# Properties

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

# Related objects

## 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: $$1$$ 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

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -2.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -2.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{12} -1.00000 q^{13} +2.00000 q^{14} +1.00000 q^{16} -1.00000 q^{18} +2.00000 q^{19} +2.00000 q^{21} +6.00000 q^{23} +1.00000 q^{24} +1.00000 q^{26} -1.00000 q^{27} -2.00000 q^{28} -4.00000 q^{31} -1.00000 q^{32} +1.00000 q^{36} -2.00000 q^{37} -2.00000 q^{38} +1.00000 q^{39} -6.00000 q^{41} -2.00000 q^{42} +4.00000 q^{43} -6.00000 q^{46} -1.00000 q^{48} -3.00000 q^{49} -1.00000 q^{52} +6.00000 q^{53} +1.00000 q^{54} +2.00000 q^{56} -2.00000 q^{57} -10.0000 q^{61} +4.00000 q^{62} -2.00000 q^{63} +1.00000 q^{64} -8.00000 q^{67} -6.00000 q^{69} -1.00000 q^{72} -8.00000 q^{73} +2.00000 q^{74} +2.00000 q^{76} -1.00000 q^{78} +8.00000 q^{79} +1.00000 q^{81} +6.00000 q^{82} +12.0000 q^{83} +2.00000 q^{84} -4.00000 q^{86} +6.00000 q^{89} +2.00000 q^{91} +6.00000 q^{92} +4.00000 q^{93} +1.00000 q^{96} -8.00000 q^{97} +3.00000 q^{98} +O(q^{100})$$

## 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
$$3$$ −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 $$-0.623376\pi$$
−0.377964 + 0.925820i $$0.623376\pi$$
$$8$$ −1.00000 −0.353553
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$12$$ −1.00000 −0.288675
$$13$$ −1.00000 −0.277350
$$14$$ 2.00000 0.534522
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$18$$ −1.00000 −0.235702
$$19$$ 2.00000 0.458831 0.229416 0.973329i $$-0.426318\pi$$
0.229416 + 0.973329i $$0.426318\pi$$
$$20$$ 0 0
$$21$$ 2.00000 0.436436
$$22$$ 0 0
$$23$$ 6.00000 1.25109 0.625543 0.780189i $$-0.284877\pi$$
0.625543 + 0.780189i $$0.284877\pi$$
$$24$$ 1.00000 0.204124
$$25$$ 0 0
$$26$$ 1.00000 0.196116
$$27$$ −1.00000 −0.192450
$$28$$ −2.00000 −0.377964
$$29$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$30$$ 0 0
$$31$$ −4.00000 −0.718421 −0.359211 0.933257i $$-0.616954\pi$$
−0.359211 + 0.933257i $$0.616954\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ −2.00000 −0.328798 −0.164399 0.986394i $$-0.552568\pi$$
−0.164399 + 0.986394i $$0.552568\pi$$
$$38$$ −2.00000 −0.324443
$$39$$ 1.00000 0.160128
$$40$$ 0 0
$$41$$ −6.00000 −0.937043 −0.468521 0.883452i $$-0.655213\pi$$
−0.468521 + 0.883452i $$0.655213\pi$$
$$42$$ −2.00000 −0.308607
$$43$$ 4.00000 0.609994 0.304997 0.952353i $$-0.401344\pi$$
0.304997 + 0.952353i $$0.401344\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ −6.00000 −0.884652
$$47$$ 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
$$51$$ 0 0
$$52$$ −1.00000 −0.138675
$$53$$ 6.00000 0.824163 0.412082 0.911147i $$-0.364802\pi$$
0.412082 + 0.911147i $$0.364802\pi$$
$$54$$ 1.00000 0.136083
$$55$$ 0 0
$$56$$ 2.00000 0.267261
$$57$$ −2.00000 −0.264906
$$58$$ 0 0
$$59$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$60$$ 0 0
$$61$$ −10.0000 −1.28037 −0.640184 0.768221i $$-0.721142\pi$$
−0.640184 + 0.768221i $$0.721142\pi$$
$$62$$ 4.00000 0.508001
$$63$$ −2.00000 −0.251976
$$64$$ 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 $$0.662521\pi$$
$$68$$ 0 0
$$69$$ −6.00000 −0.722315
$$70$$ 0 0
$$71$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$72$$ −1.00000 −0.117851
$$73$$ −8.00000 −0.936329 −0.468165 0.883641i $$-0.655085\pi$$
−0.468165 + 0.883641i $$0.655085\pi$$
$$74$$ 2.00000 0.232495
$$75$$ 0 0
$$76$$ 2.00000 0.229416
$$77$$ 0 0
$$78$$ −1.00000 −0.113228
$$79$$ 8.00000 0.900070 0.450035 0.893011i $$-0.351411\pi$$
0.450035 + 0.893011i $$0.351411\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 6.00000 0.662589
$$83$$ 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
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 6.00000 0.635999 0.317999 0.948091i $$-0.396989\pi$$
0.317999 + 0.948091i $$0.396989\pi$$
$$90$$ 0 0
$$91$$ 2.00000 0.209657
$$92$$ 6.00000 0.625543
$$93$$ 4.00000 0.414781
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 1.00000 0.102062
$$97$$ −8.00000 −0.812277 −0.406138 0.913812i $$-0.633125\pi$$
−0.406138 + 0.913812i $$0.633125\pi$$
$$98$$ 3.00000 0.303046
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −12.0000 −1.19404 −0.597022 0.802225i $$-0.703650\pi$$
−0.597022 + 0.802225i $$0.703650\pi$$
$$102$$ 0 0
$$103$$ −8.00000 −0.788263 −0.394132 0.919054i $$-0.628955\pi$$
−0.394132 + 0.919054i $$0.628955\pi$$
$$104$$ 1.00000 0.0980581
$$105$$ 0 0
$$106$$ −6.00000 −0.582772
$$107$$ −12.0000 −1.16008 −0.580042 0.814587i $$-0.696964\pi$$
−0.580042 + 0.814587i $$0.696964\pi$$
$$108$$ −1.00000 −0.0962250
$$109$$ −16.0000 −1.53252 −0.766261 0.642529i $$-0.777885\pi$$
−0.766261 + 0.642529i $$0.777885\pi$$
$$110$$ 0 0
$$111$$ 2.00000 0.189832
$$112$$ −2.00000 −0.188982
$$113$$ 12.0000 1.12887 0.564433 0.825479i $$-0.309095\pi$$
0.564433 + 0.825479i $$0.309095\pi$$
$$114$$ 2.00000 0.187317
$$115$$ 0 0
$$116$$ 0 0
$$117$$ −1.00000 −0.0924500
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −11.0000 −1.00000
$$122$$ 10.0000 0.905357
$$123$$ 6.00000 0.541002
$$124$$ −4.00000 −0.359211
$$125$$ 0 0
$$126$$ 2.00000 0.178174
$$127$$ −8.00000 −0.709885 −0.354943 0.934888i $$-0.615500\pi$$
−0.354943 + 0.934888i $$0.615500\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ −4.00000 −0.352180
$$130$$ 0 0
$$131$$ −18.0000 −1.57267 −0.786334 0.617802i $$-0.788023\pi$$
−0.786334 + 0.617802i $$0.788023\pi$$
$$132$$ 0 0
$$133$$ −4.00000 −0.346844
$$134$$ 8.00000 0.691095
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −18.0000 −1.53784 −0.768922 0.639343i $$-0.779207\pi$$
−0.768922 + 0.639343i $$0.779207\pi$$
$$138$$ 6.00000 0.510754
$$139$$ 8.00000 0.678551 0.339276 0.940687i $$-0.389818\pi$$
0.339276 + 0.940687i $$0.389818\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 1.00000 0.0833333
$$145$$ 0 0
$$146$$ 8.00000 0.662085
$$147$$ 3.00000 0.247436
$$148$$ −2.00000 −0.164399
$$149$$ 6.00000 0.491539 0.245770 0.969328i $$-0.420959\pi$$
0.245770 + 0.969328i $$0.420959\pi$$
$$150$$ 0 0
$$151$$ 20.0000 1.62758 0.813788 0.581161i $$-0.197401\pi$$
0.813788 + 0.581161i $$0.197401\pi$$
$$152$$ −2.00000 −0.162221
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 1.00000 0.0800641
$$157$$ −2.00000 −0.159617 −0.0798087 0.996810i $$-0.525431\pi$$
−0.0798087 + 0.996810i $$0.525431\pi$$
$$158$$ −8.00000 −0.636446
$$159$$ −6.00000 −0.475831
$$160$$ 0 0
$$161$$ −12.0000 −0.945732
$$162$$ −1.00000 −0.0785674
$$163$$ −20.0000 −1.56652 −0.783260 0.621694i $$-0.786445\pi$$
−0.783260 + 0.621694i $$0.786445\pi$$
$$164$$ −6.00000 −0.468521
$$165$$ 0 0
$$166$$ −12.0000 −0.931381
$$167$$ −12.0000 −0.928588 −0.464294 0.885681i $$-0.653692\pi$$
−0.464294 + 0.885681i $$0.653692\pi$$
$$168$$ −2.00000 −0.154303
$$169$$ 1.00000 0.0769231
$$170$$ 0 0
$$171$$ 2.00000 0.152944
$$172$$ 4.00000 0.304997
$$173$$ 6.00000 0.456172 0.228086 0.973641i $$-0.426753\pi$$
0.228086 + 0.973641i $$0.426753\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 0 0
$$178$$ −6.00000 −0.449719
$$179$$ 18.0000 1.34538 0.672692 0.739923i $$-0.265138\pi$$
0.672692 + 0.739923i $$0.265138\pi$$
$$180$$ 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
$$183$$ 10.0000 0.739221
$$184$$ −6.00000 −0.442326
$$185$$ 0 0
$$186$$ −4.00000 −0.293294
$$187$$ 0 0
$$188$$ 0 0
$$189$$ 2.00000 0.145479
$$190$$ 0 0
$$191$$ 24.0000 1.73658 0.868290 0.496058i $$-0.165220\pi$$
0.868290 + 0.496058i $$0.165220\pi$$
$$192$$ −1.00000 −0.0721688
$$193$$ −20.0000 −1.43963 −0.719816 0.694165i $$-0.755774\pi$$
−0.719816 + 0.694165i $$0.755774\pi$$
$$194$$ 8.00000 0.574367
$$195$$ 0 0
$$196$$ −3.00000 −0.214286
$$197$$ −6.00000 −0.427482 −0.213741 0.976890i $$-0.568565\pi$$
−0.213741 + 0.976890i $$0.568565\pi$$
$$198$$ 0 0
$$199$$ −16.0000 −1.13421 −0.567105 0.823646i $$-0.691937\pi$$
−0.567105 + 0.823646i $$0.691937\pi$$
$$200$$ 0 0
$$201$$ 8.00000 0.564276
$$202$$ 12.0000 0.844317
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 8.00000 0.557386
$$207$$ 6.00000 0.417029
$$208$$ −1.00000 −0.0693375
$$209$$ 0 0
$$210$$ 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
$$214$$ 12.0000 0.820303
$$215$$ 0 0
$$216$$ 1.00000 0.0680414
$$217$$ 8.00000 0.543075
$$218$$ 16.0000 1.08366
$$219$$ 8.00000 0.540590
$$220$$ 0 0
$$221$$ 0 0
$$222$$ −2.00000 −0.134231
$$223$$ −14.0000 −0.937509 −0.468755 0.883328i $$-0.655297\pi$$
−0.468755 + 0.883328i $$0.655297\pi$$
$$224$$ 2.00000 0.133631
$$225$$ 0 0
$$226$$ −12.0000 −0.798228
$$227$$ −12.0000 −0.796468 −0.398234 0.917284i $$-0.630377\pi$$
−0.398234 + 0.917284i $$0.630377\pi$$
$$228$$ −2.00000 −0.132453
$$229$$ −16.0000 −1.05731 −0.528655 0.848837i $$-0.677303\pi$$
−0.528655 + 0.848837i $$0.677303\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −24.0000 −1.57229 −0.786146 0.618041i $$-0.787927\pi$$
−0.786146 + 0.618041i $$0.787927\pi$$
$$234$$ 1.00000 0.0653720
$$235$$ 0 0
$$236$$ 0 0
$$237$$ −8.00000 −0.519656
$$238$$ 0 0
$$239$$ −24.0000 −1.55243 −0.776215 0.630468i $$-0.782863\pi$$
−0.776215 + 0.630468i $$0.782863\pi$$
$$240$$ 0 0
$$241$$ 2.00000 0.128831 0.0644157 0.997923i $$-0.479482\pi$$
0.0644157 + 0.997923i $$0.479482\pi$$
$$242$$ 11.0000 0.707107
$$243$$ −1.00000 −0.0641500
$$244$$ −10.0000 −0.640184
$$245$$ 0 0
$$246$$ −6.00000 −0.382546
$$247$$ −2.00000 −0.127257
$$248$$ 4.00000 0.254000
$$249$$ −12.0000 −0.760469
$$250$$ 0 0
$$251$$ 18.0000 1.13615 0.568075 0.822977i $$-0.307688\pi$$
0.568075 + 0.822977i $$0.307688\pi$$
$$252$$ −2.00000 −0.125988
$$253$$ 0 0
$$254$$ 8.00000 0.501965
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ −12.0000 −0.748539 −0.374270 0.927320i $$-0.622107\pi$$
−0.374270 + 0.927320i $$0.622107\pi$$
$$258$$ 4.00000 0.249029
$$259$$ 4.00000 0.248548
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 18.0000 1.11204
$$263$$ −6.00000 −0.369976 −0.184988 0.982741i $$-0.559225\pi$$
−0.184988 + 0.982741i $$0.559225\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 4.00000 0.245256
$$267$$ −6.00000 −0.367194
$$268$$ −8.00000 −0.488678
$$269$$ 12.0000 0.731653 0.365826 0.930683i $$-0.380786\pi$$
0.365826 + 0.930683i $$0.380786\pi$$
$$270$$ 0 0
$$271$$ −4.00000 −0.242983 −0.121491 0.992592i $$-0.538768\pi$$
−0.121491 + 0.992592i $$0.538768\pi$$
$$272$$ 0 0
$$273$$ −2.00000 −0.121046
$$274$$ 18.0000 1.08742
$$275$$ 0 0
$$276$$ −6.00000 −0.361158
$$277$$ 22.0000 1.32185 0.660926 0.750451i $$-0.270164\pi$$
0.660926 + 0.750451i $$0.270164\pi$$
$$278$$ −8.00000 −0.479808
$$279$$ −4.00000 −0.239474
$$280$$ 0 0
$$281$$ 6.00000 0.357930 0.178965 0.983855i $$-0.442725\pi$$
0.178965 + 0.983855i $$0.442725\pi$$
$$282$$ 0 0
$$283$$ 4.00000 0.237775 0.118888 0.992908i $$-0.462067\pi$$
0.118888 + 0.992908i $$0.462067\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 12.0000 0.708338
$$288$$ −1.00000 −0.0589256
$$289$$ −17.0000 −1.00000
$$290$$ 0 0
$$291$$ 8.00000 0.468968
$$292$$ −8.00000 −0.468165
$$293$$ −6.00000 −0.350524 −0.175262 0.984522i $$-0.556077\pi$$
−0.175262 + 0.984522i $$0.556077\pi$$
$$294$$ −3.00000 −0.174964
$$295$$ 0 0
$$296$$ 2.00000 0.116248
$$297$$ 0 0
$$298$$ −6.00000 −0.347571
$$299$$ −6.00000 −0.346989
$$300$$ 0 0
$$301$$ −8.00000 −0.461112
$$302$$ −20.0000 −1.15087
$$303$$ 12.0000 0.689382
$$304$$ 2.00000 0.114708
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 16.0000 0.913168 0.456584 0.889680i $$-0.349073\pi$$
0.456584 + 0.889680i $$0.349073\pi$$
$$308$$ 0 0
$$309$$ 8.00000 0.455104
$$310$$ 0 0
$$311$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$312$$ −1.00000 −0.0566139
$$313$$ −14.0000 −0.791327 −0.395663 0.918396i $$-0.629485\pi$$
−0.395663 + 0.918396i $$0.629485\pi$$
$$314$$ 2.00000 0.112867
$$315$$ 0 0
$$316$$ 8.00000 0.450035
$$317$$ 30.0000 1.68497 0.842484 0.538721i $$-0.181092\pi$$
0.842484 + 0.538721i $$0.181092\pi$$
$$318$$ 6.00000 0.336463
$$319$$ 0 0
$$320$$ 0 0
$$321$$ 12.0000 0.669775
$$322$$ 12.0000 0.668734
$$323$$ 0 0
$$324$$ 1.00000 0.0555556
$$325$$ 0 0
$$326$$ 20.0000 1.10770
$$327$$ 16.0000 0.884802
$$328$$ 6.00000 0.331295
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 26.0000 1.42909 0.714545 0.699590i $$-0.246634\pi$$
0.714545 + 0.699590i $$0.246634\pi$$
$$332$$ 12.0000 0.658586
$$333$$ −2.00000 −0.109599
$$334$$ 12.0000 0.656611
$$335$$ 0 0
$$336$$ 2.00000 0.109109
$$337$$ 34.0000 1.85210 0.926049 0.377403i $$-0.123183\pi$$
0.926049 + 0.377403i $$0.123183\pi$$
$$338$$ −1.00000 −0.0543928
$$339$$ −12.0000 −0.651751
$$340$$ 0 0
$$341$$ 0 0
$$342$$ −2.00000 −0.108148
$$343$$ 20.0000 1.07990
$$344$$ −4.00000 −0.215666
$$345$$ 0 0
$$346$$ −6.00000 −0.322562
$$347$$ −24.0000 −1.28839 −0.644194 0.764862i $$-0.722807\pi$$
−0.644194 + 0.764862i $$0.722807\pi$$
$$348$$ 0 0
$$349$$ −16.0000 −0.856460 −0.428230 0.903670i $$-0.640863\pi$$
−0.428230 + 0.903670i $$0.640863\pi$$
$$350$$ 0 0
$$351$$ 1.00000 0.0533761
$$352$$ 0 0
$$353$$ −18.0000 −0.958043 −0.479022 0.877803i $$-0.659008\pi$$
−0.479022 + 0.877803i $$0.659008\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 6.00000 0.317999
$$357$$ 0 0
$$358$$ −18.0000 −0.951330
$$359$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$360$$ 0 0
$$361$$ −15.0000 −0.789474
$$362$$ 22.0000 1.15629
$$363$$ 11.0000 0.577350
$$364$$ 2.00000 0.104828
$$365$$ 0 0
$$366$$ −10.0000 −0.522708
$$367$$ 16.0000 0.835193 0.417597 0.908633i $$-0.362873\pi$$
0.417597 + 0.908633i $$0.362873\pi$$
$$368$$ 6.00000 0.312772
$$369$$ −6.00000 −0.312348
$$370$$ 0 0
$$371$$ −12.0000 −0.623009
$$372$$ 4.00000 0.207390
$$373$$ 22.0000 1.13912 0.569558 0.821951i $$-0.307114\pi$$
0.569558 + 0.821951i $$0.307114\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 0 0
$$378$$ −2.00000 −0.102869
$$379$$ −10.0000 −0.513665 −0.256833 0.966456i $$-0.582679\pi$$
−0.256833 + 0.966456i $$0.582679\pi$$
$$380$$ 0 0
$$381$$ 8.00000 0.409852
$$382$$ −24.0000 −1.22795
$$383$$ 36.0000 1.83951 0.919757 0.392488i $$-0.128386\pi$$
0.919757 + 0.392488i $$0.128386\pi$$
$$384$$ 1.00000 0.0510310
$$385$$ 0 0
$$386$$ 20.0000 1.01797
$$387$$ 4.00000 0.203331
$$388$$ −8.00000 −0.406138
$$389$$ 12.0000 0.608424 0.304212 0.952604i $$-0.401607\pi$$
0.304212 + 0.952604i $$0.401607\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 3.00000 0.151523
$$393$$ 18.0000 0.907980
$$394$$ 6.00000 0.302276
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 22.0000 1.10415 0.552074 0.833795i $$-0.313837\pi$$
0.552074 + 0.833795i $$0.313837\pi$$
$$398$$ 16.0000 0.802008
$$399$$ 4.00000 0.200250
$$400$$ 0 0
$$401$$ −6.00000 −0.299626 −0.149813 0.988714i $$-0.547867\pi$$
−0.149813 + 0.988714i $$0.547867\pi$$
$$402$$ −8.00000 −0.399004
$$403$$ 4.00000 0.199254
$$404$$ −12.0000 −0.597022
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 0 0
$$409$$ 14.0000 0.692255 0.346128 0.938187i $$-0.387496\pi$$
0.346128 + 0.938187i $$0.387496\pi$$
$$410$$ 0 0
$$411$$ 18.0000 0.887875
$$412$$ −8.00000 −0.394132
$$413$$ 0 0
$$414$$ −6.00000 −0.294884
$$415$$ 0 0
$$416$$ 1.00000 0.0490290
$$417$$ −8.00000 −0.391762
$$418$$ 0 0
$$419$$ −18.0000 −0.879358 −0.439679 0.898155i $$-0.644908\pi$$
−0.439679 + 0.898155i $$0.644908\pi$$
$$420$$ 0 0
$$421$$ −4.00000 −0.194948 −0.0974740 0.995238i $$-0.531076\pi$$
−0.0974740 + 0.995238i $$0.531076\pi$$
$$422$$ 4.00000 0.194717
$$423$$ 0 0
$$424$$ −6.00000 −0.291386
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 20.0000 0.967868
$$428$$ −12.0000 −0.580042
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 24.0000 1.15604 0.578020 0.816023i $$-0.303826\pi$$
0.578020 + 0.816023i $$0.303826\pi$$
$$432$$ −1.00000 −0.0481125
$$433$$ 34.0000 1.63394 0.816968 0.576683i $$-0.195653\pi$$
0.816968 + 0.576683i $$0.195653\pi$$
$$434$$ −8.00000 −0.384012
$$435$$ 0 0
$$436$$ −16.0000 −0.766261
$$437$$ 12.0000 0.574038
$$438$$ −8.00000 −0.382255
$$439$$ 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
$$442$$ 0 0
$$443$$ 24.0000 1.14027 0.570137 0.821549i $$-0.306890\pi$$
0.570137 + 0.821549i $$0.306890\pi$$
$$444$$ 2.00000 0.0949158
$$445$$ 0 0
$$446$$ 14.0000 0.662919
$$447$$ −6.00000 −0.283790
$$448$$ −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
$$451$$ 0 0
$$452$$ 12.0000 0.564433
$$453$$ −20.0000 −0.939682
$$454$$ 12.0000 0.563188
$$455$$ 0 0
$$456$$ 2.00000 0.0936586
$$457$$ −32.0000 −1.49690 −0.748448 0.663193i $$-0.769201\pi$$
−0.748448 + 0.663193i $$0.769201\pi$$
$$458$$ 16.0000 0.747631
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 18.0000 0.838344 0.419172 0.907907i $$-0.362320\pi$$
0.419172 + 0.907907i $$0.362320\pi$$
$$462$$ 0 0
$$463$$ −14.0000 −0.650635 −0.325318 0.945605i $$-0.605471\pi$$
−0.325318 + 0.945605i $$0.605471\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 24.0000 1.11178
$$467$$ 12.0000 0.555294 0.277647 0.960683i $$-0.410445\pi$$
0.277647 + 0.960683i $$0.410445\pi$$
$$468$$ −1.00000 −0.0462250
$$469$$ 16.0000 0.738811
$$470$$ 0 0
$$471$$ 2.00000 0.0921551
$$472$$ 0 0
$$473$$ 0 0
$$474$$ 8.00000 0.367452
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 6.00000 0.274721
$$478$$ 24.0000 1.09773
$$479$$ −24.0000 −1.09659 −0.548294 0.836286i $$-0.684723\pi$$
−0.548294 + 0.836286i $$0.684723\pi$$
$$480$$ 0 0
$$481$$ 2.00000 0.0911922
$$482$$ −2.00000 −0.0910975
$$483$$ 12.0000 0.546019
$$484$$ −11.0000 −0.500000
$$485$$ 0 0
$$486$$ 1.00000 0.0453609
$$487$$ −2.00000 −0.0906287 −0.0453143 0.998973i $$-0.514429\pi$$
−0.0453143 + 0.998973i $$0.514429\pi$$
$$488$$ 10.0000 0.452679
$$489$$ 20.0000 0.904431
$$490$$ 0 0
$$491$$ 6.00000 0.270776 0.135388 0.990793i $$-0.456772\pi$$
0.135388 + 0.990793i $$0.456772\pi$$
$$492$$ 6.00000 0.270501
$$493$$ 0 0
$$494$$ 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 $$-0.398544\pi$$
0.313363 + 0.949633i $$0.398544\pi$$
$$500$$ 0 0
$$501$$ 12.0000 0.536120
$$502$$ −18.0000 −0.803379
$$503$$ −6.00000 −0.267527 −0.133763 0.991013i $$-0.542706\pi$$
−0.133763 + 0.991013i $$0.542706\pi$$
$$504$$ 2.00000 0.0890871
$$505$$ 0 0
$$506$$ 0 0
$$507$$ −1.00000 −0.0444116
$$508$$ −8.00000 −0.354943
$$509$$ 18.0000 0.797836 0.398918 0.916987i $$-0.369386\pi$$
0.398918 + 0.916987i $$0.369386\pi$$
$$510$$ 0 0
$$511$$ 16.0000 0.707798
$$512$$ −1.00000 −0.0441942
$$513$$ −2.00000 −0.0883022
$$514$$ 12.0000 0.529297
$$515$$ 0 0
$$516$$ −4.00000 −0.176090
$$517$$ 0 0
$$518$$ −4.00000 −0.175750
$$519$$ −6.00000 −0.263371
$$520$$ 0 0
$$521$$ −18.0000 −0.788594 −0.394297 0.918983i $$-0.629012\pi$$
−0.394297 + 0.918983i $$0.629012\pi$$
$$522$$ 0 0
$$523$$ 4.00000 0.174908 0.0874539 0.996169i $$-0.472127\pi$$
0.0874539 + 0.996169i $$0.472127\pi$$
$$524$$ −18.0000 −0.786334
$$525$$ 0 0
$$526$$ 6.00000 0.261612
$$527$$ 0 0
$$528$$ 0 0
$$529$$ 13.0000 0.565217
$$530$$ 0 0
$$531$$ 0 0
$$532$$ −4.00000 −0.173422
$$533$$ 6.00000 0.259889
$$534$$ 6.00000 0.259645
$$535$$ 0 0
$$536$$ 8.00000 0.345547
$$537$$ −18.0000 −0.776757
$$538$$ −12.0000 −0.517357
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 20.0000 0.859867 0.429934 0.902861i $$-0.358537\pi$$
0.429934 + 0.902861i $$0.358537\pi$$
$$542$$ 4.00000 0.171815
$$543$$ 22.0000 0.944110
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 2.00000 0.0855921
$$547$$ 4.00000 0.171028 0.0855138 0.996337i $$-0.472747\pi$$
0.0855138 + 0.996337i $$0.472747\pi$$
$$548$$ −18.0000 −0.768922
$$549$$ −10.0000 −0.426790
$$550$$ 0 0
$$551$$ 0 0
$$552$$ 6.00000 0.255377
$$553$$ −16.0000 −0.680389
$$554$$ −22.0000 −0.934690
$$555$$ 0 0
$$556$$ 8.00000 0.339276
$$557$$ 6.00000 0.254228 0.127114 0.991888i $$-0.459429\pi$$
0.127114 + 0.991888i $$0.459429\pi$$
$$558$$ 4.00000 0.169334
$$559$$ −4.00000 −0.169182
$$560$$ 0 0
$$561$$ 0 0
$$562$$ −6.00000 −0.253095
$$563$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ −4.00000 −0.168133
$$567$$ −2.00000 −0.0839921
$$568$$ 0 0
$$569$$ 30.0000 1.25767 0.628833 0.777541i $$-0.283533\pi$$
0.628833 + 0.777541i $$0.283533\pi$$
$$570$$ 0 0
$$571$$ −16.0000 −0.669579 −0.334790 0.942293i $$-0.608665\pi$$
−0.334790 + 0.942293i $$0.608665\pi$$
$$572$$ 0 0
$$573$$ −24.0000 −1.00261
$$574$$ −12.0000 −0.500870
$$575$$ 0 0
$$576$$ 1.00000 0.0416667
$$577$$ 4.00000 0.166522 0.0832611 0.996528i $$-0.473466\pi$$
0.0832611 + 0.996528i $$0.473466\pi$$
$$578$$ 17.0000 0.707107
$$579$$ 20.0000 0.831172
$$580$$ 0 0
$$581$$ −24.0000 −0.995688
$$582$$ −8.00000 −0.331611
$$583$$ 0 0
$$584$$ 8.00000 0.331042
$$585$$ 0 0
$$586$$ 6.00000 0.247858
$$587$$ −36.0000 −1.48588 −0.742940 0.669359i $$-0.766569\pi$$
−0.742940 + 0.669359i $$0.766569\pi$$
$$588$$ 3.00000 0.123718
$$589$$ −8.00000 −0.329634
$$590$$ 0 0
$$591$$ 6.00000 0.246807
$$592$$ −2.00000 −0.0821995
$$593$$ −6.00000 −0.246390 −0.123195 0.992382i $$-0.539314\pi$$
−0.123195 + 0.992382i $$0.539314\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 6.00000 0.245770
$$597$$ 16.0000 0.654836
$$598$$ 6.00000 0.245358
$$599$$ −24.0000 −0.980613 −0.490307 0.871550i $$-0.663115\pi$$
−0.490307 + 0.871550i $$0.663115\pi$$
$$600$$ 0 0
$$601$$ 38.0000 1.55005 0.775026 0.631929i $$-0.217737\pi$$
0.775026 + 0.631929i $$0.217737\pi$$
$$602$$ 8.00000 0.326056
$$603$$ −8.00000 −0.325785
$$604$$ 20.0000 0.813788
$$605$$ 0 0
$$606$$ −12.0000 −0.487467
$$607$$ −32.0000 −1.29884 −0.649420 0.760430i $$-0.724988\pi$$
−0.649420 + 0.760430i $$0.724988\pi$$
$$608$$ −2.00000 −0.0811107
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 0 0
$$613$$ −26.0000 −1.05013 −0.525065 0.851062i $$-0.675959\pi$$
−0.525065 + 0.851062i $$0.675959\pi$$
$$614$$ −16.0000 −0.645707
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −18.0000 −0.724653 −0.362326 0.932051i $$-0.618017\pi$$
−0.362326 + 0.932051i $$0.618017\pi$$
$$618$$ −8.00000 −0.321807
$$619$$ 14.0000 0.562708 0.281354 0.959604i $$-0.409217\pi$$
0.281354 + 0.959604i $$0.409217\pi$$
$$620$$ 0 0
$$621$$ −6.00000 −0.240772
$$622$$ 0 0
$$623$$ −12.0000 −0.480770
$$624$$ 1.00000 0.0400320
$$625$$ 0 0
$$626$$ 14.0000 0.559553
$$627$$ 0 0
$$628$$ −2.00000 −0.0798087
$$629$$ 0 0
$$630$$ 0 0
$$631$$ 8.00000 0.318475 0.159237 0.987240i $$-0.449096\pi$$
0.159237 + 0.987240i $$0.449096\pi$$
$$632$$ −8.00000 −0.318223
$$633$$ 4.00000 0.158986
$$634$$ −30.0000 −1.19145
$$635$$ 0 0
$$636$$ −6.00000 −0.237915
$$637$$ 3.00000 0.118864
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 42.0000 1.65890 0.829450 0.558581i $$-0.188654\pi$$
0.829450 + 0.558581i $$0.188654\pi$$
$$642$$ −12.0000 −0.473602
$$643$$ 40.0000 1.57745 0.788723 0.614749i $$-0.210743\pi$$
0.788723 + 0.614749i $$0.210743\pi$$
$$644$$ −12.0000 −0.472866
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 18.0000 0.707653 0.353827 0.935311i $$-0.384880\pi$$
0.353827 + 0.935311i $$0.384880\pi$$
$$648$$ −1.00000 −0.0392837
$$649$$ 0 0
$$650$$ 0 0
$$651$$ −8.00000 −0.313545
$$652$$ −20.0000 −0.783260
$$653$$ −6.00000 −0.234798 −0.117399 0.993085i $$-0.537456\pi$$
−0.117399 + 0.993085i $$0.537456\pi$$
$$654$$ −16.0000 −0.625650
$$655$$ 0 0
$$656$$ −6.00000 −0.234261
$$657$$ −8.00000 −0.312110
$$658$$ 0 0
$$659$$ −18.0000 −0.701180 −0.350590 0.936529i $$-0.614019\pi$$
−0.350590 + 0.936529i $$0.614019\pi$$
$$660$$ 0 0
$$661$$ −28.0000 −1.08907 −0.544537 0.838737i $$-0.683295\pi$$
−0.544537 + 0.838737i $$0.683295\pi$$
$$662$$ −26.0000 −1.01052
$$663$$ 0 0
$$664$$ −12.0000 −0.465690
$$665$$ 0 0
$$666$$ 2.00000 0.0774984
$$667$$ 0 0
$$668$$ −12.0000 −0.464294
$$669$$ 14.0000 0.541271
$$670$$ 0 0
$$671$$ 0 0
$$672$$ −2.00000 −0.0771517
$$673$$ −26.0000 −1.00223 −0.501113 0.865382i $$-0.667076\pi$$
−0.501113 + 0.865382i $$0.667076\pi$$
$$674$$ −34.0000 −1.30963
$$675$$ 0 0
$$676$$ 1.00000 0.0384615
$$677$$ −6.00000 −0.230599 −0.115299 0.993331i $$-0.536783\pi$$
−0.115299 + 0.993331i $$0.536783\pi$$
$$678$$ 12.0000 0.460857
$$679$$ 16.0000 0.614024
$$680$$ 0 0
$$681$$ 12.0000 0.459841
$$682$$ 0 0
$$683$$ −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
$$687$$ 16.0000 0.610438
$$688$$ 4.00000 0.152499
$$689$$ −6.00000 −0.228582
$$690$$ 0 0
$$691$$ 2.00000 0.0760836 0.0380418 0.999276i $$-0.487888\pi$$
0.0380418 + 0.999276i $$0.487888\pi$$
$$692$$ 6.00000 0.228086
$$693$$ 0 0
$$694$$ 24.0000 0.911028
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 0 0
$$698$$ 16.0000 0.605609
$$699$$ 24.0000 0.907763
$$700$$ 0 0
$$701$$ −36.0000 −1.35970 −0.679851 0.733351i $$-0.737955\pi$$
−0.679851 + 0.733351i $$0.737955\pi$$
$$702$$ −1.00000 −0.0377426
$$703$$ −4.00000 −0.150863
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 18.0000 0.677439
$$707$$ 24.0000 0.902613
$$708$$ 0 0
$$709$$ 8.00000 0.300446 0.150223 0.988652i $$-0.452001\pi$$
0.150223 + 0.988652i $$0.452001\pi$$
$$710$$ 0 0
$$711$$ 8.00000 0.300023
$$712$$ −6.00000 −0.224860
$$713$$ −24.0000 −0.898807
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 18.0000 0.672692
$$717$$ 24.0000 0.896296
$$718$$ 0 0
$$719$$ 36.0000 1.34257 0.671287 0.741198i $$-0.265742\pi$$
0.671287 + 0.741198i $$0.265742\pi$$
$$720$$ 0 0
$$721$$ 16.0000 0.595871
$$722$$ 15.0000 0.558242
$$723$$ −2.00000 −0.0743808
$$724$$ −22.0000 −0.817624
$$725$$ 0 0
$$726$$ −11.0000 −0.408248
$$727$$ 28.0000 1.03846 0.519231 0.854634i $$-0.326218\pi$$
0.519231 + 0.854634i $$0.326218\pi$$
$$728$$ −2.00000 −0.0741249
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ 0 0
$$732$$ 10.0000 0.369611
$$733$$ −26.0000 −0.960332 −0.480166 0.877178i $$-0.659424\pi$$
−0.480166 + 0.877178i $$0.659424\pi$$
$$734$$ −16.0000 −0.590571
$$735$$ 0 0
$$736$$ −6.00000 −0.221163
$$737$$ 0 0
$$738$$ 6.00000 0.220863
$$739$$ −34.0000 −1.25071 −0.625355 0.780340i $$-0.715046\pi$$
−0.625355 + 0.780340i $$0.715046\pi$$
$$740$$ 0 0
$$741$$ 2.00000 0.0734718
$$742$$ 12.0000 0.440534
$$743$$ 12.0000 0.440237 0.220119 0.975473i $$-0.429356\pi$$
0.220119 + 0.975473i $$0.429356\pi$$
$$744$$ −4.00000 −0.146647
$$745$$ 0 0
$$746$$ −22.0000 −0.805477
$$747$$ 12.0000 0.439057
$$748$$ 0 0
$$749$$ 24.0000 0.876941
$$750$$ 0 0
$$751$$ 32.0000 1.16770 0.583848 0.811863i $$-0.301546\pi$$
0.583848 + 0.811863i $$0.301546\pi$$
$$752$$ 0 0
$$753$$ −18.0000 −0.655956
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 2.00000 0.0727393
$$757$$ −14.0000 −0.508839 −0.254419 0.967094i $$-0.581884\pi$$
−0.254419 + 0.967094i $$0.581884\pi$$
$$758$$ 10.0000 0.363216
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −54.0000 −1.95750 −0.978749 0.205061i $$-0.934261\pi$$
−0.978749 + 0.205061i $$0.934261\pi$$
$$762$$ −8.00000 −0.289809
$$763$$ 32.0000 1.15848
$$764$$ 24.0000 0.868290
$$765$$ 0 0
$$766$$ −36.0000 −1.30073
$$767$$ 0 0
$$768$$ −1.00000 −0.0360844
$$769$$ −22.0000 −0.793340 −0.396670 0.917961i $$-0.629834\pi$$
−0.396670 + 0.917961i $$0.629834\pi$$
$$770$$ 0 0
$$771$$ 12.0000 0.432169
$$772$$ −20.0000 −0.719816
$$773$$ 42.0000 1.51064 0.755318 0.655359i $$-0.227483\pi$$
0.755318 + 0.655359i $$0.227483\pi$$
$$774$$ −4.00000 −0.143777
$$775$$ 0 0
$$776$$ 8.00000 0.287183
$$777$$ −4.00000 −0.143499
$$778$$ −12.0000 −0.430221
$$779$$ −12.0000 −0.429945
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ 0 0
$$784$$ −3.00000 −0.107143
$$785$$ 0 0
$$786$$ −18.0000 −0.642039
$$787$$ 4.00000 0.142585 0.0712923 0.997455i $$-0.477288\pi$$
0.0712923 + 0.997455i $$0.477288\pi$$
$$788$$ −6.00000 −0.213741
$$789$$ 6.00000 0.213606
$$790$$ 0 0
$$791$$ −24.0000 −0.853342
$$792$$ 0 0
$$793$$ 10.0000 0.355110
$$794$$ −22.0000 −0.780751
$$795$$ 0 0
$$796$$ −16.0000 −0.567105
$$797$$ −18.0000 −0.637593 −0.318796 0.947823i $$-0.603279\pi$$
−0.318796 + 0.947823i $$0.603279\pi$$
$$798$$ −4.00000 −0.141598
$$799$$ 0 0
$$800$$ 0 0
$$801$$ 6.00000 0.212000
$$802$$ 6.00000 0.211867
$$803$$ 0 0
$$804$$ 8.00000 0.282138
$$805$$ 0 0
$$806$$ −4.00000 −0.140894
$$807$$ −12.0000 −0.422420
$$808$$ 12.0000 0.422159
$$809$$ 18.0000 0.632846 0.316423 0.948618i $$-0.397518\pi$$
0.316423 + 0.948618i $$0.397518\pi$$
$$810$$ 0 0
$$811$$ −22.0000 −0.772524 −0.386262 0.922389i $$-0.626234\pi$$
−0.386262 + 0.922389i $$0.626234\pi$$
$$812$$ 0 0
$$813$$ 4.00000 0.140286
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 8.00000 0.279885
$$818$$ −14.0000 −0.489499
$$819$$ 2.00000 0.0698857
$$820$$ 0 0
$$821$$ 30.0000 1.04701 0.523504 0.852023i $$-0.324625\pi$$
0.523504 + 0.852023i $$0.324625\pi$$
$$822$$ −18.0000 −0.627822
$$823$$ 40.0000 1.39431 0.697156 0.716919i $$-0.254448\pi$$
0.697156 + 0.716919i $$0.254448\pi$$
$$824$$ 8.00000 0.278693
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 36.0000 1.25184 0.625921 0.779886i $$-0.284723\pi$$
0.625921 + 0.779886i $$0.284723\pi$$
$$828$$ 6.00000 0.208514
$$829$$ −34.0000 −1.18087 −0.590434 0.807086i $$-0.701044\pi$$
−0.590434 + 0.807086i $$0.701044\pi$$
$$830$$ 0 0
$$831$$ −22.0000 −0.763172
$$832$$ −1.00000 −0.0346688
$$833$$ 0 0
$$834$$ 8.00000 0.277017
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 4.00000 0.138260
$$838$$ 18.0000 0.621800
$$839$$ −24.0000 −0.828572 −0.414286 0.910147i $$-0.635969\pi$$
−0.414286 + 0.910147i $$0.635969\pi$$
$$840$$ 0 0
$$841$$ −29.0000 −1.00000
$$842$$ 4.00000 0.137849
$$843$$ −6.00000 −0.206651
$$844$$ −4.00000 −0.137686
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 22.0000 0.755929
$$848$$ 6.00000 0.206041
$$849$$ −4.00000 −0.137280
$$850$$ 0 0
$$851$$ −12.0000 −0.411355
$$852$$ 0 0
$$853$$ −26.0000 −0.890223 −0.445112 0.895475i $$-0.646836\pi$$
−0.445112 + 0.895475i $$0.646836\pi$$
$$854$$ −20.0000 −0.684386
$$855$$ 0 0
$$856$$ 12.0000 0.410152
$$857$$ −12.0000 −0.409912 −0.204956 0.978771i $$-0.565705\pi$$
−0.204956 + 0.978771i $$0.565705\pi$$
$$858$$ 0 0
$$859$$ 20.0000 0.682391 0.341196 0.939992i $$-0.389168\pi$$
0.341196 + 0.939992i $$0.389168\pi$$
$$860$$ 0 0
$$861$$ −12.0000 −0.408959
$$862$$ −24.0000 −0.817443
$$863$$ −36.0000 −1.22545 −0.612727 0.790295i $$-0.709928\pi$$
−0.612727 + 0.790295i $$0.709928\pi$$
$$864$$ 1.00000 0.0340207
$$865$$ 0 0
$$866$$ −34.0000 −1.15537
$$867$$ 17.0000 0.577350
$$868$$ 8.00000 0.271538
$$869$$ 0 0
$$870$$ 0 0
$$871$$ 8.00000 0.271070
$$872$$ 16.0000 0.541828
$$873$$ −8.00000 −0.270759
$$874$$ −12.0000 −0.405906
$$875$$ 0 0
$$876$$ 8.00000 0.270295
$$877$$ 34.0000 1.14810 0.574049 0.818821i $$-0.305372\pi$$
0.574049 + 0.818821i $$0.305372\pi$$
$$878$$ −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
$$883$$ 52.0000 1.74994 0.874970 0.484178i $$-0.160881\pi$$
0.874970 + 0.484178i $$0.160881\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ −24.0000 −0.806296
$$887$$ −18.0000 −0.604381 −0.302190 0.953248i $$-0.597718\pi$$
−0.302190 + 0.953248i $$0.597718\pi$$
$$888$$ −2.00000 −0.0671156
$$889$$ 16.0000 0.536623
$$890$$ 0 0
$$891$$ 0 0
$$892$$ −14.0000 −0.468755
$$893$$ 0 0
$$894$$ 6.00000 0.200670
$$895$$ 0 0
$$896$$ 2.00000 0.0668153
$$897$$ 6.00000 0.200334
$$898$$ 18.0000 0.600668
$$899$$ 0 0
$$900$$ 0 0
$$901$$ 0 0
$$902$$ 0 0
$$903$$ 8.00000 0.266223
$$904$$ −12.0000 −0.399114
$$905$$ 0 0
$$906$$ 20.0000 0.664455
$$907$$ −44.0000 −1.46100 −0.730498 0.682915i $$-0.760712\pi$$
−0.730498 + 0.682915i $$0.760712\pi$$
$$908$$ −12.0000 −0.398234
$$909$$ −12.0000 −0.398015
$$910$$ 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
$$913$$ 0 0
$$914$$ 32.0000 1.05847
$$915$$ 0 0
$$916$$ −16.0000 −0.528655
$$917$$ 36.0000 1.18882
$$918$$ 0 0
$$919$$ 56.0000 1.84727 0.923635 0.383274i $$-0.125203\pi$$
0.923635 + 0.383274i $$0.125203\pi$$
$$920$$ 0 0
$$921$$ −16.0000 −0.527218
$$922$$ −18.0000 −0.592798
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 14.0000 0.460069
$$927$$ −8.00000 −0.262754
$$928$$ 0 0
$$929$$ 30.0000 0.984268 0.492134 0.870519i $$-0.336217\pi$$
0.492134 + 0.870519i $$0.336217\pi$$
$$930$$ 0 0
$$931$$ −6.00000 −0.196642
$$932$$ −24.0000 −0.786146
$$933$$ 0 0
$$934$$ −12.0000 −0.392652
$$935$$ 0 0
$$936$$ 1.00000 0.0326860
$$937$$ −26.0000 −0.849383 −0.424691 0.905338i $$-0.639617\pi$$
−0.424691 + 0.905338i $$0.639617\pi$$
$$938$$ −16.0000 −0.522419
$$939$$ 14.0000 0.456873
$$940$$ 0 0
$$941$$ 6.00000 0.195594 0.0977972 0.995206i $$-0.468820\pi$$
0.0977972 + 0.995206i $$0.468820\pi$$
$$942$$ −2.00000 −0.0651635
$$943$$ −36.0000 −1.17232
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −36.0000 −1.16984 −0.584921 0.811090i $$-0.698875\pi$$
−0.584921 + 0.811090i $$0.698875\pi$$
$$948$$ −8.00000 −0.259828
$$949$$ 8.00000 0.259691
$$950$$ 0 0
$$951$$ −30.0000 −0.972817
$$952$$ 0 0
$$953$$ −48.0000 −1.55487 −0.777436 0.628962i $$-0.783480\pi$$
−0.777436 + 0.628962i $$0.783480\pi$$
$$954$$ −6.00000 −0.194257
$$955$$ 0 0
$$956$$ −24.0000 −0.776215
$$957$$ 0 0
$$958$$ 24.0000 0.775405
$$959$$ 36.0000 1.16250
$$960$$ 0 0
$$961$$ −15.0000 −0.483871
$$962$$ −2.00000 −0.0644826
$$963$$ −12.0000 −0.386695
$$964$$ 2.00000 0.0644157
$$965$$ 0 0
$$966$$ −12.0000 −0.386094
$$967$$ −50.0000 −1.60789 −0.803946 0.594703i $$-0.797270\pi$$
−0.803946 + 0.594703i $$0.797270\pi$$
$$968$$ 11.0000 0.353553
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 18.0000 0.577647 0.288824 0.957382i $$-0.406736\pi$$
0.288824 + 0.957382i $$0.406736\pi$$
$$972$$ −1.00000 −0.0320750
$$973$$ −16.0000 −0.512936
$$974$$ 2.00000 0.0640841
$$975$$ 0 0
$$976$$ −10.0000 −0.320092
$$977$$ −6.00000 −0.191957 −0.0959785 0.995383i $$-0.530598\pi$$
−0.0959785 + 0.995383i $$0.530598\pi$$
$$978$$ −20.0000 −0.639529
$$979$$ 0 0
$$980$$ 0 0
$$981$$ −16.0000 −0.510841
$$982$$ −6.00000 −0.191468
$$983$$ 12.0000 0.382741 0.191370 0.981518i $$-0.438707\pi$$
0.191370 + 0.981518i $$0.438707\pi$$
$$984$$ −6.00000 −0.191273
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 0 0
$$988$$ −2.00000 −0.0636285
$$989$$ 24.0000 0.763156
$$990$$ 0 0
$$991$$ 32.0000 1.01651 0.508257 0.861206i $$-0.330290\pi$$
0.508257 + 0.861206i $$0.330290\pi$$
$$992$$ 4.00000 0.127000
$$993$$ −26.0000 −0.825085
$$994$$ 0 0
$$995$$ 0 0
$$996$$ −12.0000 −0.380235
$$997$$ 46.0000 1.45683 0.728417 0.685134i $$-0.240256\pi$$
0.728417 + 0.685134i $$0.240256\pi$$
$$998$$ −14.0000 −0.443162
$$999$$ 2.00000 0.0632772
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.b.1.1 1
3.2 odd 2 5850.2.a.bk.1.1 1
5.2 odd 4 1950.2.e.k.1249.1 2
5.3 odd 4 1950.2.e.k.1249.2 2
5.4 even 2 390.2.a.g.1.1 1
15.2 even 4 5850.2.e.r.5149.2 2
15.8 even 4 5850.2.e.r.5149.1 2
15.14 odd 2 1170.2.a.g.1.1 1
20.19 odd 2 3120.2.a.b.1.1 1
60.59 even 2 9360.2.a.bg.1.1 1
65.34 odd 4 5070.2.b.n.1351.2 2
65.44 odd 4 5070.2.b.n.1351.1 2
65.64 even 2 5070.2.a.k.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
390.2.a.g.1.1 1 5.4 even 2
1170.2.a.g.1.1 1 15.14 odd 2
1950.2.a.b.1.1 1 1.1 even 1 trivial
1950.2.e.k.1249.1 2 5.2 odd 4
1950.2.e.k.1249.2 2 5.3 odd 4
3120.2.a.b.1.1 1 20.19 odd 2
5070.2.a.k.1.1 1 65.64 even 2
5070.2.b.n.1351.1 2 65.44 odd 4
5070.2.b.n.1351.2 2 65.34 odd 4
5850.2.a.bk.1.1 1 3.2 odd 2
5850.2.e.r.5149.1 2 15.8 even 4
5850.2.e.r.5149.2 2 15.2 even 4
9360.2.a.bg.1.1 1 60.59 even 2