# Properties

 Label 1950.2.a.r.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 $1$

# 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: $$0$$ 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

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} -6.00000 q^{11} -1.00000 q^{12} +1.00000 q^{13} +1.00000 q^{16} +1.00000 q^{18} +6.00000 q^{19} -6.00000 q^{22} +6.00000 q^{23} -1.00000 q^{24} +1.00000 q^{26} -1.00000 q^{27} +2.00000 q^{29} +4.00000 q^{31} +1.00000 q^{32} +6.00000 q^{33} +1.00000 q^{36} +10.0000 q^{37} +6.00000 q^{38} -1.00000 q^{39} -6.00000 q^{41} +8.00000 q^{43} -6.00000 q^{44} +6.00000 q^{46} +8.00000 q^{47} -1.00000 q^{48} -7.00000 q^{49} +1.00000 q^{52} -6.00000 q^{53} -1.00000 q^{54} -6.00000 q^{57} +2.00000 q^{58} +10.0000 q^{59} -6.00000 q^{61} +4.00000 q^{62} +1.00000 q^{64} +6.00000 q^{66} +4.00000 q^{67} -6.00000 q^{69} -8.00000 q^{71} +1.00000 q^{72} +6.00000 q^{73} +10.0000 q^{74} +6.00000 q^{76} -1.00000 q^{78} +16.0000 q^{79} +1.00000 q^{81} -6.00000 q^{82} +4.00000 q^{83} +8.00000 q^{86} -2.00000 q^{87} -6.00000 q^{88} -10.0000 q^{89} +6.00000 q^{92} -4.00000 q^{93} +8.00000 q^{94} -1.00000 q^{96} +2.00000 q^{97} -7.00000 q^{98} -6.00000 q^{99} +O(q^{100})$$

## 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
$$7$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$8$$ 1.00000 0.353553
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ −6.00000 −1.80907 −0.904534 0.426401i $$-0.859781\pi$$
−0.904534 + 0.426401i $$0.859781\pi$$
$$12$$ −1.00000 −0.288675
$$13$$ 1.00000 0.277350
$$14$$ 0 0
$$15$$ 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
$$19$$ 6.00000 1.37649 0.688247 0.725476i $$-0.258380\pi$$
0.688247 + 0.725476i $$0.258380\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ −6.00000 −1.27920
$$23$$ 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
$$28$$ 0 0
$$29$$ 2.00000 0.371391 0.185695 0.982607i $$-0.440546\pi$$
0.185695 + 0.982607i $$0.440546\pi$$
$$30$$ 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
$$33$$ 6.00000 1.04447
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ 10.0000 1.64399 0.821995 0.569495i $$-0.192861\pi$$
0.821995 + 0.569495i $$0.192861\pi$$
$$38$$ 6.00000 0.973329
$$39$$ −1.00000 −0.160128
$$40$$ 0 0
$$41$$ −6.00000 −0.937043 −0.468521 0.883452i $$-0.655213\pi$$
−0.468521 + 0.883452i $$0.655213\pi$$
$$42$$ 0 0
$$43$$ 8.00000 1.21999 0.609994 0.792406i $$-0.291172\pi$$
0.609994 + 0.792406i $$0.291172\pi$$
$$44$$ −6.00000 −0.904534
$$45$$ 0 0
$$46$$ 6.00000 0.884652
$$47$$ 8.00000 1.16692 0.583460 0.812142i $$-0.301699\pi$$
0.583460 + 0.812142i $$0.301699\pi$$
$$48$$ −1.00000 −0.144338
$$49$$ −7.00000 −1.00000
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 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
$$56$$ 0 0
$$57$$ −6.00000 −0.794719
$$58$$ 2.00000 0.262613
$$59$$ 10.0000 1.30189 0.650945 0.759125i $$-0.274373\pi$$
0.650945 + 0.759125i $$0.274373\pi$$
$$60$$ 0 0
$$61$$ −6.00000 −0.768221 −0.384111 0.923287i $$-0.625492\pi$$
−0.384111 + 0.923287i $$0.625492\pi$$
$$62$$ 4.00000 0.508001
$$63$$ 0 0
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ 6.00000 0.738549
$$67$$ 4.00000 0.488678 0.244339 0.969690i $$-0.421429\pi$$
0.244339 + 0.969690i $$0.421429\pi$$
$$68$$ 0 0
$$69$$ −6.00000 −0.722315
$$70$$ 0 0
$$71$$ −8.00000 −0.949425 −0.474713 0.880141i $$-0.657448\pi$$
−0.474713 + 0.880141i $$0.657448\pi$$
$$72$$ 1.00000 0.117851
$$73$$ 6.00000 0.702247 0.351123 0.936329i $$-0.385800\pi$$
0.351123 + 0.936329i $$0.385800\pi$$
$$74$$ 10.0000 1.16248
$$75$$ 0 0
$$76$$ 6.00000 0.688247
$$77$$ 0 0
$$78$$ −1.00000 −0.113228
$$79$$ 16.0000 1.80014 0.900070 0.435745i $$-0.143515\pi$$
0.900070 + 0.435745i $$0.143515\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ −6.00000 −0.662589
$$83$$ 4.00000 0.439057 0.219529 0.975606i $$-0.429548\pi$$
0.219529 + 0.975606i $$0.429548\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 8.00000 0.862662
$$87$$ −2.00000 −0.214423
$$88$$ −6.00000 −0.639602
$$89$$ −10.0000 −1.06000 −0.529999 0.847998i $$-0.677808\pi$$
−0.529999 + 0.847998i $$0.677808\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 6.00000 0.625543
$$93$$ −4.00000 −0.414781
$$94$$ 8.00000 0.825137
$$95$$ 0 0
$$96$$ −1.00000 −0.102062
$$97$$ 2.00000 0.203069 0.101535 0.994832i $$-0.467625\pi$$
0.101535 + 0.994832i $$0.467625\pi$$
$$98$$ −7.00000 −0.707107
$$99$$ −6.00000 −0.603023
$$100$$ 0 0
$$101$$ −2.00000 −0.199007 −0.0995037 0.995037i $$-0.531726\pi$$
−0.0995037 + 0.995037i $$0.531726\pi$$
$$102$$ 0 0
$$103$$ −10.0000 −0.985329 −0.492665 0.870219i $$-0.663977\pi$$
−0.492665 + 0.870219i $$0.663977\pi$$
$$104$$ 1.00000 0.0980581
$$105$$ 0 0
$$106$$ −6.00000 −0.582772
$$107$$ 16.0000 1.54678 0.773389 0.633932i $$-0.218560\pi$$
0.773389 + 0.633932i $$0.218560\pi$$
$$108$$ −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
$$111$$ −10.0000 −0.949158
$$112$$ 0 0
$$113$$ 8.00000 0.752577 0.376288 0.926503i $$-0.377200\pi$$
0.376288 + 0.926503i $$0.377200\pi$$
$$114$$ −6.00000 −0.561951
$$115$$ 0 0
$$116$$ 2.00000 0.185695
$$117$$ 1.00000 0.0924500
$$118$$ 10.0000 0.920575
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 25.0000 2.27273
$$122$$ −6.00000 −0.543214
$$123$$ 6.00000 0.541002
$$124$$ 4.00000 0.359211
$$125$$ 0 0
$$126$$ 0 0
$$127$$ −18.0000 −1.59724 −0.798621 0.601834i $$-0.794437\pi$$
−0.798621 + 0.601834i $$0.794437\pi$$
$$128$$ 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 $$0.324356\pi$$
$$132$$ 6.00000 0.522233
$$133$$ 0 0
$$134$$ 4.00000 0.345547
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 2.00000 0.170872 0.0854358 0.996344i $$-0.472772\pi$$
0.0854358 + 0.996344i $$0.472772\pi$$
$$138$$ −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
$$141$$ −8.00000 −0.673722
$$142$$ −8.00000 −0.671345
$$143$$ −6.00000 −0.501745
$$144$$ 1.00000 0.0833333
$$145$$ 0 0
$$146$$ 6.00000 0.496564
$$147$$ 7.00000 0.577350
$$148$$ 10.0000 0.821995
$$149$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$150$$ 0 0
$$151$$ −8.00000 −0.651031 −0.325515 0.945537i $$-0.605538\pi$$
−0.325515 + 0.945537i $$0.605538\pi$$
$$152$$ 6.00000 0.486664
$$153$$ 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 $$0.525431\pi$$
$$158$$ 16.0000 1.27289
$$159$$ 6.00000 0.475831
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 1.00000 0.0785674
$$163$$ 12.0000 0.939913 0.469956 0.882690i $$-0.344270\pi$$
0.469956 + 0.882690i $$0.344270\pi$$
$$164$$ −6.00000 −0.468521
$$165$$ 0 0
$$166$$ 4.00000 0.310460
$$167$$ 8.00000 0.619059 0.309529 0.950890i $$-0.399829\pi$$
0.309529 + 0.950890i $$0.399829\pi$$
$$168$$ 0 0
$$169$$ 1.00000 0.0769231
$$170$$ 0 0
$$171$$ 6.00000 0.458831
$$172$$ 8.00000 0.609994
$$173$$ 18.0000 1.36851 0.684257 0.729241i $$-0.260127\pi$$
0.684257 + 0.729241i $$0.260127\pi$$
$$174$$ −2.00000 −0.151620
$$175$$ 0 0
$$176$$ −6.00000 −0.452267
$$177$$ −10.0000 −0.751646
$$178$$ −10.0000 −0.749532
$$179$$ −12.0000 −0.896922 −0.448461 0.893802i $$-0.648028\pi$$
−0.448461 + 0.893802i $$0.648028\pi$$
$$180$$ 0 0
$$181$$ −2.00000 −0.148659 −0.0743294 0.997234i $$-0.523682\pi$$
−0.0743294 + 0.997234i $$0.523682\pi$$
$$182$$ 0 0
$$183$$ 6.00000 0.443533
$$184$$ 6.00000 0.442326
$$185$$ 0 0
$$186$$ −4.00000 −0.293294
$$187$$ 0 0
$$188$$ 8.00000 0.583460
$$189$$ 0 0
$$190$$ 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
$$193$$ 10.0000 0.719816 0.359908 0.932988i $$-0.382808\pi$$
0.359908 + 0.932988i $$0.382808\pi$$
$$194$$ 2.00000 0.143592
$$195$$ 0 0
$$196$$ −7.00000 −0.500000
$$197$$ −22.0000 −1.56744 −0.783718 0.621117i $$-0.786679\pi$$
−0.783718 + 0.621117i $$0.786679\pi$$
$$198$$ −6.00000 −0.426401
$$199$$ 16.0000 1.13421 0.567105 0.823646i $$-0.308063\pi$$
0.567105 + 0.823646i $$0.308063\pi$$
$$200$$ 0 0
$$201$$ −4.00000 −0.282138
$$202$$ −2.00000 −0.140720
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 0 0
$$206$$ −10.0000 −0.696733
$$207$$ 6.00000 0.417029
$$208$$ 1.00000 0.0693375
$$209$$ −36.0000 −2.49017
$$210$$ 0 0
$$211$$ 4.00000 0.275371 0.137686 0.990476i $$-0.456034\pi$$
0.137686 + 0.990476i $$0.456034\pi$$
$$212$$ −6.00000 −0.412082
$$213$$ 8.00000 0.548151
$$214$$ 16.0000 1.09374
$$215$$ 0 0
$$216$$ −1.00000 −0.0680414
$$217$$ 0 0
$$218$$ −16.0000 −1.08366
$$219$$ −6.00000 −0.405442
$$220$$ 0 0
$$221$$ 0 0
$$222$$ −10.0000 −0.671156
$$223$$ 4.00000 0.267860 0.133930 0.990991i $$-0.457240\pi$$
0.133930 + 0.990991i $$0.457240\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 8.00000 0.532152
$$227$$ −20.0000 −1.32745 −0.663723 0.747978i $$-0.731025\pi$$
−0.663723 + 0.747978i $$0.731025\pi$$
$$228$$ −6.00000 −0.397360
$$229$$ 20.0000 1.32164 0.660819 0.750546i $$-0.270209\pi$$
0.660819 + 0.750546i $$0.270209\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 2.00000 0.131306
$$233$$ −8.00000 −0.524097 −0.262049 0.965055i $$-0.584398\pi$$
−0.262049 + 0.965055i $$0.584398\pi$$
$$234$$ 1.00000 0.0653720
$$235$$ 0 0
$$236$$ 10.0000 0.650945
$$237$$ −16.0000 −1.03931
$$238$$ 0 0
$$239$$ −28.0000 −1.81117 −0.905585 0.424165i $$-0.860568\pi$$
−0.905585 + 0.424165i $$0.860568\pi$$
$$240$$ 0 0
$$241$$ −22.0000 −1.41714 −0.708572 0.705638i $$-0.750660\pi$$
−0.708572 + 0.705638i $$0.750660\pi$$
$$242$$ 25.0000 1.60706
$$243$$ −1.00000 −0.0641500
$$244$$ −6.00000 −0.384111
$$245$$ 0 0
$$246$$ 6.00000 0.382546
$$247$$ 6.00000 0.381771
$$248$$ 4.00000 0.254000
$$249$$ −4.00000 −0.253490
$$250$$ 0 0
$$251$$ 8.00000 0.504956 0.252478 0.967603i $$-0.418755\pi$$
0.252478 + 0.967603i $$0.418755\pi$$
$$252$$ 0 0
$$253$$ −36.0000 −2.26330
$$254$$ −18.0000 −1.12942
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ −24.0000 −1.49708 −0.748539 0.663090i $$-0.769245\pi$$
−0.748539 + 0.663090i $$0.769245\pi$$
$$258$$ −8.00000 −0.498058
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 2.00000 0.123797
$$262$$ 12.0000 0.741362
$$263$$ −6.00000 −0.369976 −0.184988 0.982741i $$-0.559225\pi$$
−0.184988 + 0.982741i $$0.559225\pi$$
$$264$$ 6.00000 0.369274
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 10.0000 0.611990
$$268$$ 4.00000 0.244339
$$269$$ 14.0000 0.853595 0.426798 0.904347i $$-0.359642\pi$$
0.426798 + 0.904347i $$0.359642\pi$$
$$270$$ 0 0
$$271$$ −16.0000 −0.971931 −0.485965 0.873978i $$-0.661532\pi$$
−0.485965 + 0.873978i $$0.661532\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 2.00000 0.120824
$$275$$ 0 0
$$276$$ −6.00000 −0.361158
$$277$$ 26.0000 1.56219 0.781094 0.624413i $$-0.214662\pi$$
0.781094 + 0.624413i $$0.214662\pi$$
$$278$$ 8.00000 0.479808
$$279$$ 4.00000 0.239474
$$280$$ 0 0
$$281$$ −10.0000 −0.596550 −0.298275 0.954480i $$-0.596411\pi$$
−0.298275 + 0.954480i $$0.596411\pi$$
$$282$$ −8.00000 −0.476393
$$283$$ −4.00000 −0.237775 −0.118888 0.992908i $$-0.537933\pi$$
−0.118888 + 0.992908i $$0.537933\pi$$
$$284$$ −8.00000 −0.474713
$$285$$ 0 0
$$286$$ −6.00000 −0.354787
$$287$$ 0 0
$$288$$ 1.00000 0.0589256
$$289$$ −17.0000 −1.00000
$$290$$ 0 0
$$291$$ −2.00000 −0.117242
$$292$$ 6.00000 0.351123
$$293$$ −14.0000 −0.817889 −0.408944 0.912559i $$-0.634103\pi$$
−0.408944 + 0.912559i $$0.634103\pi$$
$$294$$ 7.00000 0.408248
$$295$$ 0 0
$$296$$ 10.0000 0.581238
$$297$$ 6.00000 0.348155
$$298$$ 0 0
$$299$$ 6.00000 0.346989
$$300$$ 0 0
$$301$$ 0 0
$$302$$ −8.00000 −0.460348
$$303$$ 2.00000 0.114897
$$304$$ 6.00000 0.344124
$$305$$ 0 0
$$306$$ 0 0
$$307$$ −28.0000 −1.59804 −0.799022 0.601302i $$-0.794649\pi$$
−0.799022 + 0.601302i $$0.794649\pi$$
$$308$$ 0 0
$$309$$ 10.0000 0.568880
$$310$$ 0 0
$$311$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$312$$ −1.00000 −0.0566139
$$313$$ −16.0000 −0.904373 −0.452187 0.891923i $$-0.649356\pi$$
−0.452187 + 0.891923i $$0.649356\pi$$
$$314$$ −2.00000 −0.112867
$$315$$ 0 0
$$316$$ 16.0000 0.900070
$$317$$ 18.0000 1.01098 0.505490 0.862832i $$-0.331312\pi$$
0.505490 + 0.862832i $$0.331312\pi$$
$$318$$ 6.00000 0.336463
$$319$$ −12.0000 −0.671871
$$320$$ 0 0
$$321$$ −16.0000 −0.893033
$$322$$ 0 0
$$323$$ 0 0
$$324$$ 1.00000 0.0555556
$$325$$ 0 0
$$326$$ 12.0000 0.664619
$$327$$ 16.0000 0.884802
$$328$$ −6.00000 −0.331295
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −26.0000 −1.42909 −0.714545 0.699590i $$-0.753366\pi$$
−0.714545 + 0.699590i $$0.753366\pi$$
$$332$$ 4.00000 0.219529
$$333$$ 10.0000 0.547997
$$334$$ 8.00000 0.437741
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 8.00000 0.435788 0.217894 0.975972i $$-0.430081\pi$$
0.217894 + 0.975972i $$0.430081\pi$$
$$338$$ 1.00000 0.0543928
$$339$$ −8.00000 −0.434500
$$340$$ 0 0
$$341$$ −24.0000 −1.29967
$$342$$ 6.00000 0.324443
$$343$$ 0 0
$$344$$ 8.00000 0.431331
$$345$$ 0 0
$$346$$ 18.0000 0.967686
$$347$$ 32.0000 1.71785 0.858925 0.512101i $$-0.171133\pi$$
0.858925 + 0.512101i $$0.171133\pi$$
$$348$$ −2.00000 −0.107211
$$349$$ 8.00000 0.428230 0.214115 0.976808i $$-0.431313\pi$$
0.214115 + 0.976808i $$0.431313\pi$$
$$350$$ 0 0
$$351$$ −1.00000 −0.0533761
$$352$$ −6.00000 −0.319801
$$353$$ −10.0000 −0.532246 −0.266123 0.963939i $$-0.585743\pi$$
−0.266123 + 0.963939i $$0.585743\pi$$
$$354$$ −10.0000 −0.531494
$$355$$ 0 0
$$356$$ −10.0000 −0.529999
$$357$$ 0 0
$$358$$ −12.0000 −0.634220
$$359$$ −12.0000 −0.633336 −0.316668 0.948536i $$-0.602564\pi$$
−0.316668 + 0.948536i $$0.602564\pi$$
$$360$$ 0 0
$$361$$ 17.0000 0.894737
$$362$$ −2.00000 −0.105118
$$363$$ −25.0000 −1.31216
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 6.00000 0.313625
$$367$$ 6.00000 0.313197 0.156599 0.987662i $$-0.449947\pi$$
0.156599 + 0.987662i $$0.449947\pi$$
$$368$$ 6.00000 0.312772
$$369$$ −6.00000 −0.312348
$$370$$ 0 0
$$371$$ 0 0
$$372$$ −4.00000 −0.207390
$$373$$ −10.0000 −0.517780 −0.258890 0.965907i $$-0.583357\pi$$
−0.258890 + 0.965907i $$0.583357\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 8.00000 0.412568
$$377$$ 2.00000 0.103005
$$378$$ 0 0
$$379$$ 22.0000 1.13006 0.565032 0.825069i $$-0.308864\pi$$
0.565032 + 0.825069i $$0.308864\pi$$
$$380$$ 0 0
$$381$$ 18.0000 0.922168
$$382$$ −8.00000 −0.409316
$$383$$ −20.0000 −1.02195 −0.510976 0.859595i $$-0.670716\pi$$
−0.510976 + 0.859595i $$0.670716\pi$$
$$384$$ −1.00000 −0.0510310
$$385$$ 0 0
$$386$$ 10.0000 0.508987
$$387$$ 8.00000 0.406663
$$388$$ 2.00000 0.101535
$$389$$ −14.0000 −0.709828 −0.354914 0.934899i $$-0.615490\pi$$
−0.354914 + 0.934899i $$0.615490\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ −7.00000 −0.353553
$$393$$ −12.0000 −0.605320
$$394$$ −22.0000 −1.10834
$$395$$ 0 0
$$396$$ −6.00000 −0.301511
$$397$$ −18.0000 −0.903394 −0.451697 0.892171i $$-0.649181\pi$$
−0.451697 + 0.892171i $$0.649181\pi$$
$$398$$ 16.0000 0.802008
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 14.0000 0.699127 0.349563 0.936913i $$-0.386330\pi$$
0.349563 + 0.936913i $$0.386330\pi$$
$$402$$ −4.00000 −0.199502
$$403$$ 4.00000 0.199254
$$404$$ −2.00000 −0.0995037
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −60.0000 −2.97409
$$408$$ 0 0
$$409$$ 30.0000 1.48340 0.741702 0.670729i $$-0.234019\pi$$
0.741702 + 0.670729i $$0.234019\pi$$
$$410$$ 0 0
$$411$$ −2.00000 −0.0986527
$$412$$ −10.0000 −0.492665
$$413$$ 0 0
$$414$$ 6.00000 0.294884
$$415$$ 0 0
$$416$$ 1.00000 0.0490290
$$417$$ −8.00000 −0.391762
$$418$$ −36.0000 −1.76082
$$419$$ 28.0000 1.36789 0.683945 0.729534i $$-0.260263\pi$$
0.683945 + 0.729534i $$0.260263\pi$$
$$420$$ 0 0
$$421$$ −8.00000 −0.389896 −0.194948 0.980814i $$-0.562454\pi$$
−0.194948 + 0.980814i $$0.562454\pi$$
$$422$$ 4.00000 0.194717
$$423$$ 8.00000 0.388973
$$424$$ −6.00000 −0.291386
$$425$$ 0 0
$$426$$ 8.00000 0.387601
$$427$$ 0 0
$$428$$ 16.0000 0.773389
$$429$$ 6.00000 0.289683
$$430$$ 0 0
$$431$$ −8.00000 −0.385346 −0.192673 0.981263i $$-0.561716\pi$$
−0.192673 + 0.981263i $$0.561716\pi$$
$$432$$ −1.00000 −0.0481125
$$433$$ 24.0000 1.15337 0.576683 0.816968i $$-0.304347\pi$$
0.576683 + 0.816968i $$0.304347\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ −16.0000 −0.766261
$$437$$ 36.0000 1.72211
$$438$$ −6.00000 −0.286691
$$439$$ 16.0000 0.763638 0.381819 0.924237i $$-0.375298\pi$$
0.381819 + 0.924237i $$0.375298\pi$$
$$440$$ 0 0
$$441$$ −7.00000 −0.333333
$$442$$ 0 0
$$443$$ −32.0000 −1.52037 −0.760183 0.649709i $$-0.774891\pi$$
−0.760183 + 0.649709i $$0.774891\pi$$
$$444$$ −10.0000 −0.474579
$$445$$ 0 0
$$446$$ 4.00000 0.189405
$$447$$ 0 0
$$448$$ 0 0
$$449$$ −34.0000 −1.60456 −0.802280 0.596948i $$-0.796380\pi$$
−0.802280 + 0.596948i $$0.796380\pi$$
$$450$$ 0 0
$$451$$ 36.0000 1.69517
$$452$$ 8.00000 0.376288
$$453$$ 8.00000 0.375873
$$454$$ −20.0000 −0.938647
$$455$$ 0 0
$$456$$ −6.00000 −0.280976
$$457$$ 18.0000 0.842004 0.421002 0.907060i $$-0.361678\pi$$
0.421002 + 0.907060i $$0.361678\pi$$
$$458$$ 20.0000 0.934539
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 36.0000 1.67669 0.838344 0.545142i $$-0.183524\pi$$
0.838344 + 0.545142i $$0.183524\pi$$
$$462$$ 0 0
$$463$$ −16.0000 −0.743583 −0.371792 0.928316i $$-0.621256\pi$$
−0.371792 + 0.928316i $$0.621256\pi$$
$$464$$ 2.00000 0.0928477
$$465$$ 0 0
$$466$$ −8.00000 −0.370593
$$467$$ −40.0000 −1.85098 −0.925490 0.378773i $$-0.876346\pi$$
−0.925490 + 0.378773i $$0.876346\pi$$
$$468$$ 1.00000 0.0462250
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 2.00000 0.0921551
$$472$$ 10.0000 0.460287
$$473$$ −48.0000 −2.20704
$$474$$ −16.0000 −0.734904
$$475$$ 0 0
$$476$$ 0 0
$$477$$ −6.00000 −0.274721
$$478$$ −28.0000 −1.28069
$$479$$ 20.0000 0.913823 0.456912 0.889512i $$-0.348956\pi$$
0.456912 + 0.889512i $$0.348956\pi$$
$$480$$ 0 0
$$481$$ 10.0000 0.455961
$$482$$ −22.0000 −1.00207
$$483$$ 0 0
$$484$$ 25.0000 1.13636
$$485$$ 0 0
$$486$$ −1.00000 −0.0453609
$$487$$ −12.0000 −0.543772 −0.271886 0.962329i $$-0.587647\pi$$
−0.271886 + 0.962329i $$0.587647\pi$$
$$488$$ −6.00000 −0.271607
$$489$$ −12.0000 −0.542659
$$490$$ 0 0
$$491$$ −12.0000 −0.541552 −0.270776 0.962642i $$-0.587280\pi$$
−0.270776 + 0.962642i $$0.587280\pi$$
$$492$$ 6.00000 0.270501
$$493$$ 0 0
$$494$$ 6.00000 0.269953
$$495$$ 0 0
$$496$$ 4.00000 0.179605
$$497$$ 0 0
$$498$$ −4.00000 −0.179244
$$499$$ −6.00000 −0.268597 −0.134298 0.990941i $$-0.542878\pi$$
−0.134298 + 0.990941i $$0.542878\pi$$
$$500$$ 0 0
$$501$$ −8.00000 −0.357414
$$502$$ 8.00000 0.357057
$$503$$ −26.0000 −1.15928 −0.579641 0.814872i $$-0.696807\pi$$
−0.579641 + 0.814872i $$0.696807\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ −36.0000 −1.60040
$$507$$ −1.00000 −0.0444116
$$508$$ −18.0000 −0.798621
$$509$$ 24.0000 1.06378 0.531891 0.846813i $$-0.321482\pi$$
0.531891 + 0.846813i $$0.321482\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 1.00000 0.0441942
$$513$$ −6.00000 −0.264906
$$514$$ −24.0000 −1.05859
$$515$$ 0 0
$$516$$ −8.00000 −0.352180
$$517$$ −48.0000 −2.11104
$$518$$ 0 0
$$519$$ −18.0000 −0.790112
$$520$$ 0 0
$$521$$ −10.0000 −0.438108 −0.219054 0.975713i $$-0.570297\pi$$
−0.219054 + 0.975713i $$0.570297\pi$$
$$522$$ 2.00000 0.0875376
$$523$$ 40.0000 1.74908 0.874539 0.484955i $$-0.161164\pi$$
0.874539 + 0.484955i $$0.161164\pi$$
$$524$$ 12.0000 0.524222
$$525$$ 0 0
$$526$$ −6.00000 −0.261612
$$527$$ 0 0
$$528$$ 6.00000 0.261116
$$529$$ 13.0000 0.565217
$$530$$ 0 0
$$531$$ 10.0000 0.433963
$$532$$ 0 0
$$533$$ −6.00000 −0.259889
$$534$$ 10.0000 0.432742
$$535$$ 0 0
$$536$$ 4.00000 0.172774
$$537$$ 12.0000 0.517838
$$538$$ 14.0000 0.603583
$$539$$ 42.0000 1.80907
$$540$$ 0 0
$$541$$ 4.00000 0.171973 0.0859867 0.996296i $$-0.472596\pi$$
0.0859867 + 0.996296i $$0.472596\pi$$
$$542$$ −16.0000 −0.687259
$$543$$ 2.00000 0.0858282
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 12.0000 0.513083 0.256541 0.966533i $$-0.417417\pi$$
0.256541 + 0.966533i $$0.417417\pi$$
$$548$$ 2.00000 0.0854358
$$549$$ −6.00000 −0.256074
$$550$$ 0 0
$$551$$ 12.0000 0.511217
$$552$$ −6.00000 −0.255377
$$553$$ 0 0
$$554$$ 26.0000 1.10463
$$555$$ 0 0
$$556$$ 8.00000 0.339276
$$557$$ −2.00000 −0.0847427 −0.0423714 0.999102i $$-0.513491\pi$$
−0.0423714 + 0.999102i $$0.513491\pi$$
$$558$$ 4.00000 0.169334
$$559$$ 8.00000 0.338364
$$560$$ 0 0
$$561$$ 0 0
$$562$$ −10.0000 −0.421825
$$563$$ −24.0000 −1.01148 −0.505740 0.862686i $$-0.668780\pi$$
−0.505740 + 0.862686i $$0.668780\pi$$
$$564$$ −8.00000 −0.336861
$$565$$ 0 0
$$566$$ −4.00000 −0.168133
$$567$$ 0 0
$$568$$ −8.00000 −0.335673
$$569$$ 22.0000 0.922288 0.461144 0.887325i $$-0.347439\pi$$
0.461144 + 0.887325i $$0.347439\pi$$
$$570$$ 0 0
$$571$$ 4.00000 0.167395 0.0836974 0.996491i $$-0.473327\pi$$
0.0836974 + 0.996491i $$0.473327\pi$$
$$572$$ −6.00000 −0.250873
$$573$$ 8.00000 0.334205
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 1.00000 0.0416667
$$577$$ −2.00000 −0.0832611 −0.0416305 0.999133i $$-0.513255\pi$$
−0.0416305 + 0.999133i $$0.513255\pi$$
$$578$$ −17.0000 −0.707107
$$579$$ −10.0000 −0.415586
$$580$$ 0 0
$$581$$ 0 0
$$582$$ −2.00000 −0.0829027
$$583$$ 36.0000 1.49097
$$584$$ 6.00000 0.248282
$$585$$ 0 0
$$586$$ −14.0000 −0.578335
$$587$$ −12.0000 −0.495293 −0.247647 0.968850i $$-0.579657\pi$$
−0.247647 + 0.968850i $$0.579657\pi$$
$$588$$ 7.00000 0.288675
$$589$$ 24.0000 0.988903
$$590$$ 0 0
$$591$$ 22.0000 0.904959
$$592$$ 10.0000 0.410997
$$593$$ 6.00000 0.246390 0.123195 0.992382i $$-0.460686\pi$$
0.123195 + 0.992382i $$0.460686\pi$$
$$594$$ 6.00000 0.246183
$$595$$ 0 0
$$596$$ 0 0
$$597$$ −16.0000 −0.654836
$$598$$ 6.00000 0.245358
$$599$$ 16.0000 0.653742 0.326871 0.945069i $$-0.394006\pi$$
0.326871 + 0.945069i $$0.394006\pi$$
$$600$$ 0 0
$$601$$ −10.0000 −0.407909 −0.203954 0.978980i $$-0.565379\pi$$
−0.203954 + 0.978980i $$0.565379\pi$$
$$602$$ 0 0
$$603$$ 4.00000 0.162893
$$604$$ −8.00000 −0.325515
$$605$$ 0 0
$$606$$ 2.00000 0.0812444
$$607$$ 2.00000 0.0811775 0.0405887 0.999176i $$-0.487077\pi$$
0.0405887 + 0.999176i $$0.487077\pi$$
$$608$$ 6.00000 0.243332
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 8.00000 0.323645
$$612$$ 0 0
$$613$$ −38.0000 −1.53481 −0.767403 0.641165i $$-0.778451\pi$$
−0.767403 + 0.641165i $$0.778451\pi$$
$$614$$ −28.0000 −1.12999
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −18.0000 −0.724653 −0.362326 0.932051i $$-0.618017\pi$$
−0.362326 + 0.932051i $$0.618017\pi$$
$$618$$ 10.0000 0.402259
$$619$$ 38.0000 1.52735 0.763674 0.645601i $$-0.223393\pi$$
0.763674 + 0.645601i $$0.223393\pi$$
$$620$$ 0 0
$$621$$ −6.00000 −0.240772
$$622$$ 0 0
$$623$$ 0 0
$$624$$ −1.00000 −0.0400320
$$625$$ 0 0
$$626$$ −16.0000 −0.639489
$$627$$ 36.0000 1.43770
$$628$$ −2.00000 −0.0798087
$$629$$ 0 0
$$630$$ 0 0
$$631$$ 32.0000 1.27390 0.636950 0.770905i $$-0.280196\pi$$
0.636950 + 0.770905i $$0.280196\pi$$
$$632$$ 16.0000 0.636446
$$633$$ −4.00000 −0.158986
$$634$$ 18.0000 0.714871
$$635$$ 0 0
$$636$$ 6.00000 0.237915
$$637$$ −7.00000 −0.277350
$$638$$ −12.0000 −0.475085
$$639$$ −8.00000 −0.316475
$$640$$ 0 0
$$641$$ −42.0000 −1.65890 −0.829450 0.558581i $$-0.811346\pi$$
−0.829450 + 0.558581i $$0.811346\pi$$
$$642$$ −16.0000 −0.631470
$$643$$ 44.0000 1.73519 0.867595 0.497271i $$-0.165665\pi$$
0.867595 + 0.497271i $$0.165665\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 14.0000 0.550397 0.275198 0.961387i $$-0.411256\pi$$
0.275198 + 0.961387i $$0.411256\pi$$
$$648$$ 1.00000 0.0392837
$$649$$ −60.0000 −2.35521
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 12.0000 0.469956
$$653$$ 6.00000 0.234798 0.117399 0.993085i $$-0.462544\pi$$
0.117399 + 0.993085i $$0.462544\pi$$
$$654$$ 16.0000 0.625650
$$655$$ 0 0
$$656$$ −6.00000 −0.234261
$$657$$ 6.00000 0.234082
$$658$$ 0 0
$$659$$ −12.0000 −0.467454 −0.233727 0.972302i $$-0.575092\pi$$
−0.233727 + 0.972302i $$0.575092\pi$$
$$660$$ 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
$$664$$ 4.00000 0.155230
$$665$$ 0 0
$$666$$ 10.0000 0.387492
$$667$$ 12.0000 0.464642
$$668$$ 8.00000 0.309529
$$669$$ −4.00000 −0.154649
$$670$$ 0 0
$$671$$ 36.0000 1.38976
$$672$$ 0 0
$$673$$ 44.0000 1.69608 0.848038 0.529936i $$-0.177784\pi$$
0.848038 + 0.529936i $$0.177784\pi$$
$$674$$ 8.00000 0.308148
$$675$$ 0 0
$$676$$ 1.00000 0.0384615
$$677$$ 38.0000 1.46046 0.730229 0.683202i $$-0.239413\pi$$
0.730229 + 0.683202i $$0.239413\pi$$
$$678$$ −8.00000 −0.307238
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 20.0000 0.766402
$$682$$ −24.0000 −0.919007
$$683$$ −12.0000 −0.459167 −0.229584 0.973289i $$-0.573736\pi$$
−0.229584 + 0.973289i $$0.573736\pi$$
$$684$$ 6.00000 0.229416
$$685$$ 0 0
$$686$$ 0 0
$$687$$ −20.0000 −0.763048
$$688$$ 8.00000 0.304997
$$689$$ −6.00000 −0.228582
$$690$$ 0 0
$$691$$ 22.0000 0.836919 0.418460 0.908235i $$-0.362570\pi$$
0.418460 + 0.908235i $$0.362570\pi$$
$$692$$ 18.0000 0.684257
$$693$$ 0 0
$$694$$ 32.0000 1.21470
$$695$$ 0 0
$$696$$ −2.00000 −0.0758098
$$697$$ 0 0
$$698$$ 8.00000 0.302804
$$699$$ 8.00000 0.302588
$$700$$ 0 0
$$701$$ −18.0000 −0.679851 −0.339925 0.940452i $$-0.610402\pi$$
−0.339925 + 0.940452i $$0.610402\pi$$
$$702$$ −1.00000 −0.0377426
$$703$$ 60.0000 2.26294
$$704$$ −6.00000 −0.226134
$$705$$ 0 0
$$706$$ −10.0000 −0.376355
$$707$$ 0 0
$$708$$ −10.0000 −0.375823
$$709$$ −44.0000 −1.65245 −0.826227 0.563337i $$-0.809517\pi$$
−0.826227 + 0.563337i $$0.809517\pi$$
$$710$$ 0 0
$$711$$ 16.0000 0.600047
$$712$$ −10.0000 −0.374766
$$713$$ 24.0000 0.898807
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −12.0000 −0.448461
$$717$$ 28.0000 1.04568
$$718$$ −12.0000 −0.447836
$$719$$ 24.0000 0.895049 0.447524 0.894272i $$-0.352306\pi$$
0.447524 + 0.894272i $$0.352306\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 17.0000 0.632674
$$723$$ 22.0000 0.818189
$$724$$ −2.00000 −0.0743294
$$725$$ 0 0
$$726$$ −25.0000 −0.927837
$$727$$ 38.0000 1.40934 0.704671 0.709534i $$-0.251095\pi$$
0.704671 + 0.709534i $$0.251095\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ 0 0
$$732$$ 6.00000 0.221766
$$733$$ −26.0000 −0.960332 −0.480166 0.877178i $$-0.659424\pi$$
−0.480166 + 0.877178i $$0.659424\pi$$
$$734$$ 6.00000 0.221464
$$735$$ 0 0
$$736$$ 6.00000 0.221163
$$737$$ −24.0000 −0.884051
$$738$$ −6.00000 −0.220863
$$739$$ −22.0000 −0.809283 −0.404642 0.914475i $$-0.632604\pi$$
−0.404642 + 0.914475i $$0.632604\pi$$
$$740$$ 0 0
$$741$$ −6.00000 −0.220416
$$742$$ 0 0
$$743$$ 52.0000 1.90769 0.953847 0.300291i $$-0.0970839\pi$$
0.953847 + 0.300291i $$0.0970839\pi$$
$$744$$ −4.00000 −0.146647
$$745$$ 0 0
$$746$$ −10.0000 −0.366126
$$747$$ 4.00000 0.146352
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 32.0000 1.16770 0.583848 0.811863i $$-0.301546\pi$$
0.583848 + 0.811863i $$0.301546\pi$$
$$752$$ 8.00000 0.291730
$$753$$ −8.00000 −0.291536
$$754$$ 2.00000 0.0728357
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 10.0000 0.363456 0.181728 0.983349i $$-0.441831\pi$$
0.181728 + 0.983349i $$0.441831\pi$$
$$758$$ 22.0000 0.799076
$$759$$ 36.0000 1.30672
$$760$$ 0 0
$$761$$ 30.0000 1.08750 0.543750 0.839248i $$-0.317004\pi$$
0.543750 + 0.839248i $$0.317004\pi$$
$$762$$ 18.0000 0.652071
$$763$$ 0 0
$$764$$ −8.00000 −0.289430
$$765$$ 0 0
$$766$$ −20.0000 −0.722629
$$767$$ 10.0000 0.361079
$$768$$ −1.00000 −0.0360844
$$769$$ −54.0000 −1.94729 −0.973645 0.228069i $$-0.926759\pi$$
−0.973645 + 0.228069i $$0.926759\pi$$
$$770$$ 0 0
$$771$$ 24.0000 0.864339
$$772$$ 10.0000 0.359908
$$773$$ −18.0000 −0.647415 −0.323708 0.946157i $$-0.604929\pi$$
−0.323708 + 0.946157i $$0.604929\pi$$
$$774$$ 8.00000 0.287554
$$775$$ 0 0
$$776$$ 2.00000 0.0717958
$$777$$ 0 0
$$778$$ −14.0000 −0.501924
$$779$$ −36.0000 −1.28983
$$780$$ 0 0
$$781$$ 48.0000 1.71758
$$782$$ 0 0
$$783$$ −2.00000 −0.0714742
$$784$$ −7.00000 −0.250000
$$785$$ 0 0
$$786$$ −12.0000 −0.428026
$$787$$ −12.0000 −0.427754 −0.213877 0.976861i $$-0.568609\pi$$
−0.213877 + 0.976861i $$0.568609\pi$$
$$788$$ −22.0000 −0.783718
$$789$$ 6.00000 0.213606
$$790$$ 0 0
$$791$$ 0 0
$$792$$ −6.00000 −0.213201
$$793$$ −6.00000 −0.213066
$$794$$ −18.0000 −0.638796
$$795$$ 0 0
$$796$$ 16.0000 0.567105
$$797$$ 42.0000 1.48772 0.743858 0.668338i $$-0.232994\pi$$
0.743858 + 0.668338i $$0.232994\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ 0 0
$$801$$ −10.0000 −0.353333
$$802$$ 14.0000 0.494357
$$803$$ −36.0000 −1.27041
$$804$$ −4.00000 −0.141069
$$805$$ 0 0
$$806$$ 4.00000 0.140894
$$807$$ −14.0000 −0.492823
$$808$$ −2.00000 −0.0703598
$$809$$ 6.00000 0.210949 0.105474 0.994422i $$-0.466364\pi$$
0.105474 + 0.994422i $$0.466364\pi$$
$$810$$ 0 0
$$811$$ 2.00000 0.0702295 0.0351147 0.999383i $$-0.488820\pi$$
0.0351147 + 0.999383i $$0.488820\pi$$
$$812$$ 0 0
$$813$$ 16.0000 0.561144
$$814$$ −60.0000 −2.10300
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 48.0000 1.67931
$$818$$ 30.0000 1.04893
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 4.00000 0.139601 0.0698005 0.997561i $$-0.477764\pi$$
0.0698005 + 0.997561i $$0.477764\pi$$
$$822$$ −2.00000 −0.0697580
$$823$$ −54.0000 −1.88232 −0.941161 0.337959i $$-0.890263\pi$$
−0.941161 + 0.337959i $$0.890263\pi$$
$$824$$ −10.0000 −0.348367
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −20.0000 −0.695468 −0.347734 0.937593i $$-0.613049\pi$$
−0.347734 + 0.937593i $$0.613049\pi$$
$$828$$ 6.00000 0.208514
$$829$$ 26.0000 0.903017 0.451509 0.892267i $$-0.350886\pi$$
0.451509 + 0.892267i $$0.350886\pi$$
$$830$$ 0 0
$$831$$ −26.0000 −0.901930
$$832$$ 1.00000 0.0346688
$$833$$ 0 0
$$834$$ −8.00000 −0.277017
$$835$$ 0 0
$$836$$ −36.0000 −1.24509
$$837$$ −4.00000 −0.138260
$$838$$ 28.0000 0.967244
$$839$$ −16.0000 −0.552381 −0.276191 0.961103i $$-0.589072\pi$$
−0.276191 + 0.961103i $$0.589072\pi$$
$$840$$ 0 0
$$841$$ −25.0000 −0.862069
$$842$$ −8.00000 −0.275698
$$843$$ 10.0000 0.344418
$$844$$ 4.00000 0.137686
$$845$$ 0 0
$$846$$ 8.00000 0.275046
$$847$$ 0 0
$$848$$ −6.00000 −0.206041
$$849$$ 4.00000 0.137280
$$850$$ 0 0
$$851$$ 60.0000 2.05677
$$852$$ 8.00000 0.274075
$$853$$ 14.0000 0.479351 0.239675 0.970853i $$-0.422959\pi$$
0.239675 + 0.970853i $$0.422959\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 16.0000 0.546869
$$857$$ −52.0000 −1.77629 −0.888143 0.459567i $$-0.848005\pi$$
−0.888143 + 0.459567i $$0.848005\pi$$
$$858$$ 6.00000 0.204837
$$859$$ 12.0000 0.409435 0.204717 0.978821i $$-0.434372\pi$$
0.204717 + 0.978821i $$0.434372\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ −8.00000 −0.272481
$$863$$ −32.0000 −1.08929 −0.544646 0.838666i $$-0.683336\pi$$
−0.544646 + 0.838666i $$0.683336\pi$$
$$864$$ −1.00000 −0.0340207
$$865$$ 0 0
$$866$$ 24.0000 0.815553
$$867$$ 17.0000 0.577350
$$868$$ 0 0
$$869$$ −96.0000 −3.25658
$$870$$ 0 0
$$871$$ 4.00000 0.135535
$$872$$ −16.0000 −0.541828
$$873$$ 2.00000 0.0676897
$$874$$ 36.0000 1.21772
$$875$$ 0 0
$$876$$ −6.00000 −0.202721
$$877$$ −22.0000 −0.742887 −0.371444 0.928456i $$-0.621137\pi$$
−0.371444 + 0.928456i $$0.621137\pi$$
$$878$$ 16.0000 0.539974
$$879$$ 14.0000 0.472208
$$880$$ 0 0
$$881$$ −54.0000 −1.81931 −0.909653 0.415369i $$-0.863653\pi$$
−0.909653 + 0.415369i $$0.863653\pi$$
$$882$$ −7.00000 −0.235702
$$883$$ −16.0000 −0.538443 −0.269221 0.963078i $$-0.586766\pi$$
−0.269221 + 0.963078i $$0.586766\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ −32.0000 −1.07506
$$887$$ −26.0000 −0.872995 −0.436497 0.899706i $$-0.643781\pi$$
−0.436497 + 0.899706i $$0.643781\pi$$
$$888$$ −10.0000 −0.335578
$$889$$ 0 0
$$890$$ 0 0
$$891$$ −6.00000 −0.201008
$$892$$ 4.00000 0.133930
$$893$$ 48.0000 1.60626
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ −6.00000 −0.200334
$$898$$ −34.0000 −1.13459
$$899$$ 8.00000 0.266815
$$900$$ 0 0
$$901$$ 0 0
$$902$$ 36.0000 1.19867
$$903$$ 0 0
$$904$$ 8.00000 0.266076
$$905$$ 0 0
$$906$$ 8.00000 0.265782
$$907$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$908$$ −20.0000 −0.663723
$$909$$ −2.00000 −0.0663358
$$910$$ 0 0
$$911$$ −16.0000 −0.530104 −0.265052 0.964234i $$-0.585389\pi$$
−0.265052 + 0.964234i $$0.585389\pi$$
$$912$$ −6.00000 −0.198680
$$913$$ −24.0000 −0.794284
$$914$$ 18.0000 0.595387
$$915$$ 0 0
$$916$$ 20.0000 0.660819
$$917$$ 0 0
$$918$$ 0 0
$$919$$ −24.0000 −0.791687 −0.395843 0.918318i $$-0.629548\pi$$
−0.395843 + 0.918318i $$0.629548\pi$$
$$920$$ 0 0
$$921$$ 28.0000 0.922631
$$922$$ 36.0000 1.18560
$$923$$ −8.00000 −0.263323
$$924$$ 0 0
$$925$$ 0 0
$$926$$ −16.0000 −0.525793
$$927$$ −10.0000 −0.328443
$$928$$ 2.00000 0.0656532
$$929$$ 6.00000 0.196854 0.0984268 0.995144i $$-0.468619\pi$$
0.0984268 + 0.995144i $$0.468619\pi$$
$$930$$ 0 0
$$931$$ −42.0000 −1.37649
$$932$$ −8.00000 −0.262049
$$933$$ 0 0
$$934$$ −40.0000 −1.30884
$$935$$ 0 0
$$936$$ 1.00000 0.0326860
$$937$$ 16.0000 0.522697 0.261349 0.965244i $$-0.415833\pi$$
0.261349 + 0.965244i $$0.415833\pi$$
$$938$$ 0 0
$$939$$ 16.0000 0.522140
$$940$$ 0 0
$$941$$ −24.0000 −0.782378 −0.391189 0.920310i $$-0.627936\pi$$
−0.391189 + 0.920310i $$0.627936\pi$$
$$942$$ 2.00000 0.0651635
$$943$$ −36.0000 −1.17232
$$944$$ 10.0000 0.325472
$$945$$ 0 0
$$946$$ −48.0000 −1.56061
$$947$$ 4.00000 0.129983 0.0649913 0.997886i $$-0.479298\pi$$
0.0649913 + 0.997886i $$0.479298\pi$$
$$948$$ −16.0000 −0.519656
$$949$$ 6.00000 0.194768
$$950$$ 0 0
$$951$$ −18.0000 −0.583690
$$952$$ 0 0
$$953$$ 44.0000 1.42530 0.712650 0.701520i $$-0.247495\pi$$
0.712650 + 0.701520i $$0.247495\pi$$
$$954$$ −6.00000 −0.194257
$$955$$ 0 0
$$956$$ −28.0000 −0.905585
$$957$$ 12.0000 0.387905
$$958$$ 20.0000 0.646171
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −15.0000 −0.483871
$$962$$ 10.0000 0.322413
$$963$$ 16.0000 0.515593
$$964$$ −22.0000 −0.708572
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 32.0000 1.02905 0.514525 0.857475i $$-0.327968\pi$$
0.514525 + 0.857475i $$0.327968\pi$$
$$968$$ 25.0000 0.803530
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 40.0000 1.28366 0.641831 0.766846i $$-0.278175\pi$$
0.641831 + 0.766846i $$0.278175\pi$$
$$972$$ −1.00000 −0.0320750
$$973$$ 0 0
$$974$$ −12.0000 −0.384505
$$975$$ 0 0
$$976$$ −6.00000 −0.192055
$$977$$ 50.0000 1.59964 0.799821 0.600239i $$-0.204928\pi$$
0.799821 + 0.600239i $$0.204928\pi$$
$$978$$ −12.0000 −0.383718
$$979$$ 60.0000 1.91761
$$980$$ 0 0
$$981$$ −16.0000 −0.510841
$$982$$ −12.0000 −0.382935
$$983$$ 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
$$988$$ 6.00000 0.190885
$$989$$ 48.0000 1.52631
$$990$$ 0 0
$$991$$ 8.00000 0.254128 0.127064 0.991894i $$-0.459445\pi$$
0.127064 + 0.991894i $$0.459445\pi$$
$$992$$ 4.00000 0.127000
$$993$$ 26.0000 0.825085
$$994$$ 0 0
$$995$$ 0 0
$$996$$ −4.00000 −0.126745
$$997$$ 14.0000 0.443384 0.221692 0.975117i $$-0.428842\pi$$
0.221692 + 0.975117i $$0.428842\pi$$
$$998$$ −6.00000 −0.189927
$$999$$ −10.0000 −0.316386
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.r.1.1 1
3.2 odd 2 5850.2.a.o.1.1 1
5.2 odd 4 390.2.e.a.79.2 yes 2
5.3 odd 4 390.2.e.a.79.1 2
5.4 even 2 1950.2.a.i.1.1 1
15.2 even 4 1170.2.e.d.469.1 2
15.8 even 4 1170.2.e.d.469.2 2
15.14 odd 2 5850.2.a.bs.1.1 1
20.3 even 4 3120.2.l.a.1249.2 2
20.7 even 4 3120.2.l.a.1249.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
390.2.e.a.79.1 2 5.3 odd 4
390.2.e.a.79.2 yes 2 5.2 odd 4
1170.2.e.d.469.1 2 15.2 even 4
1170.2.e.d.469.2 2 15.8 even 4
1950.2.a.i.1.1 1 5.4 even 2
1950.2.a.r.1.1 1 1.1 even 1 trivial
3120.2.l.a.1249.1 2 20.7 even 4
3120.2.l.a.1249.2 2 20.3 even 4
5850.2.a.o.1.1 1 3.2 odd 2
5850.2.a.bs.1.1 1 15.14 odd 2