# Properties

 Label 3150.2.a.bk.1.1 Level $3150$ Weight $2$ Character 3150.1 Self dual yes Analytic conductor $25.153$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3150 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3150.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$25.1528766367$$ 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 $$\chi$$ $$=$$ 3150.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{7} +1.00000 q^{8} +O(q^{10})$$ $$q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{7} +1.00000 q^{8} -2.00000 q^{11} -6.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} +4.00000 q^{17} -6.00000 q^{19} -2.00000 q^{22} -8.00000 q^{23} -6.00000 q^{26} +1.00000 q^{28} -6.00000 q^{29} -2.00000 q^{31} +1.00000 q^{32} +4.00000 q^{34} -4.00000 q^{37} -6.00000 q^{38} -2.00000 q^{41} -4.00000 q^{43} -2.00000 q^{44} -8.00000 q^{46} +8.00000 q^{47} +1.00000 q^{49} -6.00000 q^{52} +6.00000 q^{53} +1.00000 q^{56} -6.00000 q^{58} +8.00000 q^{59} -10.0000 q^{61} -2.00000 q^{62} +1.00000 q^{64} -8.00000 q^{67} +4.00000 q^{68} +6.00000 q^{71} +14.0000 q^{73} -4.00000 q^{74} -6.00000 q^{76} -2.00000 q^{77} -12.0000 q^{79} -2.00000 q^{82} -8.00000 q^{83} -4.00000 q^{86} -2.00000 q^{88} +10.0000 q^{89} -6.00000 q^{91} -8.00000 q^{92} +8.00000 q^{94} +10.0000 q^{97} +1.00000 q^{98} +O(q^{100})$$

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.00000 0.707107
$$3$$ 0 0
$$4$$ 1.00000 0.500000
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 1.00000 0.377964
$$8$$ 1.00000 0.353553
$$9$$ 0 0
$$10$$ 0 0
$$11$$ −2.00000 −0.603023 −0.301511 0.953463i $$-0.597491\pi$$
−0.301511 + 0.953463i $$0.597491\pi$$
$$12$$ 0 0
$$13$$ −6.00000 −1.66410 −0.832050 0.554700i $$-0.812833\pi$$
−0.832050 + 0.554700i $$0.812833\pi$$
$$14$$ 1.00000 0.267261
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 4.00000 0.970143 0.485071 0.874475i $$-0.338794\pi$$
0.485071 + 0.874475i $$0.338794\pi$$
$$18$$ 0 0
$$19$$ −6.00000 −1.37649 −0.688247 0.725476i $$-0.741620\pi$$
−0.688247 + 0.725476i $$0.741620\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ −2.00000 −0.426401
$$23$$ −8.00000 −1.66812 −0.834058 0.551677i $$-0.813988\pi$$
−0.834058 + 0.551677i $$0.813988\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ −6.00000 −1.17670
$$27$$ 0 0
$$28$$ 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
$$31$$ −2.00000 −0.359211 −0.179605 0.983739i $$-0.557482\pi$$
−0.179605 + 0.983739i $$0.557482\pi$$
$$32$$ 1.00000 0.176777
$$33$$ 0 0
$$34$$ 4.00000 0.685994
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −4.00000 −0.657596 −0.328798 0.944400i $$-0.606644\pi$$
−0.328798 + 0.944400i $$0.606644\pi$$
$$38$$ −6.00000 −0.973329
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −2.00000 −0.312348 −0.156174 0.987730i $$-0.549916\pi$$
−0.156174 + 0.987730i $$0.549916\pi$$
$$42$$ 0 0
$$43$$ −4.00000 −0.609994 −0.304997 0.952353i $$-0.598656\pi$$
−0.304997 + 0.952353i $$0.598656\pi$$
$$44$$ −2.00000 −0.301511
$$45$$ 0 0
$$46$$ −8.00000 −1.17954
$$47$$ 8.00000 1.16692 0.583460 0.812142i $$-0.301699\pi$$
0.583460 + 0.812142i $$0.301699\pi$$
$$48$$ 0 0
$$49$$ 1.00000 0.142857
$$50$$ 0 0
$$51$$ 0 0
$$52$$ −6.00000 −0.832050
$$53$$ 6.00000 0.824163 0.412082 0.911147i $$-0.364802\pi$$
0.412082 + 0.911147i $$0.364802\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 1.00000 0.133631
$$57$$ 0 0
$$58$$ −6.00000 −0.787839
$$59$$ 8.00000 1.04151 0.520756 0.853706i $$-0.325650\pi$$
0.520756 + 0.853706i $$0.325650\pi$$
$$60$$ 0 0
$$61$$ −10.0000 −1.28037 −0.640184 0.768221i $$-0.721142\pi$$
−0.640184 + 0.768221i $$0.721142\pi$$
$$62$$ −2.00000 −0.254000
$$63$$ 0 0
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −8.00000 −0.977356 −0.488678 0.872464i $$-0.662521\pi$$
−0.488678 + 0.872464i $$0.662521\pi$$
$$68$$ 4.00000 0.485071
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 6.00000 0.712069 0.356034 0.934473i $$-0.384129\pi$$
0.356034 + 0.934473i $$0.384129\pi$$
$$72$$ 0 0
$$73$$ 14.0000 1.63858 0.819288 0.573382i $$-0.194369\pi$$
0.819288 + 0.573382i $$0.194369\pi$$
$$74$$ −4.00000 −0.464991
$$75$$ 0 0
$$76$$ −6.00000 −0.688247
$$77$$ −2.00000 −0.227921
$$78$$ 0 0
$$79$$ −12.0000 −1.35011 −0.675053 0.737769i $$-0.735879\pi$$
−0.675053 + 0.737769i $$0.735879\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ −2.00000 −0.220863
$$83$$ −8.00000 −0.878114 −0.439057 0.898459i $$-0.644687\pi$$
−0.439057 + 0.898459i $$0.644687\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ −4.00000 −0.431331
$$87$$ 0 0
$$88$$ −2.00000 −0.213201
$$89$$ 10.0000 1.06000 0.529999 0.847998i $$-0.322192\pi$$
0.529999 + 0.847998i $$0.322192\pi$$
$$90$$ 0 0
$$91$$ −6.00000 −0.628971
$$92$$ −8.00000 −0.834058
$$93$$ 0 0
$$94$$ 8.00000 0.825137
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 10.0000 1.01535 0.507673 0.861550i $$-0.330506\pi$$
0.507673 + 0.861550i $$0.330506\pi$$
$$98$$ 1.00000 0.101015
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −10.0000 −0.995037 −0.497519 0.867453i $$-0.665755\pi$$
−0.497519 + 0.867453i $$0.665755\pi$$
$$102$$ 0 0
$$103$$ −8.00000 −0.788263 −0.394132 0.919054i $$-0.628955\pi$$
−0.394132 + 0.919054i $$0.628955\pi$$
$$104$$ −6.00000 −0.588348
$$105$$ 0 0
$$106$$ 6.00000 0.582772
$$107$$ 12.0000 1.16008 0.580042 0.814587i $$-0.303036\pi$$
0.580042 + 0.814587i $$0.303036\pi$$
$$108$$ 0 0
$$109$$ −14.0000 −1.34096 −0.670478 0.741929i $$-0.733911\pi$$
−0.670478 + 0.741929i $$0.733911\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 1.00000 0.0944911
$$113$$ 6.00000 0.564433 0.282216 0.959351i $$-0.408930\pi$$
0.282216 + 0.959351i $$0.408930\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ −6.00000 −0.557086
$$117$$ 0 0
$$118$$ 8.00000 0.736460
$$119$$ 4.00000 0.366679
$$120$$ 0 0
$$121$$ −7.00000 −0.636364
$$122$$ −10.0000 −0.905357
$$123$$ 0 0
$$124$$ −2.00000 −0.179605
$$125$$ 0 0
$$126$$ 0 0
$$127$$ −4.00000 −0.354943 −0.177471 0.984126i $$-0.556792\pi$$
−0.177471 + 0.984126i $$0.556792\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 12.0000 1.04844 0.524222 0.851581i $$-0.324356\pi$$
0.524222 + 0.851581i $$0.324356\pi$$
$$132$$ 0 0
$$133$$ −6.00000 −0.520266
$$134$$ −8.00000 −0.691095
$$135$$ 0 0
$$136$$ 4.00000 0.342997
$$137$$ −6.00000 −0.512615 −0.256307 0.966595i $$-0.582506\pi$$
−0.256307 + 0.966595i $$0.582506\pi$$
$$138$$ 0 0
$$139$$ −14.0000 −1.18746 −0.593732 0.804663i $$-0.702346\pi$$
−0.593732 + 0.804663i $$0.702346\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 6.00000 0.503509
$$143$$ 12.0000 1.00349
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 14.0000 1.15865
$$147$$ 0 0
$$148$$ −4.00000 −0.328798
$$149$$ −10.0000 −0.819232 −0.409616 0.912258i $$-0.634337\pi$$
−0.409616 + 0.912258i $$0.634337\pi$$
$$150$$ 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
$$154$$ −2.00000 −0.161165
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 22.0000 1.75579 0.877896 0.478852i $$-0.158947\pi$$
0.877896 + 0.478852i $$0.158947\pi$$
$$158$$ −12.0000 −0.954669
$$159$$ 0 0
$$160$$ 0 0
$$161$$ −8.00000 −0.630488
$$162$$ 0 0
$$163$$ 4.00000 0.313304 0.156652 0.987654i $$-0.449930\pi$$
0.156652 + 0.987654i $$0.449930\pi$$
$$164$$ −2.00000 −0.156174
$$165$$ 0 0
$$166$$ −8.00000 −0.620920
$$167$$ −12.0000 −0.928588 −0.464294 0.885681i $$-0.653692\pi$$
−0.464294 + 0.885681i $$0.653692\pi$$
$$168$$ 0 0
$$169$$ 23.0000 1.76923
$$170$$ 0 0
$$171$$ 0 0
$$172$$ −4.00000 −0.304997
$$173$$ 8.00000 0.608229 0.304114 0.952636i $$-0.401639\pi$$
0.304114 + 0.952636i $$0.401639\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ −2.00000 −0.150756
$$177$$ 0 0
$$178$$ 10.0000 0.749532
$$179$$ −10.0000 −0.747435 −0.373718 0.927543i $$-0.621917\pi$$
−0.373718 + 0.927543i $$0.621917\pi$$
$$180$$ 0 0
$$181$$ 2.00000 0.148659 0.0743294 0.997234i $$-0.476318\pi$$
0.0743294 + 0.997234i $$0.476318\pi$$
$$182$$ −6.00000 −0.444750
$$183$$ 0 0
$$184$$ −8.00000 −0.589768
$$185$$ 0 0
$$186$$ 0 0
$$187$$ −8.00000 −0.585018
$$188$$ 8.00000 0.583460
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 18.0000 1.30243 0.651217 0.758891i $$-0.274259\pi$$
0.651217 + 0.758891i $$0.274259\pi$$
$$192$$ 0 0
$$193$$ 8.00000 0.575853 0.287926 0.957653i $$-0.407034\pi$$
0.287926 + 0.957653i $$0.407034\pi$$
$$194$$ 10.0000 0.717958
$$195$$ 0 0
$$196$$ 1.00000 0.0714286
$$197$$ 2.00000 0.142494 0.0712470 0.997459i $$-0.477302\pi$$
0.0712470 + 0.997459i $$0.477302\pi$$
$$198$$ 0 0
$$199$$ 6.00000 0.425329 0.212664 0.977125i $$-0.431786\pi$$
0.212664 + 0.977125i $$0.431786\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ −10.0000 −0.703598
$$203$$ −6.00000 −0.421117
$$204$$ 0 0
$$205$$ 0 0
$$206$$ −8.00000 −0.557386
$$207$$ 0 0
$$208$$ −6.00000 −0.416025
$$209$$ 12.0000 0.830057
$$210$$ 0 0
$$211$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$212$$ 6.00000 0.412082
$$213$$ 0 0
$$214$$ 12.0000 0.820303
$$215$$ 0 0
$$216$$ 0 0
$$217$$ −2.00000 −0.135769
$$218$$ −14.0000 −0.948200
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −24.0000 −1.61441
$$222$$ 0 0
$$223$$ 8.00000 0.535720 0.267860 0.963458i $$-0.413684\pi$$
0.267860 + 0.963458i $$0.413684\pi$$
$$224$$ 1.00000 0.0668153
$$225$$ 0 0
$$226$$ 6.00000 0.399114
$$227$$ −8.00000 −0.530979 −0.265489 0.964114i $$-0.585534\pi$$
−0.265489 + 0.964114i $$0.585534\pi$$
$$228$$ 0 0
$$229$$ 14.0000 0.925146 0.462573 0.886581i $$-0.346926\pi$$
0.462573 + 0.886581i $$0.346926\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ −6.00000 −0.393919
$$233$$ −10.0000 −0.655122 −0.327561 0.944830i $$-0.606227\pi$$
−0.327561 + 0.944830i $$0.606227\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 8.00000 0.520756
$$237$$ 0 0
$$238$$ 4.00000 0.259281
$$239$$ −26.0000 −1.68180 −0.840900 0.541190i $$-0.817974\pi$$
−0.840900 + 0.541190i $$0.817974\pi$$
$$240$$ 0 0
$$241$$ −26.0000 −1.67481 −0.837404 0.546585i $$-0.815928\pi$$
−0.837404 + 0.546585i $$0.815928\pi$$
$$242$$ −7.00000 −0.449977
$$243$$ 0 0
$$244$$ −10.0000 −0.640184
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 36.0000 2.29063
$$248$$ −2.00000 −0.127000
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$252$$ 0 0
$$253$$ 16.0000 1.00591
$$254$$ −4.00000 −0.250982
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$258$$ 0 0
$$259$$ −4.00000 −0.248548
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 12.0000 0.741362
$$263$$ 16.0000 0.986602 0.493301 0.869859i $$-0.335790\pi$$
0.493301 + 0.869859i $$0.335790\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ −6.00000 −0.367884
$$267$$ 0 0
$$268$$ −8.00000 −0.488678
$$269$$ −18.0000 −1.09748 −0.548740 0.835993i $$-0.684892\pi$$
−0.548740 + 0.835993i $$0.684892\pi$$
$$270$$ 0 0
$$271$$ 10.0000 0.607457 0.303728 0.952759i $$-0.401768\pi$$
0.303728 + 0.952759i $$0.401768\pi$$
$$272$$ 4.00000 0.242536
$$273$$ 0 0
$$274$$ −6.00000 −0.362473
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −28.0000 −1.68236 −0.841178 0.540758i $$-0.818138\pi$$
−0.841178 + 0.540758i $$0.818138\pi$$
$$278$$ −14.0000 −0.839664
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −30.0000 −1.78965 −0.894825 0.446417i $$-0.852700\pi$$
−0.894825 + 0.446417i $$0.852700\pi$$
$$282$$ 0 0
$$283$$ −20.0000 −1.18888 −0.594438 0.804141i $$-0.702626\pi$$
−0.594438 + 0.804141i $$0.702626\pi$$
$$284$$ 6.00000 0.356034
$$285$$ 0 0
$$286$$ 12.0000 0.709575
$$287$$ −2.00000 −0.118056
$$288$$ 0 0
$$289$$ −1.00000 −0.0588235
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 14.0000 0.819288
$$293$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ −4.00000 −0.232495
$$297$$ 0 0
$$298$$ −10.0000 −0.579284
$$299$$ 48.0000 2.77591
$$300$$ 0 0
$$301$$ −4.00000 −0.230556
$$302$$ −8.00000 −0.460348
$$303$$ 0 0
$$304$$ −6.00000 −0.344124
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 12.0000 0.684876 0.342438 0.939540i $$-0.388747\pi$$
0.342438 + 0.939540i $$0.388747\pi$$
$$308$$ −2.00000 −0.113961
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −24.0000 −1.36092 −0.680458 0.732787i $$-0.738219\pi$$
−0.680458 + 0.732787i $$0.738219\pi$$
$$312$$ 0 0
$$313$$ 2.00000 0.113047 0.0565233 0.998401i $$-0.481998\pi$$
0.0565233 + 0.998401i $$0.481998\pi$$
$$314$$ 22.0000 1.24153
$$315$$ 0 0
$$316$$ −12.0000 −0.675053
$$317$$ −2.00000 −0.112331 −0.0561656 0.998421i $$-0.517887\pi$$
−0.0561656 + 0.998421i $$0.517887\pi$$
$$318$$ 0 0
$$319$$ 12.0000 0.671871
$$320$$ 0 0
$$321$$ 0 0
$$322$$ −8.00000 −0.445823
$$323$$ −24.0000 −1.33540
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 4.00000 0.221540
$$327$$ 0 0
$$328$$ −2.00000 −0.110432
$$329$$ 8.00000 0.441054
$$330$$ 0 0
$$331$$ −8.00000 −0.439720 −0.219860 0.975531i $$-0.570560\pi$$
−0.219860 + 0.975531i $$0.570560\pi$$
$$332$$ −8.00000 −0.439057
$$333$$ 0 0
$$334$$ −12.0000 −0.656611
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 8.00000 0.435788 0.217894 0.975972i $$-0.430081\pi$$
0.217894 + 0.975972i $$0.430081\pi$$
$$338$$ 23.0000 1.25104
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 4.00000 0.216612
$$342$$ 0 0
$$343$$ 1.00000 0.0539949
$$344$$ −4.00000 −0.215666
$$345$$ 0 0
$$346$$ 8.00000 0.430083
$$347$$ 36.0000 1.93258 0.966291 0.257454i $$-0.0828835\pi$$
0.966291 + 0.257454i $$0.0828835\pi$$
$$348$$ 0 0
$$349$$ −26.0000 −1.39175 −0.695874 0.718164i $$-0.744983\pi$$
−0.695874 + 0.718164i $$0.744983\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ −2.00000 −0.106600
$$353$$ 20.0000 1.06449 0.532246 0.846590i $$-0.321348\pi$$
0.532246 + 0.846590i $$0.321348\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 10.0000 0.529999
$$357$$ 0 0
$$358$$ −10.0000 −0.528516
$$359$$ −6.00000 −0.316668 −0.158334 0.987386i $$-0.550612\pi$$
−0.158334 + 0.987386i $$0.550612\pi$$
$$360$$ 0 0
$$361$$ 17.0000 0.894737
$$362$$ 2.00000 0.105118
$$363$$ 0 0
$$364$$ −6.00000 −0.314485
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 32.0000 1.67039 0.835193 0.549957i $$-0.185356\pi$$
0.835193 + 0.549957i $$0.185356\pi$$
$$368$$ −8.00000 −0.417029
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 6.00000 0.311504
$$372$$ 0 0
$$373$$ 24.0000 1.24267 0.621336 0.783544i $$-0.286590\pi$$
0.621336 + 0.783544i $$0.286590\pi$$
$$374$$ −8.00000 −0.413670
$$375$$ 0 0
$$376$$ 8.00000 0.412568
$$377$$ 36.0000 1.85409
$$378$$ 0 0
$$379$$ 20.0000 1.02733 0.513665 0.857991i $$-0.328287\pi$$
0.513665 + 0.857991i $$0.328287\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 18.0000 0.920960
$$383$$ 20.0000 1.02195 0.510976 0.859595i $$-0.329284\pi$$
0.510976 + 0.859595i $$0.329284\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 8.00000 0.407189
$$387$$ 0 0
$$388$$ 10.0000 0.507673
$$389$$ −2.00000 −0.101404 −0.0507020 0.998714i $$-0.516146\pi$$
−0.0507020 + 0.998714i $$0.516146\pi$$
$$390$$ 0 0
$$391$$ −32.0000 −1.61831
$$392$$ 1.00000 0.0505076
$$393$$ 0 0
$$394$$ 2.00000 0.100759
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 2.00000 0.100377 0.0501886 0.998740i $$-0.484018\pi$$
0.0501886 + 0.998740i $$0.484018\pi$$
$$398$$ 6.00000 0.300753
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 14.0000 0.699127 0.349563 0.936913i $$-0.386330\pi$$
0.349563 + 0.936913i $$0.386330\pi$$
$$402$$ 0 0
$$403$$ 12.0000 0.597763
$$404$$ −10.0000 −0.497519
$$405$$ 0 0
$$406$$ −6.00000 −0.297775
$$407$$ 8.00000 0.396545
$$408$$ 0 0
$$409$$ 26.0000 1.28562 0.642809 0.766027i $$-0.277769\pi$$
0.642809 + 0.766027i $$0.277769\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ −8.00000 −0.394132
$$413$$ 8.00000 0.393654
$$414$$ 0 0
$$415$$ 0 0
$$416$$ −6.00000 −0.294174
$$417$$ 0 0
$$418$$ 12.0000 0.586939
$$419$$ −20.0000 −0.977064 −0.488532 0.872546i $$-0.662467\pi$$
−0.488532 + 0.872546i $$0.662467\pi$$
$$420$$ 0 0
$$421$$ 34.0000 1.65706 0.828529 0.559946i $$-0.189178\pi$$
0.828529 + 0.559946i $$0.189178\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 6.00000 0.291386
$$425$$ 0 0
$$426$$ 0 0
$$427$$ −10.0000 −0.483934
$$428$$ 12.0000 0.580042
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −2.00000 −0.0963366 −0.0481683 0.998839i $$-0.515338\pi$$
−0.0481683 + 0.998839i $$0.515338\pi$$
$$432$$ 0 0
$$433$$ 26.0000 1.24948 0.624740 0.780833i $$-0.285205\pi$$
0.624740 + 0.780833i $$0.285205\pi$$
$$434$$ −2.00000 −0.0960031
$$435$$ 0 0
$$436$$ −14.0000 −0.670478
$$437$$ 48.0000 2.29615
$$438$$ 0 0
$$439$$ −18.0000 −0.859093 −0.429547 0.903045i $$-0.641327\pi$$
−0.429547 + 0.903045i $$0.641327\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ −24.0000 −1.14156
$$443$$ −12.0000 −0.570137 −0.285069 0.958507i $$-0.592016\pi$$
−0.285069 + 0.958507i $$0.592016\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 8.00000 0.378811
$$447$$ 0 0
$$448$$ 1.00000 0.0472456
$$449$$ 6.00000 0.283158 0.141579 0.989927i $$-0.454782\pi$$
0.141579 + 0.989927i $$0.454782\pi$$
$$450$$ 0 0
$$451$$ 4.00000 0.188353
$$452$$ 6.00000 0.282216
$$453$$ 0 0
$$454$$ −8.00000 −0.375459
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −28.0000 −1.30978 −0.654892 0.755722i $$-0.727286\pi$$
−0.654892 + 0.755722i $$0.727286\pi$$
$$458$$ 14.0000 0.654177
$$459$$ 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
$$463$$ −36.0000 −1.67306 −0.836531 0.547920i $$-0.815420\pi$$
−0.836531 + 0.547920i $$0.815420\pi$$
$$464$$ −6.00000 −0.278543
$$465$$ 0 0
$$466$$ −10.0000 −0.463241
$$467$$ −24.0000 −1.11059 −0.555294 0.831654i $$-0.687394\pi$$
−0.555294 + 0.831654i $$0.687394\pi$$
$$468$$ 0 0
$$469$$ −8.00000 −0.369406
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 8.00000 0.368230
$$473$$ 8.00000 0.367840
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 4.00000 0.183340
$$477$$ 0 0
$$478$$ −26.0000 −1.18921
$$479$$ −8.00000 −0.365529 −0.182765 0.983157i $$-0.558505\pi$$
−0.182765 + 0.983157i $$0.558505\pi$$
$$480$$ 0 0
$$481$$ 24.0000 1.09431
$$482$$ −26.0000 −1.18427
$$483$$ 0 0
$$484$$ −7.00000 −0.318182
$$485$$ 0 0
$$486$$ 0 0
$$487$$ −12.0000 −0.543772 −0.271886 0.962329i $$-0.587647\pi$$
−0.271886 + 0.962329i $$0.587647\pi$$
$$488$$ −10.0000 −0.452679
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 30.0000 1.35388 0.676941 0.736038i $$-0.263305\pi$$
0.676941 + 0.736038i $$0.263305\pi$$
$$492$$ 0 0
$$493$$ −24.0000 −1.08091
$$494$$ 36.0000 1.61972
$$495$$ 0 0
$$496$$ −2.00000 −0.0898027
$$497$$ 6.00000 0.269137
$$498$$ 0 0
$$499$$ −12.0000 −0.537194 −0.268597 0.963253i $$-0.586560\pi$$
−0.268597 + 0.963253i $$0.586560\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ −12.0000 −0.535054 −0.267527 0.963550i $$-0.586206\pi$$
−0.267527 + 0.963550i $$0.586206\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 16.0000 0.711287
$$507$$ 0 0
$$508$$ −4.00000 −0.177471
$$509$$ −30.0000 −1.32973 −0.664863 0.746965i $$-0.731510\pi$$
−0.664863 + 0.746965i $$0.731510\pi$$
$$510$$ 0 0
$$511$$ 14.0000 0.619324
$$512$$ 1.00000 0.0441942
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ −16.0000 −0.703679
$$518$$ −4.00000 −0.175750
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 26.0000 1.13908 0.569540 0.821963i $$-0.307121\pi$$
0.569540 + 0.821963i $$0.307121\pi$$
$$522$$ 0 0
$$523$$ 28.0000 1.22435 0.612177 0.790721i $$-0.290294\pi$$
0.612177 + 0.790721i $$0.290294\pi$$
$$524$$ 12.0000 0.524222
$$525$$ 0 0
$$526$$ 16.0000 0.697633
$$527$$ −8.00000 −0.348485
$$528$$ 0 0
$$529$$ 41.0000 1.78261
$$530$$ 0 0
$$531$$ 0 0
$$532$$ −6.00000 −0.260133
$$533$$ 12.0000 0.519778
$$534$$ 0 0
$$535$$ 0 0
$$536$$ −8.00000 −0.345547
$$537$$ 0 0
$$538$$ −18.0000 −0.776035
$$539$$ −2.00000 −0.0861461
$$540$$ 0 0
$$541$$ 22.0000 0.945854 0.472927 0.881102i $$-0.343197\pi$$
0.472927 + 0.881102i $$0.343197\pi$$
$$542$$ 10.0000 0.429537
$$543$$ 0 0
$$544$$ 4.00000 0.171499
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −40.0000 −1.71028 −0.855138 0.518400i $$-0.826528\pi$$
−0.855138 + 0.518400i $$0.826528\pi$$
$$548$$ −6.00000 −0.256307
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 36.0000 1.53365
$$552$$ 0 0
$$553$$ −12.0000 −0.510292
$$554$$ −28.0000 −1.18961
$$555$$ 0 0
$$556$$ −14.0000 −0.593732
$$557$$ −42.0000 −1.77960 −0.889799 0.456354i $$-0.849155\pi$$
−0.889799 + 0.456354i $$0.849155\pi$$
$$558$$ 0 0
$$559$$ 24.0000 1.01509
$$560$$ 0 0
$$561$$ 0 0
$$562$$ −30.0000 −1.26547
$$563$$ −4.00000 −0.168580 −0.0842900 0.996441i $$-0.526862\pi$$
−0.0842900 + 0.996441i $$0.526862\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ −20.0000 −0.840663
$$567$$ 0 0
$$568$$ 6.00000 0.251754
$$569$$ 46.0000 1.92842 0.964210 0.265139i $$-0.0854179\pi$$
0.964210 + 0.265139i $$0.0854179\pi$$
$$570$$ 0 0
$$571$$ 4.00000 0.167395 0.0836974 0.996491i $$-0.473327\pi$$
0.0836974 + 0.996491i $$0.473327\pi$$
$$572$$ 12.0000 0.501745
$$573$$ 0 0
$$574$$ −2.00000 −0.0834784
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 26.0000 1.08239 0.541197 0.840896i $$-0.317971\pi$$
0.541197 + 0.840896i $$0.317971\pi$$
$$578$$ −1.00000 −0.0415945
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −8.00000 −0.331896
$$582$$ 0 0
$$583$$ −12.0000 −0.496989
$$584$$ 14.0000 0.579324
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −12.0000 −0.495293 −0.247647 0.968850i $$-0.579657\pi$$
−0.247647 + 0.968850i $$0.579657\pi$$
$$588$$ 0 0
$$589$$ 12.0000 0.494451
$$590$$ 0 0
$$591$$ 0 0
$$592$$ −4.00000 −0.164399
$$593$$ 12.0000 0.492781 0.246390 0.969171i $$-0.420755\pi$$
0.246390 + 0.969171i $$0.420755\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −10.0000 −0.409616
$$597$$ 0 0
$$598$$ 48.0000 1.96287
$$599$$ −30.0000 −1.22577 −0.612883 0.790173i $$-0.709990\pi$$
−0.612883 + 0.790173i $$0.709990\pi$$
$$600$$ 0 0
$$601$$ 38.0000 1.55005 0.775026 0.631929i $$-0.217737\pi$$
0.775026 + 0.631929i $$0.217737\pi$$
$$602$$ −4.00000 −0.163028
$$603$$ 0 0
$$604$$ −8.00000 −0.325515
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −8.00000 −0.324710 −0.162355 0.986732i $$-0.551909\pi$$
−0.162355 + 0.986732i $$0.551909\pi$$
$$608$$ −6.00000 −0.243332
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −48.0000 −1.94187
$$612$$ 0 0
$$613$$ −28.0000 −1.13091 −0.565455 0.824779i $$-0.691299\pi$$
−0.565455 + 0.824779i $$0.691299\pi$$
$$614$$ 12.0000 0.484281
$$615$$ 0 0
$$616$$ −2.00000 −0.0805823
$$617$$ −26.0000 −1.04672 −0.523360 0.852111i $$-0.675322\pi$$
−0.523360 + 0.852111i $$0.675322\pi$$
$$618$$ 0 0
$$619$$ −22.0000 −0.884255 −0.442127 0.896952i $$-0.645776\pi$$
−0.442127 + 0.896952i $$0.645776\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ −24.0000 −0.962312
$$623$$ 10.0000 0.400642
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 2.00000 0.0799361
$$627$$ 0 0
$$628$$ 22.0000 0.877896
$$629$$ −16.0000 −0.637962
$$630$$ 0 0
$$631$$ −4.00000 −0.159237 −0.0796187 0.996825i $$-0.525370\pi$$
−0.0796187 + 0.996825i $$0.525370\pi$$
$$632$$ −12.0000 −0.477334
$$633$$ 0 0
$$634$$ −2.00000 −0.0794301
$$635$$ 0 0
$$636$$ 0 0
$$637$$ −6.00000 −0.237729
$$638$$ 12.0000 0.475085
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −14.0000 −0.552967 −0.276483 0.961019i $$-0.589169\pi$$
−0.276483 + 0.961019i $$0.589169\pi$$
$$642$$ 0 0
$$643$$ −12.0000 −0.473234 −0.236617 0.971603i $$-0.576039\pi$$
−0.236617 + 0.971603i $$0.576039\pi$$
$$644$$ −8.00000 −0.315244
$$645$$ 0 0
$$646$$ −24.0000 −0.944267
$$647$$ 36.0000 1.41531 0.707653 0.706560i $$-0.249754\pi$$
0.707653 + 0.706560i $$0.249754\pi$$
$$648$$ 0 0
$$649$$ −16.0000 −0.628055
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 4.00000 0.156652
$$653$$ 34.0000 1.33052 0.665261 0.746611i $$-0.268320\pi$$
0.665261 + 0.746611i $$0.268320\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ −2.00000 −0.0780869
$$657$$ 0 0
$$658$$ 8.00000 0.311872
$$659$$ 26.0000 1.01282 0.506408 0.862294i $$-0.330973\pi$$
0.506408 + 0.862294i $$0.330973\pi$$
$$660$$ 0 0
$$661$$ 38.0000 1.47803 0.739014 0.673690i $$-0.235292\pi$$
0.739014 + 0.673690i $$0.235292\pi$$
$$662$$ −8.00000 −0.310929
$$663$$ 0 0
$$664$$ −8.00000 −0.310460
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 48.0000 1.85857
$$668$$ −12.0000 −0.464294
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 20.0000 0.772091
$$672$$ 0 0
$$673$$ −12.0000 −0.462566 −0.231283 0.972887i $$-0.574292\pi$$
−0.231283 + 0.972887i $$0.574292\pi$$
$$674$$ 8.00000 0.308148
$$675$$ 0 0
$$676$$ 23.0000 0.884615
$$677$$ 40.0000 1.53732 0.768662 0.639655i $$-0.220923\pi$$
0.768662 + 0.639655i $$0.220923\pi$$
$$678$$ 0 0
$$679$$ 10.0000 0.383765
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 4.00000 0.153168
$$683$$ 4.00000 0.153056 0.0765279 0.997067i $$-0.475617\pi$$
0.0765279 + 0.997067i $$0.475617\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 1.00000 0.0381802
$$687$$ 0 0
$$688$$ −4.00000 −0.152499
$$689$$ −36.0000 −1.37149
$$690$$ 0 0
$$691$$ −2.00000 −0.0760836 −0.0380418 0.999276i $$-0.512112\pi$$
−0.0380418 + 0.999276i $$0.512112\pi$$
$$692$$ 8.00000 0.304114
$$693$$ 0 0
$$694$$ 36.0000 1.36654
$$695$$ 0 0
$$696$$ 0 0
$$697$$ −8.00000 −0.303022
$$698$$ −26.0000 −0.984115
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 34.0000 1.28416 0.642081 0.766637i $$-0.278071\pi$$
0.642081 + 0.766637i $$0.278071\pi$$
$$702$$ 0 0
$$703$$ 24.0000 0.905177
$$704$$ −2.00000 −0.0753778
$$705$$ 0 0
$$706$$ 20.0000 0.752710
$$707$$ −10.0000 −0.376089
$$708$$ 0 0
$$709$$ −6.00000 −0.225335 −0.112667 0.993633i $$-0.535939\pi$$
−0.112667 + 0.993633i $$0.535939\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 10.0000 0.374766
$$713$$ 16.0000 0.599205
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −10.0000 −0.373718
$$717$$ 0 0
$$718$$ −6.00000 −0.223918
$$719$$ 4.00000 0.149175 0.0745874 0.997214i $$-0.476236\pi$$
0.0745874 + 0.997214i $$0.476236\pi$$
$$720$$ 0 0
$$721$$ −8.00000 −0.297936
$$722$$ 17.0000 0.632674
$$723$$ 0 0
$$724$$ 2.00000 0.0743294
$$725$$ 0 0
$$726$$ 0 0
$$727$$ −24.0000 −0.890111 −0.445055 0.895503i $$-0.646816\pi$$
−0.445055 + 0.895503i $$0.646816\pi$$
$$728$$ −6.00000 −0.222375
$$729$$ 0 0
$$730$$ 0 0
$$731$$ −16.0000 −0.591781
$$732$$ 0 0
$$733$$ −14.0000 −0.517102 −0.258551 0.965998i $$-0.583245\pi$$
−0.258551 + 0.965998i $$0.583245\pi$$
$$734$$ 32.0000 1.18114
$$735$$ 0 0
$$736$$ −8.00000 −0.294884
$$737$$ 16.0000 0.589368
$$738$$ 0 0
$$739$$ −44.0000 −1.61857 −0.809283 0.587419i $$-0.800144\pi$$
−0.809283 + 0.587419i $$0.800144\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 6.00000 0.220267
$$743$$ −48.0000 −1.76095 −0.880475 0.474093i $$-0.842776\pi$$
−0.880475 + 0.474093i $$0.842776\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 24.0000 0.878702
$$747$$ 0 0
$$748$$ −8.00000 −0.292509
$$749$$ 12.0000 0.438470
$$750$$ 0 0
$$751$$ 20.0000 0.729810 0.364905 0.931045i $$-0.381101\pi$$
0.364905 + 0.931045i $$0.381101\pi$$
$$752$$ 8.00000 0.291730
$$753$$ 0 0
$$754$$ 36.0000 1.31104
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 32.0000 1.16306 0.581530 0.813525i $$-0.302454\pi$$
0.581530 + 0.813525i $$0.302454\pi$$
$$758$$ 20.0000 0.726433
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 6.00000 0.217500 0.108750 0.994069i $$-0.465315\pi$$
0.108750 + 0.994069i $$0.465315\pi$$
$$762$$ 0 0
$$763$$ −14.0000 −0.506834
$$764$$ 18.0000 0.651217
$$765$$ 0 0
$$766$$ 20.0000 0.722629
$$767$$ −48.0000 −1.73318
$$768$$ 0 0
$$769$$ 30.0000 1.08183 0.540914 0.841078i $$-0.318079\pi$$
0.540914 + 0.841078i $$0.318079\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 8.00000 0.287926
$$773$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 10.0000 0.358979
$$777$$ 0 0
$$778$$ −2.00000 −0.0717035
$$779$$ 12.0000 0.429945
$$780$$ 0 0
$$781$$ −12.0000 −0.429394
$$782$$ −32.0000 −1.14432
$$783$$ 0 0
$$784$$ 1.00000 0.0357143
$$785$$ 0 0
$$786$$ 0 0
$$787$$ −28.0000 −0.998092 −0.499046 0.866575i $$-0.666316\pi$$
−0.499046 + 0.866575i $$0.666316\pi$$
$$788$$ 2.00000 0.0712470
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 6.00000 0.213335
$$792$$ 0 0
$$793$$ 60.0000 2.13066
$$794$$ 2.00000 0.0709773
$$795$$ 0 0
$$796$$ 6.00000 0.212664
$$797$$ −8.00000 −0.283375 −0.141687 0.989911i $$-0.545253\pi$$
−0.141687 + 0.989911i $$0.545253\pi$$
$$798$$ 0 0
$$799$$ 32.0000 1.13208
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 14.0000 0.494357
$$803$$ −28.0000 −0.988099
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 12.0000 0.422682
$$807$$ 0 0
$$808$$ −10.0000 −0.351799
$$809$$ −30.0000 −1.05474 −0.527372 0.849635i $$-0.676823\pi$$
−0.527372 + 0.849635i $$0.676823\pi$$
$$810$$ 0 0
$$811$$ −14.0000 −0.491606 −0.245803 0.969320i $$-0.579052\pi$$
−0.245803 + 0.969320i $$0.579052\pi$$
$$812$$ −6.00000 −0.210559
$$813$$ 0 0
$$814$$ 8.00000 0.280400
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 24.0000 0.839654
$$818$$ 26.0000 0.909069
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −22.0000 −0.767805 −0.383903 0.923374i $$-0.625420\pi$$
−0.383903 + 0.923374i $$0.625420\pi$$
$$822$$ 0 0
$$823$$ −12.0000 −0.418294 −0.209147 0.977884i $$-0.567069\pi$$
−0.209147 + 0.977884i $$0.567069\pi$$
$$824$$ −8.00000 −0.278693
$$825$$ 0 0
$$826$$ 8.00000 0.278356
$$827$$ −20.0000 −0.695468 −0.347734 0.937593i $$-0.613049\pi$$
−0.347734 + 0.937593i $$0.613049\pi$$
$$828$$ 0 0
$$829$$ 18.0000 0.625166 0.312583 0.949890i $$-0.398806\pi$$
0.312583 + 0.949890i $$0.398806\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ −6.00000 −0.208013
$$833$$ 4.00000 0.138592
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 12.0000 0.415029
$$837$$ 0 0
$$838$$ −20.0000 −0.690889
$$839$$ 20.0000 0.690477 0.345238 0.938515i $$-0.387798\pi$$
0.345238 + 0.938515i $$0.387798\pi$$
$$840$$ 0 0
$$841$$ 7.00000 0.241379
$$842$$ 34.0000 1.17172
$$843$$ 0 0
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ −7.00000 −0.240523
$$848$$ 6.00000 0.206041
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 32.0000 1.09695
$$852$$ 0 0
$$853$$ −38.0000 −1.30110 −0.650548 0.759465i $$-0.725461\pi$$
−0.650548 + 0.759465i $$0.725461\pi$$
$$854$$ −10.0000 −0.342193
$$855$$ 0 0
$$856$$ 12.0000 0.410152
$$857$$ 24.0000 0.819824 0.409912 0.912125i $$-0.365559\pi$$
0.409912 + 0.912125i $$0.365559\pi$$
$$858$$ 0 0
$$859$$ −10.0000 −0.341196 −0.170598 0.985341i $$-0.554570\pi$$
−0.170598 + 0.985341i $$0.554570\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ −2.00000 −0.0681203
$$863$$ −48.0000 −1.63394 −0.816970 0.576681i $$-0.804348\pi$$
−0.816970 + 0.576681i $$0.804348\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 26.0000 0.883516
$$867$$ 0 0
$$868$$ −2.00000 −0.0678844
$$869$$ 24.0000 0.814144
$$870$$ 0 0
$$871$$ 48.0000 1.62642
$$872$$ −14.0000 −0.474100
$$873$$ 0 0
$$874$$ 48.0000 1.62362
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 28.0000 0.945493 0.472746 0.881199i $$-0.343263\pi$$
0.472746 + 0.881199i $$0.343263\pi$$
$$878$$ −18.0000 −0.607471
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −30.0000 −1.01073 −0.505363 0.862907i $$-0.668641\pi$$
−0.505363 + 0.862907i $$0.668641\pi$$
$$882$$ 0 0
$$883$$ −24.0000 −0.807664 −0.403832 0.914833i $$-0.632322\pi$$
−0.403832 + 0.914833i $$0.632322\pi$$
$$884$$ −24.0000 −0.807207
$$885$$ 0 0
$$886$$ −12.0000 −0.403148
$$887$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$888$$ 0 0
$$889$$ −4.00000 −0.134156
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 8.00000 0.267860
$$893$$ −48.0000 −1.60626
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 1.00000 0.0334077
$$897$$ 0 0
$$898$$ 6.00000 0.200223
$$899$$ 12.0000 0.400222
$$900$$ 0 0
$$901$$ 24.0000 0.799556
$$902$$ 4.00000 0.133185
$$903$$ 0 0
$$904$$ 6.00000 0.199557
$$905$$ 0 0
$$906$$ 0 0
$$907$$ −24.0000 −0.796907 −0.398453 0.917189i $$-0.630453\pi$$
−0.398453 + 0.917189i $$0.630453\pi$$
$$908$$ −8.00000 −0.265489
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −6.00000 −0.198789 −0.0993944 0.995048i $$-0.531691\pi$$
−0.0993944 + 0.995048i $$0.531691\pi$$
$$912$$ 0 0
$$913$$ 16.0000 0.529523
$$914$$ −28.0000 −0.926158
$$915$$ 0 0
$$916$$ 14.0000 0.462573
$$917$$ 12.0000 0.396275
$$918$$ 0 0
$$919$$ 60.0000 1.97922 0.989609 0.143787i $$-0.0459280\pi$$
0.989609 + 0.143787i $$0.0459280\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 18.0000 0.592798
$$923$$ −36.0000 −1.18495
$$924$$ 0 0
$$925$$ 0 0
$$926$$ −36.0000 −1.18303
$$927$$ 0 0
$$928$$ −6.00000 −0.196960
$$929$$ 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
$$932$$ −10.0000 −0.327561
$$933$$ 0 0
$$934$$ −24.0000 −0.785304
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −26.0000 −0.849383 −0.424691 0.905338i $$-0.639617\pi$$
−0.424691 + 0.905338i $$0.639617\pi$$
$$938$$ −8.00000 −0.261209
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 10.0000 0.325991 0.162995 0.986627i $$-0.447884\pi$$
0.162995 + 0.986627i $$0.447884\pi$$
$$942$$ 0 0
$$943$$ 16.0000 0.521032
$$944$$ 8.00000 0.260378
$$945$$ 0 0
$$946$$ 8.00000 0.260102
$$947$$ −12.0000 −0.389948 −0.194974 0.980808i $$-0.562462\pi$$
−0.194974 + 0.980808i $$0.562462\pi$$
$$948$$ 0 0
$$949$$ −84.0000 −2.72676
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 4.00000 0.129641
$$953$$ −10.0000 −0.323932 −0.161966 0.986796i $$-0.551783\pi$$
−0.161966 + 0.986796i $$0.551783\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ −26.0000 −0.840900
$$957$$ 0 0
$$958$$ −8.00000 −0.258468
$$959$$ −6.00000 −0.193750
$$960$$ 0 0
$$961$$ −27.0000 −0.870968
$$962$$ 24.0000 0.773791
$$963$$ 0 0
$$964$$ −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 $$0.856740\pi$$
$$968$$ −7.00000 −0.224989
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 24.0000 0.770197 0.385098 0.922876i $$-0.374168\pi$$
0.385098 + 0.922876i $$0.374168\pi$$
$$972$$ 0 0
$$973$$ −14.0000 −0.448819
$$974$$ −12.0000 −0.384505
$$975$$ 0 0
$$976$$ −10.0000 −0.320092
$$977$$ −18.0000 −0.575871 −0.287936 0.957650i $$-0.592969\pi$$
−0.287936 + 0.957650i $$0.592969\pi$$
$$978$$ 0 0
$$979$$ −20.0000 −0.639203
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 30.0000 0.957338
$$983$$ −16.0000 −0.510321 −0.255160 0.966899i $$-0.582128\pi$$
−0.255160 + 0.966899i $$0.582128\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ −24.0000 −0.764316
$$987$$ 0 0
$$988$$ 36.0000 1.14531
$$989$$ 32.0000 1.01754
$$990$$ 0 0
$$991$$ 4.00000 0.127064 0.0635321 0.997980i $$-0.479763\pi$$
0.0635321 + 0.997980i $$0.479763\pi$$
$$992$$ −2.00000 −0.0635001
$$993$$ 0 0
$$994$$ 6.00000 0.190308
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 42.0000 1.33015 0.665077 0.746775i $$-0.268399\pi$$
0.665077 + 0.746775i $$0.268399\pi$$
$$998$$ −12.0000 −0.379853
$$999$$ 0 0
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 3150.2.a.bk.1.1 1
3.2 odd 2 1050.2.a.d.1.1 1
5.2 odd 4 630.2.g.c.379.2 2
5.3 odd 4 630.2.g.c.379.1 2
5.4 even 2 3150.2.a.d.1.1 1
12.11 even 2 8400.2.a.bp.1.1 1
15.2 even 4 210.2.g.b.169.1 2
15.8 even 4 210.2.g.b.169.2 yes 2
15.14 odd 2 1050.2.a.p.1.1 1
20.3 even 4 5040.2.t.h.1009.2 2
20.7 even 4 5040.2.t.h.1009.1 2
21.20 even 2 7350.2.a.bk.1.1 1
60.23 odd 4 1680.2.t.e.1009.2 2
60.47 odd 4 1680.2.t.e.1009.1 2
60.59 even 2 8400.2.a.w.1.1 1
105.2 even 12 1470.2.n.b.949.1 4
105.17 odd 12 1470.2.n.f.79.2 4
105.23 even 12 1470.2.n.b.949.2 4
105.32 even 12 1470.2.n.b.79.2 4
105.38 odd 12 1470.2.n.f.79.1 4
105.47 odd 12 1470.2.n.f.949.1 4
105.53 even 12 1470.2.n.b.79.1 4
105.62 odd 4 1470.2.g.b.589.1 2
105.68 odd 12 1470.2.n.f.949.2 4
105.83 odd 4 1470.2.g.b.589.2 2
105.104 even 2 7350.2.a.bz.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
210.2.g.b.169.1 2 15.2 even 4
210.2.g.b.169.2 yes 2 15.8 even 4
630.2.g.c.379.1 2 5.3 odd 4
630.2.g.c.379.2 2 5.2 odd 4
1050.2.a.d.1.1 1 3.2 odd 2
1050.2.a.p.1.1 1 15.14 odd 2
1470.2.g.b.589.1 2 105.62 odd 4
1470.2.g.b.589.2 2 105.83 odd 4
1470.2.n.b.79.1 4 105.53 even 12
1470.2.n.b.79.2 4 105.32 even 12
1470.2.n.b.949.1 4 105.2 even 12
1470.2.n.b.949.2 4 105.23 even 12
1470.2.n.f.79.1 4 105.38 odd 12
1470.2.n.f.79.2 4 105.17 odd 12
1470.2.n.f.949.1 4 105.47 odd 12
1470.2.n.f.949.2 4 105.68 odd 12
1680.2.t.e.1009.1 2 60.47 odd 4
1680.2.t.e.1009.2 2 60.23 odd 4
3150.2.a.d.1.1 1 5.4 even 2
3150.2.a.bk.1.1 1 1.1 even 1 trivial
5040.2.t.h.1009.1 2 20.7 even 4
5040.2.t.h.1009.2 2 20.3 even 4
7350.2.a.bk.1.1 1 21.20 even 2
7350.2.a.bz.1.1 1 105.104 even 2
8400.2.a.w.1.1 1 60.59 even 2
8400.2.a.bp.1.1 1 12.11 even 2