# Properties

 Label 1050.2.a.a.1.1 Level $1050$ Weight $2$ Character 1050.1 Self dual yes Analytic conductor $8.384$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1050.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$8.38429221223$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 210) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 1050.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{12} -2.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} +6.00000 q^{17} -1.00000 q^{18} -4.00000 q^{19} +1.00000 q^{21} +1.00000 q^{24} +2.00000 q^{26} -1.00000 q^{27} -1.00000 q^{28} -6.00000 q^{29} -4.00000 q^{31} -1.00000 q^{32} -6.00000 q^{34} +1.00000 q^{36} -2.00000 q^{37} +4.00000 q^{38} +2.00000 q^{39} +6.00000 q^{41} -1.00000 q^{42} -8.00000 q^{43} +12.0000 q^{47} -1.00000 q^{48} +1.00000 q^{49} -6.00000 q^{51} -2.00000 q^{52} -6.00000 q^{53} +1.00000 q^{54} +1.00000 q^{56} +4.00000 q^{57} +6.00000 q^{58} -12.0000 q^{59} +2.00000 q^{61} +4.00000 q^{62} -1.00000 q^{63} +1.00000 q^{64} -8.00000 q^{67} +6.00000 q^{68} -1.00000 q^{72} -14.0000 q^{73} +2.00000 q^{74} -4.00000 q^{76} -2.00000 q^{78} -16.0000 q^{79} +1.00000 q^{81} -6.00000 q^{82} -12.0000 q^{83} +1.00000 q^{84} +8.00000 q^{86} +6.00000 q^{87} +6.00000 q^{89} +2.00000 q^{91} +4.00000 q^{93} -12.0000 q^{94} +1.00000 q^{96} -14.0000 q^{97} -1.00000 q^{98} +O(q^{100})$$

## 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
$$3$$ −1.00000 −0.577350
$$4$$ 1.00000 0.500000
$$5$$ 0 0
$$6$$ 1.00000 0.408248
$$7$$ −1.00000 −0.377964
$$8$$ −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
$$13$$ −2.00000 −0.554700 −0.277350 0.960769i $$-0.589456\pi$$
−0.277350 + 0.960769i $$0.589456\pi$$
$$14$$ 1.00000 0.267261
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 6.00000 1.45521 0.727607 0.685994i $$-0.240633\pi$$
0.727607 + 0.685994i $$0.240633\pi$$
$$18$$ −1.00000 −0.235702
$$19$$ −4.00000 −0.917663 −0.458831 0.888523i $$-0.651732\pi$$
−0.458831 + 0.888523i $$0.651732\pi$$
$$20$$ 0 0
$$21$$ 1.00000 0.218218
$$22$$ 0 0
$$23$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$24$$ 1.00000 0.204124
$$25$$ 0 0
$$26$$ 2.00000 0.392232
$$27$$ −1.00000 −0.192450
$$28$$ −1.00000 −0.188982
$$29$$ −6.00000 −1.11417 −0.557086 0.830455i $$-0.688081\pi$$
−0.557086 + 0.830455i $$0.688081\pi$$
$$30$$ 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
$$34$$ −6.00000 −1.02899
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ −2.00000 −0.328798 −0.164399 0.986394i $$-0.552568\pi$$
−0.164399 + 0.986394i $$0.552568\pi$$
$$38$$ 4.00000 0.648886
$$39$$ 2.00000 0.320256
$$40$$ 0 0
$$41$$ 6.00000 0.937043 0.468521 0.883452i $$-0.344787\pi$$
0.468521 + 0.883452i $$0.344787\pi$$
$$42$$ −1.00000 −0.154303
$$43$$ −8.00000 −1.21999 −0.609994 0.792406i $$-0.708828\pi$$
−0.609994 + 0.792406i $$0.708828\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 12.0000 1.75038 0.875190 0.483779i $$-0.160736\pi$$
0.875190 + 0.483779i $$0.160736\pi$$
$$48$$ −1.00000 −0.144338
$$49$$ 1.00000 0.142857
$$50$$ 0 0
$$51$$ −6.00000 −0.840168
$$52$$ −2.00000 −0.277350
$$53$$ −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
$$56$$ 1.00000 0.133631
$$57$$ 4.00000 0.529813
$$58$$ 6.00000 0.787839
$$59$$ −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 $$-0.459132\pi$$
0.128037 + 0.991769i $$0.459132\pi$$
$$62$$ 4.00000 0.508001
$$63$$ −1.00000 −0.125988
$$64$$ 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 $$0.662521\pi$$
$$68$$ 6.00000 0.727607
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$72$$ −1.00000 −0.117851
$$73$$ −14.0000 −1.63858 −0.819288 0.573382i $$-0.805631\pi$$
−0.819288 + 0.573382i $$0.805631\pi$$
$$74$$ 2.00000 0.232495
$$75$$ 0 0
$$76$$ −4.00000 −0.458831
$$77$$ 0 0
$$78$$ −2.00000 −0.226455
$$79$$ −16.0000 −1.80014 −0.900070 0.435745i $$-0.856485\pi$$
−0.900070 + 0.435745i $$0.856485\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ −6.00000 −0.662589
$$83$$ −12.0000 −1.31717 −0.658586 0.752506i $$-0.728845\pi$$
−0.658586 + 0.752506i $$0.728845\pi$$
$$84$$ 1.00000 0.109109
$$85$$ 0 0
$$86$$ 8.00000 0.862662
$$87$$ 6.00000 0.643268
$$88$$ 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
$$92$$ 0 0
$$93$$ 4.00000 0.414781
$$94$$ −12.0000 −1.23771
$$95$$ 0 0
$$96$$ 1.00000 0.102062
$$97$$ −14.0000 −1.42148 −0.710742 0.703452i $$-0.751641\pi$$
−0.710742 + 0.703452i $$0.751641\pi$$
$$98$$ −1.00000 −0.101015
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −6.00000 −0.597022 −0.298511 0.954406i $$-0.596490\pi$$
−0.298511 + 0.954406i $$0.596490\pi$$
$$102$$ 6.00000 0.594089
$$103$$ 16.0000 1.57653 0.788263 0.615338i $$-0.210980\pi$$
0.788263 + 0.615338i $$0.210980\pi$$
$$104$$ 2.00000 0.196116
$$105$$ 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
$$109$$ 14.0000 1.34096 0.670478 0.741929i $$-0.266089\pi$$
0.670478 + 0.741929i $$0.266089\pi$$
$$110$$ 0 0
$$111$$ 2.00000 0.189832
$$112$$ −1.00000 −0.0944911
$$113$$ −18.0000 −1.69330 −0.846649 0.532152i $$-0.821383\pi$$
−0.846649 + 0.532152i $$0.821383\pi$$
$$114$$ −4.00000 −0.374634
$$115$$ 0 0
$$116$$ −6.00000 −0.557086
$$117$$ −2.00000 −0.184900
$$118$$ 12.0000 1.10469
$$119$$ −6.00000 −0.550019
$$120$$ 0 0
$$121$$ −11.0000 −1.00000
$$122$$ −2.00000 −0.181071
$$123$$ −6.00000 −0.541002
$$124$$ −4.00000 −0.359211
$$125$$ 0 0
$$126$$ 1.00000 0.0890871
$$127$$ −8.00000 −0.709885 −0.354943 0.934888i $$-0.615500\pi$$
−0.354943 + 0.934888i $$0.615500\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ 8.00000 0.704361
$$130$$ 0 0
$$131$$ 12.0000 1.04844 0.524222 0.851581i $$-0.324356\pi$$
0.524222 + 0.851581i $$0.324356\pi$$
$$132$$ 0 0
$$133$$ 4.00000 0.346844
$$134$$ 8.00000 0.691095
$$135$$ 0 0
$$136$$ −6.00000 −0.514496
$$137$$ 6.00000 0.512615 0.256307 0.966595i $$-0.417494\pi$$
0.256307 + 0.966595i $$0.417494\pi$$
$$138$$ 0 0
$$139$$ −4.00000 −0.339276 −0.169638 0.985506i $$-0.554260\pi$$
−0.169638 + 0.985506i $$0.554260\pi$$
$$140$$ 0 0
$$141$$ −12.0000 −1.01058
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 1.00000 0.0833333
$$145$$ 0 0
$$146$$ 14.0000 1.15865
$$147$$ −1.00000 −0.0824786
$$148$$ −2.00000 −0.164399
$$149$$ 18.0000 1.47462 0.737309 0.675556i $$-0.236096\pi$$
0.737309 + 0.675556i $$0.236096\pi$$
$$150$$ 0 0
$$151$$ −16.0000 −1.30206 −0.651031 0.759051i $$-0.725663\pi$$
−0.651031 + 0.759051i $$0.725663\pi$$
$$152$$ 4.00000 0.324443
$$153$$ 6.00000 0.485071
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 2.00000 0.160128
$$157$$ −2.00000 −0.159617 −0.0798087 0.996810i $$-0.525431\pi$$
−0.0798087 + 0.996810i $$0.525431\pi$$
$$158$$ 16.0000 1.27289
$$159$$ 6.00000 0.475831
$$160$$ 0 0
$$161$$ 0 0
$$162$$ −1.00000 −0.0785674
$$163$$ 16.0000 1.25322 0.626608 0.779334i $$-0.284443\pi$$
0.626608 + 0.779334i $$0.284443\pi$$
$$164$$ 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 $$0.653692\pi$$
$$168$$ −1.00000 −0.0771517
$$169$$ −9.00000 −0.692308
$$170$$ 0 0
$$171$$ −4.00000 −0.305888
$$172$$ −8.00000 −0.609994
$$173$$ 6.00000 0.456172 0.228086 0.973641i $$-0.426753\pi$$
0.228086 + 0.973641i $$0.426753\pi$$
$$174$$ −6.00000 −0.454859
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 12.0000 0.901975
$$178$$ −6.00000 −0.449719
$$179$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$180$$ 0 0
$$181$$ 2.00000 0.148659 0.0743294 0.997234i $$-0.476318\pi$$
0.0743294 + 0.997234i $$0.476318\pi$$
$$182$$ −2.00000 −0.148250
$$183$$ −2.00000 −0.147844
$$184$$ 0 0
$$185$$ 0 0
$$186$$ −4.00000 −0.293294
$$187$$ 0 0
$$188$$ 12.0000 0.875190
$$189$$ 1.00000 0.0727393
$$190$$ 0 0
$$191$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$192$$ −1.00000 −0.0721688
$$193$$ 22.0000 1.58359 0.791797 0.610784i $$-0.209146\pi$$
0.791797 + 0.610784i $$0.209146\pi$$
$$194$$ 14.0000 1.00514
$$195$$ 0 0
$$196$$ 1.00000 0.0714286
$$197$$ 18.0000 1.28245 0.641223 0.767354i $$-0.278427\pi$$
0.641223 + 0.767354i $$0.278427\pi$$
$$198$$ 0 0
$$199$$ −4.00000 −0.283552 −0.141776 0.989899i $$-0.545281\pi$$
−0.141776 + 0.989899i $$0.545281\pi$$
$$200$$ 0 0
$$201$$ 8.00000 0.564276
$$202$$ 6.00000 0.422159
$$203$$ 6.00000 0.421117
$$204$$ −6.00000 −0.420084
$$205$$ 0 0
$$206$$ −16.0000 −1.11477
$$207$$ 0 0
$$208$$ −2.00000 −0.138675
$$209$$ 0 0
$$210$$ 0 0
$$211$$ −28.0000 −1.92760 −0.963800 0.266627i $$-0.914091\pi$$
−0.963800 + 0.266627i $$0.914091\pi$$
$$212$$ −6.00000 −0.412082
$$213$$ 0 0
$$214$$ 12.0000 0.820303
$$215$$ 0 0
$$216$$ 1.00000 0.0680414
$$217$$ 4.00000 0.271538
$$218$$ −14.0000 −0.948200
$$219$$ 14.0000 0.946032
$$220$$ 0 0
$$221$$ −12.0000 −0.807207
$$222$$ −2.00000 −0.134231
$$223$$ 16.0000 1.07144 0.535720 0.844396i $$-0.320040\pi$$
0.535720 + 0.844396i $$0.320040\pi$$
$$224$$ 1.00000 0.0668153
$$225$$ 0 0
$$226$$ 18.0000 1.19734
$$227$$ 12.0000 0.796468 0.398234 0.917284i $$-0.369623\pi$$
0.398234 + 0.917284i $$0.369623\pi$$
$$228$$ 4.00000 0.264906
$$229$$ 2.00000 0.132164 0.0660819 0.997814i $$-0.478950\pi$$
0.0660819 + 0.997814i $$0.478950\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 6.00000 0.393919
$$233$$ 6.00000 0.393073 0.196537 0.980497i $$-0.437031\pi$$
0.196537 + 0.980497i $$0.437031\pi$$
$$234$$ 2.00000 0.130744
$$235$$ 0 0
$$236$$ −12.0000 −0.781133
$$237$$ 16.0000 1.03931
$$238$$ 6.00000 0.388922
$$239$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$240$$ 0 0
$$241$$ 26.0000 1.67481 0.837404 0.546585i $$-0.184072\pi$$
0.837404 + 0.546585i $$0.184072\pi$$
$$242$$ 11.0000 0.707107
$$243$$ −1.00000 −0.0641500
$$244$$ 2.00000 0.128037
$$245$$ 0 0
$$246$$ 6.00000 0.382546
$$247$$ 8.00000 0.509028
$$248$$ 4.00000 0.254000
$$249$$ 12.0000 0.760469
$$250$$ 0 0
$$251$$ 12.0000 0.757433 0.378717 0.925513i $$-0.376365\pi$$
0.378717 + 0.925513i $$0.376365\pi$$
$$252$$ −1.00000 −0.0629941
$$253$$ 0 0
$$254$$ 8.00000 0.501965
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 6.00000 0.374270 0.187135 0.982334i $$-0.440080\pi$$
0.187135 + 0.982334i $$0.440080\pi$$
$$258$$ −8.00000 −0.498058
$$259$$ 2.00000 0.124274
$$260$$ 0 0
$$261$$ −6.00000 −0.371391
$$262$$ −12.0000 −0.741362
$$263$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ −4.00000 −0.245256
$$267$$ −6.00000 −0.367194
$$268$$ −8.00000 −0.488678
$$269$$ 18.0000 1.09748 0.548740 0.835993i $$-0.315108\pi$$
0.548740 + 0.835993i $$0.315108\pi$$
$$270$$ 0 0
$$271$$ 20.0000 1.21491 0.607457 0.794353i $$-0.292190\pi$$
0.607457 + 0.794353i $$0.292190\pi$$
$$272$$ 6.00000 0.363803
$$273$$ −2.00000 −0.121046
$$274$$ −6.00000 −0.362473
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 22.0000 1.32185 0.660926 0.750451i $$-0.270164\pi$$
0.660926 + 0.750451i $$0.270164\pi$$
$$278$$ 4.00000 0.239904
$$279$$ −4.00000 −0.239474
$$280$$ 0 0
$$281$$ −30.0000 −1.78965 −0.894825 0.446417i $$-0.852700\pi$$
−0.894825 + 0.446417i $$0.852700\pi$$
$$282$$ 12.0000 0.714590
$$283$$ −20.0000 −1.18888 −0.594438 0.804141i $$-0.702626\pi$$
−0.594438 + 0.804141i $$0.702626\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −6.00000 −0.354169
$$288$$ −1.00000 −0.0589256
$$289$$ 19.0000 1.11765
$$290$$ 0 0
$$291$$ 14.0000 0.820695
$$292$$ −14.0000 −0.819288
$$293$$ 6.00000 0.350524 0.175262 0.984522i $$-0.443923\pi$$
0.175262 + 0.984522i $$0.443923\pi$$
$$294$$ 1.00000 0.0583212
$$295$$ 0 0
$$296$$ 2.00000 0.116248
$$297$$ 0 0
$$298$$ −18.0000 −1.04271
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 8.00000 0.461112
$$302$$ 16.0000 0.920697
$$303$$ 6.00000 0.344691
$$304$$ −4.00000 −0.229416
$$305$$ 0 0
$$306$$ −6.00000 −0.342997
$$307$$ −20.0000 −1.14146 −0.570730 0.821138i $$-0.693340\pi$$
−0.570730 + 0.821138i $$0.693340\pi$$
$$308$$ 0 0
$$309$$ −16.0000 −0.910208
$$310$$ 0 0
$$311$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$312$$ −2.00000 −0.113228
$$313$$ 10.0000 0.565233 0.282617 0.959233i $$-0.408798\pi$$
0.282617 + 0.959233i $$0.408798\pi$$
$$314$$ 2.00000 0.112867
$$315$$ 0 0
$$316$$ −16.0000 −0.900070
$$317$$ −30.0000 −1.68497 −0.842484 0.538721i $$-0.818908\pi$$
−0.842484 + 0.538721i $$0.818908\pi$$
$$318$$ −6.00000 −0.336463
$$319$$ 0 0
$$320$$ 0 0
$$321$$ 12.0000 0.669775
$$322$$ 0 0
$$323$$ −24.0000 −1.33540
$$324$$ 1.00000 0.0555556
$$325$$ 0 0
$$326$$ −16.0000 −0.886158
$$327$$ −14.0000 −0.774202
$$328$$ −6.00000 −0.331295
$$329$$ −12.0000 −0.661581
$$330$$ 0 0
$$331$$ −4.00000 −0.219860 −0.109930 0.993939i $$-0.535063\pi$$
−0.109930 + 0.993939i $$0.535063\pi$$
$$332$$ −12.0000 −0.658586
$$333$$ −2.00000 −0.109599
$$334$$ 12.0000 0.656611
$$335$$ 0 0
$$336$$ 1.00000 0.0545545
$$337$$ 22.0000 1.19842 0.599208 0.800593i $$-0.295482\pi$$
0.599208 + 0.800593i $$0.295482\pi$$
$$338$$ 9.00000 0.489535
$$339$$ 18.0000 0.977626
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 4.00000 0.216295
$$343$$ −1.00000 −0.0539949
$$344$$ 8.00000 0.431331
$$345$$ 0 0
$$346$$ −6.00000 −0.322562
$$347$$ −12.0000 −0.644194 −0.322097 0.946707i $$-0.604388\pi$$
−0.322097 + 0.946707i $$0.604388\pi$$
$$348$$ 6.00000 0.321634
$$349$$ 26.0000 1.39175 0.695874 0.718164i $$-0.255017\pi$$
0.695874 + 0.718164i $$0.255017\pi$$
$$350$$ 0 0
$$351$$ 2.00000 0.106752
$$352$$ 0 0
$$353$$ −18.0000 −0.958043 −0.479022 0.877803i $$-0.659008\pi$$
−0.479022 + 0.877803i $$0.659008\pi$$
$$354$$ −12.0000 −0.637793
$$355$$ 0 0
$$356$$ 6.00000 0.317999
$$357$$ 6.00000 0.317554
$$358$$ 0 0
$$359$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$360$$ 0 0
$$361$$ −3.00000 −0.157895
$$362$$ −2.00000 −0.105118
$$363$$ 11.0000 0.577350
$$364$$ 2.00000 0.104828
$$365$$ 0 0
$$366$$ 2.00000 0.104542
$$367$$ −8.00000 −0.417597 −0.208798 0.977959i $$-0.566955\pi$$
−0.208798 + 0.977959i $$0.566955\pi$$
$$368$$ 0 0
$$369$$ 6.00000 0.312348
$$370$$ 0 0
$$371$$ 6.00000 0.311504
$$372$$ 4.00000 0.207390
$$373$$ −26.0000 −1.34623 −0.673114 0.739538i $$-0.735044\pi$$
−0.673114 + 0.739538i $$0.735044\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ −12.0000 −0.618853
$$377$$ 12.0000 0.618031
$$378$$ −1.00000 −0.0514344
$$379$$ 20.0000 1.02733 0.513665 0.857991i $$-0.328287\pi$$
0.513665 + 0.857991i $$0.328287\pi$$
$$380$$ 0 0
$$381$$ 8.00000 0.409852
$$382$$ 0 0
$$383$$ 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
$$386$$ −22.0000 −1.11977
$$387$$ −8.00000 −0.406663
$$388$$ −14.0000 −0.710742
$$389$$ −30.0000 −1.52106 −0.760530 0.649303i $$-0.775061\pi$$
−0.760530 + 0.649303i $$0.775061\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ −1.00000 −0.0505076
$$393$$ −12.0000 −0.605320
$$394$$ −18.0000 −0.906827
$$395$$ 0 0
$$396$$ 0 0
$$397$$ −2.00000 −0.100377 −0.0501886 0.998740i $$-0.515982\pi$$
−0.0501886 + 0.998740i $$0.515982\pi$$
$$398$$ 4.00000 0.200502
$$399$$ −4.00000 −0.200250
$$400$$ 0 0
$$401$$ 18.0000 0.898877 0.449439 0.893311i $$-0.351624\pi$$
0.449439 + 0.893311i $$0.351624\pi$$
$$402$$ −8.00000 −0.399004
$$403$$ 8.00000 0.398508
$$404$$ −6.00000 −0.298511
$$405$$ 0 0
$$406$$ −6.00000 −0.297775
$$407$$ 0 0
$$408$$ 6.00000 0.297044
$$409$$ −22.0000 −1.08783 −0.543915 0.839140i $$-0.683059\pi$$
−0.543915 + 0.839140i $$0.683059\pi$$
$$410$$ 0 0
$$411$$ −6.00000 −0.295958
$$412$$ 16.0000 0.788263
$$413$$ 12.0000 0.590481
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 2.00000 0.0980581
$$417$$ 4.00000 0.195881
$$418$$ 0 0
$$419$$ −12.0000 −0.586238 −0.293119 0.956076i $$-0.594693\pi$$
−0.293119 + 0.956076i $$0.594693\pi$$
$$420$$ 0 0
$$421$$ −10.0000 −0.487370 −0.243685 0.969854i $$-0.578356\pi$$
−0.243685 + 0.969854i $$0.578356\pi$$
$$422$$ 28.0000 1.36302
$$423$$ 12.0000 0.583460
$$424$$ 6.00000 0.291386
$$425$$ 0 0
$$426$$ 0 0
$$427$$ −2.00000 −0.0967868
$$428$$ −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 $$0.195653\pi$$
$$434$$ −4.00000 −0.192006
$$435$$ 0 0
$$436$$ 14.0000 0.670478
$$437$$ 0 0
$$438$$ −14.0000 −0.668946
$$439$$ −28.0000 −1.33637 −0.668184 0.743996i $$-0.732928\pi$$
−0.668184 + 0.743996i $$0.732928\pi$$
$$440$$ 0 0
$$441$$ 1.00000 0.0476190
$$442$$ 12.0000 0.570782
$$443$$ −12.0000 −0.570137 −0.285069 0.958507i $$-0.592016\pi$$
−0.285069 + 0.958507i $$0.592016\pi$$
$$444$$ 2.00000 0.0949158
$$445$$ 0 0
$$446$$ −16.0000 −0.757622
$$447$$ −18.0000 −0.851371
$$448$$ −1.00000 −0.0472456
$$449$$ −6.00000 −0.283158 −0.141579 0.989927i $$-0.545218\pi$$
−0.141579 + 0.989927i $$0.545218\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ −18.0000 −0.846649
$$453$$ 16.0000 0.751746
$$454$$ −12.0000 −0.563188
$$455$$ 0 0
$$456$$ −4.00000 −0.187317
$$457$$ −2.00000 −0.0935561 −0.0467780 0.998905i $$-0.514895\pi$$
−0.0467780 + 0.998905i $$0.514895\pi$$
$$458$$ −2.00000 −0.0934539
$$459$$ −6.00000 −0.280056
$$460$$ 0 0
$$461$$ −30.0000 −1.39724 −0.698620 0.715493i $$-0.746202\pi$$
−0.698620 + 0.715493i $$0.746202\pi$$
$$462$$ 0 0
$$463$$ 16.0000 0.743583 0.371792 0.928316i $$-0.378744\pi$$
0.371792 + 0.928316i $$0.378744\pi$$
$$464$$ −6.00000 −0.278543
$$465$$ 0 0
$$466$$ −6.00000 −0.277945
$$467$$ −12.0000 −0.555294 −0.277647 0.960683i $$-0.589555\pi$$
−0.277647 + 0.960683i $$0.589555\pi$$
$$468$$ −2.00000 −0.0924500
$$469$$ 8.00000 0.369406
$$470$$ 0 0
$$471$$ 2.00000 0.0921551
$$472$$ 12.0000 0.552345
$$473$$ 0 0
$$474$$ −16.0000 −0.734904
$$475$$ 0 0
$$476$$ −6.00000 −0.275010
$$477$$ −6.00000 −0.274721
$$478$$ 0 0
$$479$$ −24.0000 −1.09659 −0.548294 0.836286i $$-0.684723\pi$$
−0.548294 + 0.836286i $$0.684723\pi$$
$$480$$ 0 0
$$481$$ 4.00000 0.182384
$$482$$ −26.0000 −1.18427
$$483$$ 0 0
$$484$$ −11.0000 −0.500000
$$485$$ 0 0
$$486$$ 1.00000 0.0453609
$$487$$ 16.0000 0.725029 0.362515 0.931978i $$-0.381918\pi$$
0.362515 + 0.931978i $$0.381918\pi$$
$$488$$ −2.00000 −0.0905357
$$489$$ −16.0000 −0.723545
$$490$$ 0 0
$$491$$ 24.0000 1.08310 0.541552 0.840667i $$-0.317837\pi$$
0.541552 + 0.840667i $$0.317837\pi$$
$$492$$ −6.00000 −0.270501
$$493$$ −36.0000 −1.62136
$$494$$ −8.00000 −0.359937
$$495$$ 0 0
$$496$$ −4.00000 −0.179605
$$497$$ 0 0
$$498$$ −12.0000 −0.537733
$$499$$ −4.00000 −0.179065 −0.0895323 0.995984i $$-0.528537\pi$$
−0.0895323 + 0.995984i $$0.528537\pi$$
$$500$$ 0 0
$$501$$ 12.0000 0.536120
$$502$$ −12.0000 −0.535586
$$503$$ −12.0000 −0.535054 −0.267527 0.963550i $$-0.586206\pi$$
−0.267527 + 0.963550i $$0.586206\pi$$
$$504$$ 1.00000 0.0445435
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 9.00000 0.399704
$$508$$ −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
$$511$$ 14.0000 0.619324
$$512$$ −1.00000 −0.0441942
$$513$$ 4.00000 0.176604
$$514$$ −6.00000 −0.264649
$$515$$ 0 0
$$516$$ 8.00000 0.352180
$$517$$ 0 0
$$518$$ −2.00000 −0.0878750
$$519$$ −6.00000 −0.263371
$$520$$ 0 0
$$521$$ 6.00000 0.262865 0.131432 0.991325i $$-0.458042\pi$$
0.131432 + 0.991325i $$0.458042\pi$$
$$522$$ 6.00000 0.262613
$$523$$ −20.0000 −0.874539 −0.437269 0.899331i $$-0.644054\pi$$
−0.437269 + 0.899331i $$0.644054\pi$$
$$524$$ 12.0000 0.524222
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −24.0000 −1.04546
$$528$$ 0 0
$$529$$ −23.0000 −1.00000
$$530$$ 0 0
$$531$$ −12.0000 −0.520756
$$532$$ 4.00000 0.173422
$$533$$ −12.0000 −0.519778
$$534$$ 6.00000 0.259645
$$535$$ 0 0
$$536$$ 8.00000 0.345547
$$537$$ 0 0
$$538$$ −18.0000 −0.776035
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 38.0000 1.63375 0.816874 0.576816i $$-0.195705\pi$$
0.816874 + 0.576816i $$0.195705\pi$$
$$542$$ −20.0000 −0.859074
$$543$$ −2.00000 −0.0858282
$$544$$ −6.00000 −0.257248
$$545$$ 0 0
$$546$$ 2.00000 0.0855921
$$547$$ −8.00000 −0.342055 −0.171028 0.985266i $$-0.554709\pi$$
−0.171028 + 0.985266i $$0.554709\pi$$
$$548$$ 6.00000 0.256307
$$549$$ 2.00000 0.0853579
$$550$$ 0 0
$$551$$ 24.0000 1.02243
$$552$$ 0 0
$$553$$ 16.0000 0.680389
$$554$$ −22.0000 −0.934690
$$555$$ 0 0
$$556$$ −4.00000 −0.169638
$$557$$ −30.0000 −1.27114 −0.635570 0.772043i $$-0.719235\pi$$
−0.635570 + 0.772043i $$0.719235\pi$$
$$558$$ 4.00000 0.169334
$$559$$ 16.0000 0.676728
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 30.0000 1.26547
$$563$$ 36.0000 1.51722 0.758610 0.651546i $$-0.225879\pi$$
0.758610 + 0.651546i $$0.225879\pi$$
$$564$$ −12.0000 −0.505291
$$565$$ 0 0
$$566$$ 20.0000 0.840663
$$567$$ −1.00000 −0.0419961
$$568$$ 0 0
$$569$$ −6.00000 −0.251533 −0.125767 0.992060i $$-0.540139\pi$$
−0.125767 + 0.992060i $$0.540139\pi$$
$$570$$ 0 0
$$571$$ −4.00000 −0.167395 −0.0836974 0.996491i $$-0.526673\pi$$
−0.0836974 + 0.996491i $$0.526673\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 6.00000 0.250435
$$575$$ 0 0
$$576$$ 1.00000 0.0416667
$$577$$ 34.0000 1.41544 0.707719 0.706494i $$-0.249724\pi$$
0.707719 + 0.706494i $$0.249724\pi$$
$$578$$ −19.0000 −0.790296
$$579$$ −22.0000 −0.914289
$$580$$ 0 0
$$581$$ 12.0000 0.497844
$$582$$ −14.0000 −0.580319
$$583$$ 0 0
$$584$$ 14.0000 0.579324
$$585$$ 0 0
$$586$$ −6.00000 −0.247858
$$587$$ 12.0000 0.495293 0.247647 0.968850i $$-0.420343\pi$$
0.247647 + 0.968850i $$0.420343\pi$$
$$588$$ −1.00000 −0.0412393
$$589$$ 16.0000 0.659269
$$590$$ 0 0
$$591$$ −18.0000 −0.740421
$$592$$ −2.00000 −0.0821995
$$593$$ −42.0000 −1.72473 −0.862367 0.506284i $$-0.831019\pi$$
−0.862367 + 0.506284i $$0.831019\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 18.0000 0.737309
$$597$$ 4.00000 0.163709
$$598$$ 0 0
$$599$$ −24.0000 −0.980613 −0.490307 0.871550i $$-0.663115\pi$$
−0.490307 + 0.871550i $$0.663115\pi$$
$$600$$ 0 0
$$601$$ 2.00000 0.0815817 0.0407909 0.999168i $$-0.487012\pi$$
0.0407909 + 0.999168i $$0.487012\pi$$
$$602$$ −8.00000 −0.326056
$$603$$ −8.00000 −0.325785
$$604$$ −16.0000 −0.651031
$$605$$ 0 0
$$606$$ −6.00000 −0.243733
$$607$$ −32.0000 −1.29884 −0.649420 0.760430i $$-0.724988\pi$$
−0.649420 + 0.760430i $$0.724988\pi$$
$$608$$ 4.00000 0.162221
$$609$$ −6.00000 −0.243132
$$610$$ 0 0
$$611$$ −24.0000 −0.970936
$$612$$ 6.00000 0.242536
$$613$$ −2.00000 −0.0807792 −0.0403896 0.999184i $$-0.512860\pi$$
−0.0403896 + 0.999184i $$0.512860\pi$$
$$614$$ 20.0000 0.807134
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −42.0000 −1.69086 −0.845428 0.534089i $$-0.820655\pi$$
−0.845428 + 0.534089i $$0.820655\pi$$
$$618$$ 16.0000 0.643614
$$619$$ −28.0000 −1.12542 −0.562708 0.826656i $$-0.690240\pi$$
−0.562708 + 0.826656i $$0.690240\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ −6.00000 −0.240385
$$624$$ 2.00000 0.0800641
$$625$$ 0 0
$$626$$ −10.0000 −0.399680
$$627$$ 0 0
$$628$$ −2.00000 −0.0798087
$$629$$ −12.0000 −0.478471
$$630$$ 0 0
$$631$$ −16.0000 −0.636950 −0.318475 0.947931i $$-0.603171\pi$$
−0.318475 + 0.947931i $$0.603171\pi$$
$$632$$ 16.0000 0.636446
$$633$$ 28.0000 1.11290
$$634$$ 30.0000 1.19145
$$635$$ 0 0
$$636$$ 6.00000 0.237915
$$637$$ −2.00000 −0.0792429
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 18.0000 0.710957 0.355479 0.934684i $$-0.384318\pi$$
0.355479 + 0.934684i $$0.384318\pi$$
$$642$$ −12.0000 −0.473602
$$643$$ 4.00000 0.157745 0.0788723 0.996885i $$-0.474868\pi$$
0.0788723 + 0.996885i $$0.474868\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 24.0000 0.944267
$$647$$ 12.0000 0.471769 0.235884 0.971781i $$-0.424201\pi$$
0.235884 + 0.971781i $$0.424201\pi$$
$$648$$ −1.00000 −0.0392837
$$649$$ 0 0
$$650$$ 0 0
$$651$$ −4.00000 −0.156772
$$652$$ 16.0000 0.626608
$$653$$ 42.0000 1.64359 0.821794 0.569785i $$-0.192974\pi$$
0.821794 + 0.569785i $$0.192974\pi$$
$$654$$ 14.0000 0.547443
$$655$$ 0 0
$$656$$ 6.00000 0.234261
$$657$$ −14.0000 −0.546192
$$658$$ 12.0000 0.467809
$$659$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$660$$ 0 0
$$661$$ 26.0000 1.01128 0.505641 0.862744i $$-0.331256\pi$$
0.505641 + 0.862744i $$0.331256\pi$$
$$662$$ 4.00000 0.155464
$$663$$ 12.0000 0.466041
$$664$$ 12.0000 0.465690
$$665$$ 0 0
$$666$$ 2.00000 0.0774984
$$667$$ 0 0
$$668$$ −12.0000 −0.464294
$$669$$ −16.0000 −0.618596
$$670$$ 0 0
$$671$$ 0 0
$$672$$ −1.00000 −0.0385758
$$673$$ −50.0000 −1.92736 −0.963679 0.267063i $$-0.913947\pi$$
−0.963679 + 0.267063i $$0.913947\pi$$
$$674$$ −22.0000 −0.847408
$$675$$ 0 0
$$676$$ −9.00000 −0.346154
$$677$$ 30.0000 1.15299 0.576497 0.817099i $$-0.304419\pi$$
0.576497 + 0.817099i $$0.304419\pi$$
$$678$$ −18.0000 −0.691286
$$679$$ 14.0000 0.537271
$$680$$ 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 $$0.741841\pi$$
$$684$$ −4.00000 −0.152944
$$685$$ 0 0
$$686$$ 1.00000 0.0381802
$$687$$ −2.00000 −0.0763048
$$688$$ −8.00000 −0.304997
$$689$$ 12.0000 0.457164
$$690$$ 0 0
$$691$$ 20.0000 0.760836 0.380418 0.924815i $$-0.375780\pi$$
0.380418 + 0.924815i $$0.375780\pi$$
$$692$$ 6.00000 0.228086
$$693$$ 0 0
$$694$$ 12.0000 0.455514
$$695$$ 0 0
$$696$$ −6.00000 −0.227429
$$697$$ 36.0000 1.36360
$$698$$ −26.0000 −0.984115
$$699$$ −6.00000 −0.226941
$$700$$ 0 0
$$701$$ −6.00000 −0.226617 −0.113308 0.993560i $$-0.536145\pi$$
−0.113308 + 0.993560i $$0.536145\pi$$
$$702$$ −2.00000 −0.0754851
$$703$$ 8.00000 0.301726
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 18.0000 0.677439
$$707$$ 6.00000 0.225653
$$708$$ 12.0000 0.450988
$$709$$ 14.0000 0.525781 0.262891 0.964826i $$-0.415324\pi$$
0.262891 + 0.964826i $$0.415324\pi$$
$$710$$ 0 0
$$711$$ −16.0000 −0.600047
$$712$$ −6.00000 −0.224860
$$713$$ 0 0
$$714$$ −6.00000 −0.224544
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$720$$ 0 0
$$721$$ −16.0000 −0.595871
$$722$$ 3.00000 0.111648
$$723$$ −26.0000 −0.966950
$$724$$ 2.00000 0.0743294
$$725$$ 0 0
$$726$$ −11.0000 −0.408248
$$727$$ −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
$$731$$ −48.0000 −1.77534
$$732$$ −2.00000 −0.0739221
$$733$$ −2.00000 −0.0738717 −0.0369358 0.999318i $$-0.511760\pi$$
−0.0369358 + 0.999318i $$0.511760\pi$$
$$734$$ 8.00000 0.295285
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 0 0
$$738$$ −6.00000 −0.220863
$$739$$ 44.0000 1.61857 0.809283 0.587419i $$-0.199856\pi$$
0.809283 + 0.587419i $$0.199856\pi$$
$$740$$ 0 0
$$741$$ −8.00000 −0.293887
$$742$$ −6.00000 −0.220267
$$743$$ 24.0000 0.880475 0.440237 0.897881i $$-0.354894\pi$$
0.440237 + 0.897881i $$0.354894\pi$$
$$744$$ −4.00000 −0.146647
$$745$$ 0 0
$$746$$ 26.0000 0.951928
$$747$$ −12.0000 −0.439057
$$748$$ 0 0
$$749$$ 12.0000 0.438470
$$750$$ 0 0
$$751$$ −40.0000 −1.45962 −0.729810 0.683650i $$-0.760392\pi$$
−0.729810 + 0.683650i $$0.760392\pi$$
$$752$$ 12.0000 0.437595
$$753$$ −12.0000 −0.437304
$$754$$ −12.0000 −0.437014
$$755$$ 0 0
$$756$$ 1.00000 0.0363696
$$757$$ 22.0000 0.799604 0.399802 0.916602i $$-0.369079\pi$$
0.399802 + 0.916602i $$0.369079\pi$$
$$758$$ −20.0000 −0.726433
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 30.0000 1.08750 0.543750 0.839248i $$-0.317004\pi$$
0.543750 + 0.839248i $$0.317004\pi$$
$$762$$ −8.00000 −0.289809
$$763$$ −14.0000 −0.506834
$$764$$ 0 0
$$765$$ 0 0
$$766$$ −36.0000 −1.30073
$$767$$ 24.0000 0.866590
$$768$$ −1.00000 −0.0360844
$$769$$ 50.0000 1.80305 0.901523 0.432731i $$-0.142450\pi$$
0.901523 + 0.432731i $$0.142450\pi$$
$$770$$ 0 0
$$771$$ −6.00000 −0.216085
$$772$$ 22.0000 0.791797
$$773$$ 6.00000 0.215805 0.107903 0.994161i $$-0.465587\pi$$
0.107903 + 0.994161i $$0.465587\pi$$
$$774$$ 8.00000 0.287554
$$775$$ 0 0
$$776$$ 14.0000 0.502571
$$777$$ −2.00000 −0.0717496
$$778$$ 30.0000 1.07555
$$779$$ −24.0000 −0.859889
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ 6.00000 0.214423
$$784$$ 1.00000 0.0357143
$$785$$ 0 0
$$786$$ 12.0000 0.428026
$$787$$ 28.0000 0.998092 0.499046 0.866575i $$-0.333684\pi$$
0.499046 + 0.866575i $$0.333684\pi$$
$$788$$ 18.0000 0.641223
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 18.0000 0.640006
$$792$$ 0 0
$$793$$ −4.00000 −0.142044
$$794$$ 2.00000 0.0709773
$$795$$ 0 0
$$796$$ −4.00000 −0.141776
$$797$$ 6.00000 0.212531 0.106265 0.994338i $$-0.466111\pi$$
0.106265 + 0.994338i $$0.466111\pi$$
$$798$$ 4.00000 0.141598
$$799$$ 72.0000 2.54718
$$800$$ 0 0
$$801$$ 6.00000 0.212000
$$802$$ −18.0000 −0.635602
$$803$$ 0 0
$$804$$ 8.00000 0.282138
$$805$$ 0 0
$$806$$ −8.00000 −0.281788
$$807$$ −18.0000 −0.633630
$$808$$ 6.00000 0.211079
$$809$$ −30.0000 −1.05474 −0.527372 0.849635i $$-0.676823\pi$$
−0.527372 + 0.849635i $$0.676823\pi$$
$$810$$ 0 0
$$811$$ −4.00000 −0.140459 −0.0702295 0.997531i $$-0.522373\pi$$
−0.0702295 + 0.997531i $$0.522373\pi$$
$$812$$ 6.00000 0.210559
$$813$$ −20.0000 −0.701431
$$814$$ 0 0
$$815$$ 0 0
$$816$$ −6.00000 −0.210042
$$817$$ 32.0000 1.11954
$$818$$ 22.0000 0.769212
$$819$$ 2.00000 0.0698857
$$820$$ 0 0
$$821$$ −30.0000 −1.04701 −0.523504 0.852023i $$-0.675375\pi$$
−0.523504 + 0.852023i $$0.675375\pi$$
$$822$$ 6.00000 0.209274
$$823$$ −8.00000 −0.278862 −0.139431 0.990232i $$-0.544527\pi$$
−0.139431 + 0.990232i $$0.544527\pi$$
$$824$$ −16.0000 −0.557386
$$825$$ 0 0
$$826$$ −12.0000 −0.417533
$$827$$ 12.0000 0.417281 0.208640 0.977992i $$-0.433096\pi$$
0.208640 + 0.977992i $$0.433096\pi$$
$$828$$ 0 0
$$829$$ −46.0000 −1.59765 −0.798823 0.601566i $$-0.794544\pi$$
−0.798823 + 0.601566i $$0.794544\pi$$
$$830$$ 0 0
$$831$$ −22.0000 −0.763172
$$832$$ −2.00000 −0.0693375
$$833$$ 6.00000 0.207888
$$834$$ −4.00000 −0.138509
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 4.00000 0.138260
$$838$$ 12.0000 0.414533
$$839$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$840$$ 0 0
$$841$$ 7.00000 0.241379
$$842$$ 10.0000 0.344623
$$843$$ 30.0000 1.03325
$$844$$ −28.0000 −0.963800
$$845$$ 0 0
$$846$$ −12.0000 −0.412568
$$847$$ 11.0000 0.377964
$$848$$ −6.00000 −0.206041
$$849$$ 20.0000 0.686398
$$850$$ 0 0
$$851$$ 0 0
$$852$$ 0 0
$$853$$ −26.0000 −0.890223 −0.445112 0.895475i $$-0.646836\pi$$
−0.445112 + 0.895475i $$0.646836\pi$$
$$854$$ 2.00000 0.0684386
$$855$$ 0 0
$$856$$ 12.0000 0.410152
$$857$$ −18.0000 −0.614868 −0.307434 0.951569i $$-0.599470\pi$$
−0.307434 + 0.951569i $$0.599470\pi$$
$$858$$ 0 0
$$859$$ −4.00000 −0.136478 −0.0682391 0.997669i $$-0.521738\pi$$
−0.0682391 + 0.997669i $$0.521738\pi$$
$$860$$ 0 0
$$861$$ 6.00000 0.204479
$$862$$ −24.0000 −0.817443
$$863$$ 48.0000 1.63394 0.816970 0.576681i $$-0.195652\pi$$
0.816970 + 0.576681i $$0.195652\pi$$
$$864$$ 1.00000 0.0340207
$$865$$ 0 0
$$866$$ −34.0000 −1.15537
$$867$$ −19.0000 −0.645274
$$868$$ 4.00000 0.135769
$$869$$ 0 0
$$870$$ 0 0
$$871$$ 16.0000 0.542139
$$872$$ −14.0000 −0.474100
$$873$$ −14.0000 −0.473828
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 14.0000 0.473016
$$877$$ −50.0000 −1.68838 −0.844190 0.536044i $$-0.819918\pi$$
−0.844190 + 0.536044i $$0.819918\pi$$
$$878$$ 28.0000 0.944954
$$879$$ −6.00000 −0.202375
$$880$$ 0 0
$$881$$ −42.0000 −1.41502 −0.707508 0.706705i $$-0.750181\pi$$
−0.707508 + 0.706705i $$0.750181\pi$$
$$882$$ −1.00000 −0.0336718
$$883$$ 40.0000 1.34611 0.673054 0.739594i $$-0.264982\pi$$
0.673054 + 0.739594i $$0.264982\pi$$
$$884$$ −12.0000 −0.403604
$$885$$ 0 0
$$886$$ 12.0000 0.403148
$$887$$ 12.0000 0.402921 0.201460 0.979497i $$-0.435431\pi$$
0.201460 + 0.979497i $$0.435431\pi$$
$$888$$ −2.00000 −0.0671156
$$889$$ 8.00000 0.268311
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 16.0000 0.535720
$$893$$ −48.0000 −1.60626
$$894$$ 18.0000 0.602010
$$895$$ 0 0
$$896$$ 1.00000 0.0334077
$$897$$ 0 0
$$898$$ 6.00000 0.200223
$$899$$ 24.0000 0.800445
$$900$$ 0 0
$$901$$ −36.0000 −1.19933
$$902$$ 0 0
$$903$$ −8.00000 −0.266223
$$904$$ 18.0000 0.598671
$$905$$ 0 0
$$906$$ −16.0000 −0.531564
$$907$$ 16.0000 0.531271 0.265636 0.964073i $$-0.414418\pi$$
0.265636 + 0.964073i $$0.414418\pi$$
$$908$$ 12.0000 0.398234
$$909$$ −6.00000 −0.199007
$$910$$ 0 0
$$911$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$912$$ 4.00000 0.132453
$$913$$ 0 0
$$914$$ 2.00000 0.0661541
$$915$$ 0 0
$$916$$ 2.00000 0.0660819
$$917$$ −12.0000 −0.396275
$$918$$ 6.00000 0.198030
$$919$$ −16.0000 −0.527791 −0.263896 0.964551i $$-0.585007\pi$$
−0.263896 + 0.964551i $$0.585007\pi$$
$$920$$ 0 0
$$921$$ 20.0000 0.659022
$$922$$ 30.0000 0.987997
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 0 0
$$926$$ −16.0000 −0.525793
$$927$$ 16.0000 0.525509
$$928$$ 6.00000 0.196960
$$929$$ 54.0000 1.77168 0.885841 0.463988i $$-0.153582\pi$$
0.885841 + 0.463988i $$0.153582\pi$$
$$930$$ 0 0
$$931$$ −4.00000 −0.131095
$$932$$ 6.00000 0.196537
$$933$$ 0 0
$$934$$ 12.0000 0.392652
$$935$$ 0 0
$$936$$ 2.00000 0.0653720
$$937$$ 34.0000 1.11073 0.555366 0.831606i $$-0.312578\pi$$
0.555366 + 0.831606i $$0.312578\pi$$
$$938$$ −8.00000 −0.261209
$$939$$ −10.0000 −0.326338
$$940$$ 0 0
$$941$$ 18.0000 0.586783 0.293392 0.955992i $$-0.405216\pi$$
0.293392 + 0.955992i $$0.405216\pi$$
$$942$$ −2.00000 −0.0651635
$$943$$ 0 0
$$944$$ −12.0000 −0.390567
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 36.0000 1.16984 0.584921 0.811090i $$-0.301125\pi$$
0.584921 + 0.811090i $$0.301125\pi$$
$$948$$ 16.0000 0.519656
$$949$$ 28.0000 0.908918
$$950$$ 0 0
$$951$$ 30.0000 0.972817
$$952$$ 6.00000 0.194461
$$953$$ 6.00000 0.194359 0.0971795 0.995267i $$-0.469018\pi$$
0.0971795 + 0.995267i $$0.469018\pi$$
$$954$$ 6.00000 0.194257
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 24.0000 0.775405
$$959$$ −6.00000 −0.193750
$$960$$ 0 0
$$961$$ −15.0000 −0.483871
$$962$$ −4.00000 −0.128965
$$963$$ −12.0000 −0.386695
$$964$$ 26.0000 0.837404
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −56.0000 −1.80084 −0.900419 0.435023i $$-0.856740\pi$$
−0.900419 + 0.435023i $$0.856740\pi$$
$$968$$ 11.0000 0.353553
$$969$$ 24.0000 0.770991
$$970$$ 0 0
$$971$$ 12.0000 0.385098 0.192549 0.981287i $$-0.438325\pi$$
0.192549 + 0.981287i $$0.438325\pi$$
$$972$$ −1.00000 −0.0320750
$$973$$ 4.00000 0.128234
$$974$$ −16.0000 −0.512673
$$975$$ 0 0
$$976$$ 2.00000 0.0640184
$$977$$ −18.0000 −0.575871 −0.287936 0.957650i $$-0.592969\pi$$
−0.287936 + 0.957650i $$0.592969\pi$$
$$978$$ 16.0000 0.511624
$$979$$ 0 0
$$980$$ 0 0
$$981$$ 14.0000 0.446986
$$982$$ −24.0000 −0.765871
$$983$$ 36.0000 1.14822 0.574111 0.818778i $$-0.305348\pi$$
0.574111 + 0.818778i $$0.305348\pi$$
$$984$$ 6.00000 0.191273
$$985$$ 0 0
$$986$$ 36.0000 1.14647
$$987$$ 12.0000 0.381964
$$988$$ 8.00000 0.254514
$$989$$ 0 0
$$990$$ 0 0
$$991$$ 56.0000 1.77890 0.889449 0.457034i $$-0.151088\pi$$
0.889449 + 0.457034i $$0.151088\pi$$
$$992$$ 4.00000 0.127000
$$993$$ 4.00000 0.126936
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 12.0000 0.380235
$$997$$ −26.0000 −0.823428 −0.411714 0.911313i $$-0.635070\pi$$
−0.411714 + 0.911313i $$0.635070\pi$$
$$998$$ 4.00000 0.126618
$$999$$ 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 1050.2.a.a.1.1 1
3.2 odd 2 3150.2.a.ba.1.1 1
4.3 odd 2 8400.2.a.cn.1.1 1
5.2 odd 4 1050.2.g.h.799.1 2
5.3 odd 4 1050.2.g.h.799.2 2
5.4 even 2 210.2.a.d.1.1 1
7.6 odd 2 7350.2.a.bd.1.1 1
15.2 even 4 3150.2.g.o.2899.2 2
15.8 even 4 3150.2.g.o.2899.1 2
15.14 odd 2 630.2.a.f.1.1 1
20.19 odd 2 1680.2.a.b.1.1 1
35.4 even 6 1470.2.i.d.961.1 2
35.9 even 6 1470.2.i.d.361.1 2
35.19 odd 6 1470.2.i.h.361.1 2
35.24 odd 6 1470.2.i.h.961.1 2
35.34 odd 2 1470.2.a.m.1.1 1
40.19 odd 2 6720.2.a.cc.1.1 1
40.29 even 2 6720.2.a.bb.1.1 1
60.59 even 2 5040.2.a.ba.1.1 1
105.104 even 2 4410.2.a.f.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
210.2.a.d.1.1 1 5.4 even 2
630.2.a.f.1.1 1 15.14 odd 2
1050.2.a.a.1.1 1 1.1 even 1 trivial
1050.2.g.h.799.1 2 5.2 odd 4
1050.2.g.h.799.2 2 5.3 odd 4
1470.2.a.m.1.1 1 35.34 odd 2
1470.2.i.d.361.1 2 35.9 even 6
1470.2.i.d.961.1 2 35.4 even 6
1470.2.i.h.361.1 2 35.19 odd 6
1470.2.i.h.961.1 2 35.24 odd 6
1680.2.a.b.1.1 1 20.19 odd 2
3150.2.a.ba.1.1 1 3.2 odd 2
3150.2.g.o.2899.1 2 15.8 even 4
3150.2.g.o.2899.2 2 15.2 even 4
4410.2.a.f.1.1 1 105.104 even 2
5040.2.a.ba.1.1 1 60.59 even 2
6720.2.a.bb.1.1 1 40.29 even 2
6720.2.a.cc.1.1 1 40.19 odd 2
7350.2.a.bd.1.1 1 7.6 odd 2
8400.2.a.cn.1.1 1 4.3 odd 2