.34257 0.671287 0.741198i $$-0.265742\pi$$
0.671287 + 0.741198i $$0.265742\pi$$
$$720$$ 0 0
$$721$$ −8.00000 −0.297936
$$722$$ −15.0000 −0.558242
$$723$$ −22.0000 −0.818189
$$724$$ −22.0000 −0.817624
$$725$$ 0 0
$$726$$ 5.00000 0.185567
$$727$$ −8.00000 −0.296704 −0.148352 0.988935i $$-0.547397\pi$$
−0.148352 + 0.988935i $$0.547397\pi$$
$$728$$ 2.00000 0.0741249
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ 16.0000 0.591781
$$732$$ −2.00000 −0.0739221
$$733$$ −6.00000 −0.221615 −0.110808 0.993842i $$-0.535344\pi$$
−0.110808 + 0.993842i $$0.535344\pi$$
$$734$$ 4.00000 0.147643
$$735$$ 0 0
$$736$$ −2.00000 −0.0737210
$$737$$ 32.0000 1.17874
$$738$$ 10.0000 0.368105
$$739$$ 50.0000 1.83928 0.919640 0.392763i $$-0.128481\pi$$
0.919640 + 0.392763i $$0.128481\pi$$
$$740$$ 0 0
$$741$$ −2.00000 −0.0734718
$$742$$ −12.0000 −0.440534
$$743$$ −36.0000 −1.32071 −0.660356 0.750953i $$-0.729595\pi$$
−0.660356 + 0.750953i $$0.729595\pi$$
$$744$$ 4.00000 0.146647
$$745$$ 0 0
$$746$$ −14.0000 −0.512576
$$747$$ 12.0000 0.439057
$$748$$ −16.0000 −0.585018
$$749$$ −8.00000 −0.292314
$$750$$ 0 0
$$751$$ 40.0000 1.45962 0.729810 0.683650i $$-0.239608\pi$$
0.729810 + 0.683650i $$0.239608\pi$$
$$752$$ 0 0
$$753$$ 10.0000 0.364420
$$754$$ 8.00000 0.291343
$$755$$ 0 0
$$756$$ 2.00000 0.0727393
$$757$$ 38.0000 1.38113 0.690567 0.723269i $$-0.257361\pi$$
0.690567 + 0.723269i $$0.257361\pi$$
$$758$$ −14.0000 −0.508503
$$759$$ −8.00000 −0.290382
$$760$$ 0 0
$$761$$ −46.0000 −1.66750 −0.833749 0.552143i $$-0.813810\pi$$
−0.833749 + 0.552143i $$0.813810\pi$$
$$762$$ −20.0000 −0.724524
$$763$$ 8.00000 0.289619
$$764$$ −8.00000 −0.289430
$$765$$ 0 0
$$766$$ 20.0000 0.722629
$$767$$ −12.0000 −0.433295
$$768$$ 1.00000 0.0360844
$$769$$ 10.0000 0.360609 0.180305 0.983611i $$-0.442292\pi$$
0.180305 + 0.983611i $$0.442292\pi$$
$$770$$ 0 0
$$771$$ 8.00000 0.288113
$$772$$ −12.0000 −0.431889
$$773$$ −26.0000 −0.935155 −0.467578 0.883952i $$-0.654873\pi$$
−0.467578 + 0.883952i $$0.654873\pi$$
$$774$$ −4.00000 −0.143777
$$775$$ 0 0
$$776$$ 8.00000 0.287183
$$777$$ −12.0000 −0.430498
$$778$$ 20.0000 0.717035
$$779$$ −20.0000 −0.716574
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 8.00000 0.286079
$$783$$ 8.00000 0.285897
$$784$$ −3.00000 −0.107143
$$785$$ 0 0
$$786$$ 14.0000 0.499363
$$787$$ 28.0000 0.998092 0.499046 0.866575i $$-0.333684\pi$$
0.499046 + 0.866575i $$0.333684\pi$$
$$788$$ −10.0000 −0.356235
$$789$$ 18.0000 0.640817
$$790$$ 0 0
$$791$$ 32.0000 1.13779
$$792$$ 4.00000 0.142134
$$793$$ −2.00000 −0.0710221
$$794$$ −6.00000 −0.212932
$$795$$ 0 0
$$796$$ 24.0000 0.850657
$$797$$ 42.0000 1.48772 0.743858 0.668338i $$-0.232994\pi$$
0.743858 + 0.668338i $$0.232994\pi$$
$$798$$ −4.00000 −0.141598
$$799$$ 0 0
$$800$$ 0 0
$$801$$ −10.0000 −0.353333
$$802$$ −30.0000 −1.05934
$$803$$ 0 0
$$804$$ 8.00000 0.282138
$$805$$ 0 0
$$806$$ 4.00000 0.140894
$$807$$ −4.00000 −0.140807
$$808$$ 20.0000 0.703598
$$809$$ −6.00000 −0.210949 −0.105474 0.994422i $$-0.533636\pi$$
−0.105474 + 0.994422i $$0.533636\pi$$
$$810$$ 0 0
$$811$$ −34.0000 −1.19390 −0.596951 0.802278i $$-0.703621\pi$$
−0.596951 + 0.802278i $$0.703621\pi$$
$$812$$ 16.0000 0.561490
$$813$$ −28.0000 −0.982003
$$814$$ −24.0000 −0.841200
$$815$$ 0 0
$$816$$ −4.00000 −0.140028
$$817$$ 8.00000 0.279885
$$818$$ −18.0000 −0.629355
$$819$$ 2.00000 0.0698857
$$820$$ 0 0
$$821$$ 42.0000 1.46581 0.732905 0.680331i $$-0.238164\pi$$
0.732905 + 0.680331i $$0.238164\pi$$
$$822$$ 2.00000 0.0697580
$$823$$ 20.0000 0.697156 0.348578 0.937280i $$-0.386665\pi$$
0.348578 + 0.937280i $$0.386665\pi$$
$$824$$ −4.00000 −0.139347
$$825$$ 0 0
$$826$$ −24.0000 −0.835067
$$827$$ 44.0000 1.53003 0.765015 0.644013i $$-0.222732\pi$$
0.765015 + 0.644013i $$0.222732\pi$$
$$828$$ −2.00000 −0.0695048
$$829$$ −10.0000 −0.347314 −0.173657 0.984806i $$-0.555558\pi$$
−0.173657 + 0.984806i $$0.555558\pi$$
$$830$$ 0 0
$$831$$ −22.0000 −0.763172
$$832$$ 1.00000 0.0346688
$$833$$ 12.0000 0.415775
$$834$$ −16.0000 −0.554035
$$835$$ 0 0
$$836$$ −8.00000 −0.276686
$$837$$ 4.00000 0.138260
$$838$$ 14.0000 0.483622
$$839$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$840$$ 0 0
$$841$$ 35.0000 1.20690
$$842$$ −16.0000 −0.551396
$$843$$ −10.0000 −0.344418
$$844$$ −4.00000 −0.137686
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 10.0000 0.343604
$$848$$ −6.00000 −0.206041
$$849$$ −28.0000 −0.960958
$$850$$ 0 0
$$851$$ 12.0000 0.411355
$$852$$ 0 0
$$853$$ 10.0000 0.342393 0.171197 0.985237i $$-0.445237\pi$$
0.171197 + 0.985237i $$0.445237\pi$$
$$854$$ −4.00000 −0.136877
$$855$$ 0 0
$$856$$ −4.00000 −0.136717
$$857$$ 16.0000 0.546550 0.273275 0.961936i $$-0.411893\pi$$
0.273275 + 0.961936i $$0.411893\pi$$
$$858$$ 4.00000 0.136558
$$859$$ −44.0000 −1.50126 −0.750630 0.660722i $$-0.770250\pi$$
−0.750630 + 0.660722i $$0.770250\pi$$
$$860$$ 0 0
$$861$$ 20.0000 0.681598
$$862$$ 16.0000 0.544962
$$863$$ 36.0000 1.22545 0.612727 0.790295i $$-0.290072\pi$$
0.612727 + 0.790295i $$0.290072\pi$$
$$864$$ 1.00000 0.0340207
$$865$$ 0 0
$$866$$ 26.0000 0.883516
$$867$$ −1.00000 −0.0339618
$$868$$ 8.00000 0.271538
$$869$$ −32.0000 −1.08553
$$870$$ 0 0
$$871$$ 8.00000 0.271070
$$872$$ 4.00000 0.135457
$$873$$ 8.00000 0.270759
$$874$$ 4.00000 0.135302
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 22.0000 0.742887 0.371444 0.928456i $$-0.378863\pi$$
0.371444 + 0.928456i $$0.378863\pi$$
$$878$$ 8.00000 0.269987
$$879$$ 6.00000 0.202375
$$880$$ 0 0
$$881$$ −18.0000 −0.606435 −0.303218 0.952921i $$-0.598061\pi$$
−0.303218 + 0.952921i $$0.598061\pi$$
$$882$$ −3.00000 −0.101015
$$883$$ −44.0000 −1.48072 −0.740359 0.672212i $$-0.765344\pi$$
−0.740359 + 0.672212i $$0.765344\pi$$
$$884$$ −4.00000 −0.134535
$$885$$ 0 0
$$886$$ 16.0000 0.537531
$$887$$ 6.00000 0.201460 0.100730 0.994914i $$-0.467882\pi$$
0.100730 + 0.994914i $$0.467882\pi$$
$$888$$ −6.00000 −0.201347
$$889$$ −40.0000 −1.34156
$$890$$ 0 0
$$891$$ 4.00000 0.134005
$$892$$ 22.0000 0.736614
$$893$$ 0 0
$$894$$ −22.0000 −0.735790
$$895$$ 0 0
$$896$$ 2.00000 0.0668153
$$897$$ −2.00000 −0.0667781
$$898$$ −18.0000 −0.600668
$$899$$ 32.0000 1.06726
$$900$$ 0 0
$$901$$ 24.0000 0.799556
$$902$$ 40.0000 1.33185
$$903$$ −8.00000 −0.266223
$$904$$ 16.0000 0.532152
$$905$$ 0 0
$$906$$ −4.00000 −0.132891
$$907$$ 44.0000 1.46100 0.730498 0.682915i $$-0.239288\pi$$
0.730498 + 0.682915i $$0.239288\pi$$
$$908$$ −12.0000 −0.398234
$$909$$ 20.0000 0.663358
$$910$$ 0 0
$$911$$ −36.0000 −1.19273 −0.596367 0.802712i $$-0.703390\pi$$
−0.596367 + 0.802712i $$0.703390\pi$$
$$912$$ −2.00000 −0.0662266
$$913$$ 48.0000 1.58857
$$914$$ 16.0000 0.529233
$$915$$ 0 0
$$916$$ 28.0000 0.925146
$$917$$ 28.0000 0.924641
$$918$$ −4.00000 −0.132020
$$919$$ 32.0000 1.05558 0.527791 0.849374i $$-0.323020\pi$$
0.527791 + 0.849374i $$0.323020\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 30.0000 0.987997
$$923$$ 0 0
$$924$$ 8.00000 0.263181
$$925$$ 0 0
$$926$$ −2.00000 −0.0657241
$$927$$ −4.00000 −0.131377
$$928$$ 8.00000 0.262613
$$929$$ −34.0000 −1.11550 −0.557752 0.830008i $$-0.688336\pi$$
−0.557752 + 0.830008i $$0.688336\pi$$
$$930$$ 0 0
$$931$$ 6.00000 0.196642
$$932$$ −28.0000 −0.917170
$$933$$ −16.0000 −0.523816
$$934$$ −12.0000 −0.392652
$$935$$ 0 0
$$936$$ 1.00000 0.0326860
$$937$$ −10.0000 −0.326686 −0.163343 0.986569i $$-0.552228\pi$$
−0.163343 + 0.986569i $$0.552228\pi$$
$$938$$ 16.0000 0.522419
$$939$$ 10.0000 0.326338
$$940$$ 0 0
$$941$$ −38.0000 −1.23876 −0.619382 0.785090i $$-0.712617\pi$$
−0.619382 + 0.785090i $$0.712617\pi$$
$$942$$ 10.0000 0.325818
$$943$$ −20.0000 −0.651290
$$944$$ −12.0000 −0.390567
$$945$$ 0 0
$$946$$ −16.0000 −0.520205
$$947$$ 4.00000 0.129983 0.0649913 0.997886i $$-0.479298\pi$$
0.0649913 + 0.997886i $$0.479298\pi$$
$$948$$ −8.00000 −0.259828
$$949$$ 0 0
$$950$$ 0 0
$$951$$ 2.00000 0.0648544
$$952$$ −8.00000 −0.259281
$$953$$ 36.0000 1.16615 0.583077 0.812417i $$-0.301849\pi$$
0.583077 + 0.812417i $$0.301849\pi$$
$$954$$ −6.00000 −0.194257
$$955$$ 0 0
$$956$$ −8.00000 −0.258738
$$957$$ 32.0000 1.03441
$$958$$ 24.0000 0.775405
$$959$$ 4.00000 0.129167
$$960$$ 0 0
$$961$$ −15.0000 −0.483871
$$962$$ −6.00000 −0.193448
$$963$$ −4.00000 −0.128898
$$964$$ −22.0000 −0.708572
$$965$$ 0 0
$$966$$ −4.00000 −0.128698
$$967$$ −14.0000 −0.450210 −0.225105 0.974335i $$-0.572272\pi$$
−0.225105 + 0.974335i $$0.572272\pi$$
$$968$$ 5.00000 0.160706
$$969$$ 8.00000 0.256997
$$970$$ 0 0
$$971$$ 42.0000 1.34784 0.673922 0.738802i $$-0.264608\pi$$
0.673922 + 0.738802i $$0.264608\pi$$
$$972$$ 1.00000 0.0320750
$$973$$ −32.0000 −1.02587
$$974$$ −38.0000 −1.21760
$$975$$ 0 0
$$976$$ −2.00000 −0.0640184
$$977$$ −42.0000 −1.34370 −0.671850 0.740688i $$-0.734500\pi$$
−0.671850 + 0.740688i $$0.734500\pi$$
$$978$$ −12.0000 −0.383718
$$979$$ −40.0000 −1.27841
$$980$$ 0 0
$$981$$ 4.00000 0.127710
$$982$$ 38.0000 1.21263
$$983$$ 12.0000 0.382741 0.191370 0.981518i $$-0.438707\pi$$
0.191370 + 0.981518i $$0.438707\pi$$
$$984$$ 10.0000 0.318788
$$985$$ 0 0
$$986$$ −32.0000 −1.01909
$$987$$ 0 0
$$988$$ −2.00000 −0.0636285
$$989$$ 8.00000 0.254385
$$990$$ 0 0
$$991$$ 40.0000 1.27064 0.635321 0.772248i $$-0.280868\pi$$
0.635321 + 0.772248i $$0.280868\pi$$
$$992$$ 4.00000 0.127000
$$993$$ −10.0000 −0.317340
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 12.0000 0.380235
$$997$$ 26.0000 0.823428 0.411714 0.911313i $$-0.364930\pi$$
0.411714 + 0.911313i $$0.364930\pi$$
$$998$$ −14.0000 −0.443162
$$999$$ −6.00000 −0.189832
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 1950.2.a.ba.1.1 1
3.2 odd 2 5850.2.a.s.1.1 1
5.2 odd 4 1950.2.e.m.1249.2 2
5.3 odd 4 1950.2.e.m.1249.1 2
5.4 even 2 390.2.a.b.1.1 1
15.2 even 4 5850.2.e.h.5149.1 2
15.8 even 4 5850.2.e.h.5149.2 2
15.14 odd 2 1170.2.a.j.1.1 1
20.19 odd 2 3120.2.a.y.1.1 1
60.59 even 2 9360.2.a.v.1.1 1
65.34 odd 4 5070.2.b.f.1351.1 2
65.44 odd 4 5070.2.b.f.1351.2 2
65.64 even 2 5070.2.a.n.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
390.2.a.b.1.1 1 5.4 even 2
1170.2.a.j.1.1 1 15.14 odd 2
1950.2.a.ba.1.1 1 1.1 even 1 trivial
1950.2.e.m.1249.1 2 5.3 odd 4
1950.2.e.m.1249.2 2 5.2 odd 4
3120.2.a.y.1.1 1 20.19 odd 2
5070.2.a.n.1.1 1 65.64 even 2
5070.2.b.f.1351.1 2 65.34 odd 4
5070.2.b.f.1351.2 2 65.44 odd 4
5850.2.a.s.1.1 1 3.2 odd 2
5850.2.e.h.5149.1 2 15.2 even 4
5850.2.e.h.5149.2 2 15.8 even 4
9360.2.a.v.1.1 1 60.59 even 2