# Properties

 Label 3042.2.a.n.1.1 Level $3042$ Weight $2$ Character 3042.1 Self dual yes Analytic conductor $24.290$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3042 = 2 \cdot 3^{2} \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3042.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$24.2904922949$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 78) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 3042.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} +1.00000 q^{4} +2.00000 q^{5} -2.00000 q^{7} +1.00000 q^{8} +O(q^{10})$$ $$q+1.00000 q^{2} +1.00000 q^{4} +2.00000 q^{5} -2.00000 q^{7} +1.00000 q^{8} +2.00000 q^{10} -2.00000 q^{14} +1.00000 q^{16} -2.00000 q^{17} +6.00000 q^{19} +2.00000 q^{20} +4.00000 q^{23} -1.00000 q^{25} -2.00000 q^{28} +10.0000 q^{29} -10.0000 q^{31} +1.00000 q^{32} -2.00000 q^{34} -4.00000 q^{35} +8.00000 q^{37} +6.00000 q^{38} +2.00000 q^{40} +10.0000 q^{41} -4.00000 q^{43} +4.00000 q^{46} +12.0000 q^{47} -3.00000 q^{49} -1.00000 q^{50} +6.00000 q^{53} -2.00000 q^{56} +10.0000 q^{58} -4.00000 q^{59} +2.00000 q^{61} -10.0000 q^{62} +1.00000 q^{64} +2.00000 q^{67} -2.00000 q^{68} -4.00000 q^{70} -4.00000 q^{73} +8.00000 q^{74} +6.00000 q^{76} +2.00000 q^{80} +10.0000 q^{82} -4.00000 q^{83} -4.00000 q^{85} -4.00000 q^{86} +6.00000 q^{89} +4.00000 q^{92} +12.0000 q^{94} +12.0000 q^{95} +12.0000 q^{97} -3.00000 q^{98} +O(q^{100})$$

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.00000 0.707107
$$3$$ 0 0
$$4$$ 1.00000 0.500000
$$5$$ 2.00000 0.894427 0.447214 0.894427i $$-0.352416\pi$$
0.447214 + 0.894427i $$0.352416\pi$$
$$6$$ 0 0
$$7$$ −2.00000 −0.755929 −0.377964 0.925820i $$-0.623376\pi$$
−0.377964 + 0.925820i $$0.623376\pi$$
$$8$$ 1.00000 0.353553
$$9$$ 0 0
$$10$$ 2.00000 0.632456
$$11$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$12$$ 0 0
$$13$$ 0 0
$$14$$ −2.00000 −0.534522
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ −2.00000 −0.485071 −0.242536 0.970143i $$-0.577979\pi$$
−0.242536 + 0.970143i $$0.577979\pi$$
$$18$$ 0 0
$$19$$ 6.00000 1.37649 0.688247 0.725476i $$-0.258380\pi$$
0.688247 + 0.725476i $$0.258380\pi$$
$$20$$ 2.00000 0.447214
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 4.00000 0.834058 0.417029 0.908893i $$-0.363071\pi$$
0.417029 + 0.908893i $$0.363071\pi$$
$$24$$ 0 0
$$25$$ −1.00000 −0.200000
$$26$$ 0 0
$$27$$ 0 0
$$28$$ −2.00000 −0.377964
$$29$$ 10.0000 1.85695 0.928477 0.371391i $$-0.121119\pi$$
0.928477 + 0.371391i $$0.121119\pi$$
$$30$$ 0 0
$$31$$ −10.0000 −1.79605 −0.898027 0.439941i $$-0.854999\pi$$
−0.898027 + 0.439941i $$0.854999\pi$$
$$32$$ 1.00000 0.176777
$$33$$ 0 0
$$34$$ −2.00000 −0.342997
$$35$$ −4.00000 −0.676123
$$36$$ 0 0
$$37$$ 8.00000 1.31519 0.657596 0.753371i $$-0.271573\pi$$
0.657596 + 0.753371i $$0.271573\pi$$
$$38$$ 6.00000 0.973329
$$39$$ 0 0
$$40$$ 2.00000 0.316228
$$41$$ 10.0000 1.56174 0.780869 0.624695i $$-0.214777\pi$$
0.780869 + 0.624695i $$0.214777\pi$$
$$42$$ 0 0
$$43$$ −4.00000 −0.609994 −0.304997 0.952353i $$-0.598656\pi$$
−0.304997 + 0.952353i $$0.598656\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 4.00000 0.589768
$$47$$ 12.0000 1.75038 0.875190 0.483779i $$-0.160736\pi$$
0.875190 + 0.483779i $$0.160736\pi$$
$$48$$ 0 0
$$49$$ −3.00000 −0.428571
$$50$$ −1.00000 −0.141421
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 6.00000 0.824163 0.412082 0.911147i $$-0.364802\pi$$
0.412082 + 0.911147i $$0.364802\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ −2.00000 −0.267261
$$57$$ 0 0
$$58$$ 10.0000 1.31306
$$59$$ −4.00000 −0.520756 −0.260378 0.965507i $$-0.583847\pi$$
−0.260378 + 0.965507i $$0.583847\pi$$
$$60$$ 0 0
$$61$$ 2.00000 0.256074 0.128037 0.991769i $$-0.459132\pi$$
0.128037 + 0.991769i $$0.459132\pi$$
$$62$$ −10.0000 −1.27000
$$63$$ 0 0
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 2.00000 0.244339 0.122169 0.992509i $$-0.461015\pi$$
0.122169 + 0.992509i $$0.461015\pi$$
$$68$$ −2.00000 −0.242536
$$69$$ 0 0
$$70$$ −4.00000 −0.478091
$$71$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$72$$ 0 0
$$73$$ −4.00000 −0.468165 −0.234082 0.972217i $$-0.575209\pi$$
−0.234082 + 0.972217i $$0.575209\pi$$
$$74$$ 8.00000 0.929981
$$75$$ 0 0
$$76$$ 6.00000 0.688247
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$80$$ 2.00000 0.223607
$$81$$ 0 0
$$82$$ 10.0000 1.10432
$$83$$ −4.00000 −0.439057 −0.219529 0.975606i $$-0.570452\pi$$
−0.219529 + 0.975606i $$0.570452\pi$$
$$84$$ 0 0
$$85$$ −4.00000 −0.433861
$$86$$ −4.00000 −0.431331
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 6.00000 0.635999 0.317999 0.948091i $$-0.396989\pi$$
0.317999 + 0.948091i $$0.396989\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 4.00000 0.417029
$$93$$ 0 0
$$94$$ 12.0000 1.23771
$$95$$ 12.0000 1.23117
$$96$$ 0 0
$$97$$ 12.0000 1.21842 0.609208 0.793011i $$-0.291488\pi$$
0.609208 + 0.793011i $$0.291488\pi$$
$$98$$ −3.00000 −0.303046
$$99$$ 0 0
$$100$$ −1.00000 −0.100000
$$101$$ 2.00000 0.199007 0.0995037 0.995037i $$-0.468274\pi$$
0.0995037 + 0.995037i $$0.468274\pi$$
$$102$$ 0 0
$$103$$ 16.0000 1.57653 0.788263 0.615338i $$-0.210980\pi$$
0.788263 + 0.615338i $$0.210980\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 6.00000 0.582772
$$107$$ −8.00000 −0.773389 −0.386695 0.922208i $$-0.626383\pi$$
−0.386695 + 0.922208i $$0.626383\pi$$
$$108$$ 0 0
$$109$$ −4.00000 −0.383131 −0.191565 0.981480i $$-0.561356\pi$$
−0.191565 + 0.981480i $$0.561356\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ −2.00000 −0.188982
$$113$$ −14.0000 −1.31701 −0.658505 0.752577i $$-0.728811\pi$$
−0.658505 + 0.752577i $$0.728811\pi$$
$$114$$ 0 0
$$115$$ 8.00000 0.746004
$$116$$ 10.0000 0.928477
$$117$$ 0 0
$$118$$ −4.00000 −0.368230
$$119$$ 4.00000 0.366679
$$120$$ 0 0
$$121$$ −11.0000 −1.00000
$$122$$ 2.00000 0.181071
$$123$$ 0 0
$$124$$ −10.0000 −0.898027
$$125$$ −12.0000 −1.07331
$$126$$ 0 0
$$127$$ −8.00000 −0.709885 −0.354943 0.934888i $$-0.615500\pi$$
−0.354943 + 0.934888i $$0.615500\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 8.00000 0.698963 0.349482 0.936943i $$-0.386358\pi$$
0.349482 + 0.936943i $$0.386358\pi$$
$$132$$ 0 0
$$133$$ −12.0000 −1.04053
$$134$$ 2.00000 0.172774
$$135$$ 0 0
$$136$$ −2.00000 −0.171499
$$137$$ 2.00000 0.170872 0.0854358 0.996344i $$-0.472772\pi$$
0.0854358 + 0.996344i $$0.472772\pi$$
$$138$$ 0 0
$$139$$ −20.0000 −1.69638 −0.848189 0.529694i $$-0.822307\pi$$
−0.848189 + 0.529694i $$0.822307\pi$$
$$140$$ −4.00000 −0.338062
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ 20.0000 1.66091
$$146$$ −4.00000 −0.331042
$$147$$ 0 0
$$148$$ 8.00000 0.657596
$$149$$ 14.0000 1.14692 0.573462 0.819232i $$-0.305600\pi$$
0.573462 + 0.819232i $$0.305600\pi$$
$$150$$ 0 0
$$151$$ 10.0000 0.813788 0.406894 0.913475i $$-0.366612\pi$$
0.406894 + 0.913475i $$0.366612\pi$$
$$152$$ 6.00000 0.486664
$$153$$ 0 0
$$154$$ 0 0
$$155$$ −20.0000 −1.60644
$$156$$ 0 0
$$157$$ −2.00000 −0.159617 −0.0798087 0.996810i $$-0.525431\pi$$
−0.0798087 + 0.996810i $$0.525431\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 2.00000 0.158114
$$161$$ −8.00000 −0.630488
$$162$$ 0 0
$$163$$ −14.0000 −1.09656 −0.548282 0.836293i $$-0.684718\pi$$
−0.548282 + 0.836293i $$0.684718\pi$$
$$164$$ 10.0000 0.780869
$$165$$ 0 0
$$166$$ −4.00000 −0.310460
$$167$$ 12.0000 0.928588 0.464294 0.885681i $$-0.346308\pi$$
0.464294 + 0.885681i $$0.346308\pi$$
$$168$$ 0 0
$$169$$ 0 0
$$170$$ −4.00000 −0.306786
$$171$$ 0 0
$$172$$ −4.00000 −0.304997
$$173$$ −6.00000 −0.456172 −0.228086 0.973641i $$-0.573247\pi$$
−0.228086 + 0.973641i $$0.573247\pi$$
$$174$$ 0 0
$$175$$ 2.00000 0.151186
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 6.00000 0.449719
$$179$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$180$$ 0 0
$$181$$ −22.0000 −1.63525 −0.817624 0.575753i $$-0.804709\pi$$
−0.817624 + 0.575753i $$0.804709\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 4.00000 0.294884
$$185$$ 16.0000 1.17634
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 12.0000 0.875190
$$189$$ 0 0
$$190$$ 12.0000 0.870572
$$191$$ −12.0000 −0.868290 −0.434145 0.900843i $$-0.642949\pi$$
−0.434145 + 0.900843i $$0.642949\pi$$
$$192$$ 0 0
$$193$$ 16.0000 1.15171 0.575853 0.817554i $$-0.304670\pi$$
0.575853 + 0.817554i $$0.304670\pi$$
$$194$$ 12.0000 0.861550
$$195$$ 0 0
$$196$$ −3.00000 −0.214286
$$197$$ −22.0000 −1.56744 −0.783718 0.621117i $$-0.786679\pi$$
−0.783718 + 0.621117i $$0.786679\pi$$
$$198$$ 0 0
$$199$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$200$$ −1.00000 −0.0707107
$$201$$ 0 0
$$202$$ 2.00000 0.140720
$$203$$ −20.0000 −1.40372
$$204$$ 0 0
$$205$$ 20.0000 1.39686
$$206$$ 16.0000 1.11477
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 0 0
$$210$$ 0 0
$$211$$ 12.0000 0.826114 0.413057 0.910705i $$-0.364461\pi$$
0.413057 + 0.910705i $$0.364461\pi$$
$$212$$ 6.00000 0.412082
$$213$$ 0 0
$$214$$ −8.00000 −0.546869
$$215$$ −8.00000 −0.545595
$$216$$ 0 0
$$217$$ 20.0000 1.35769
$$218$$ −4.00000 −0.270914
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 0 0
$$223$$ 14.0000 0.937509 0.468755 0.883328i $$-0.344703\pi$$
0.468755 + 0.883328i $$0.344703\pi$$
$$224$$ −2.00000 −0.133631
$$225$$ 0 0
$$226$$ −14.0000 −0.931266
$$227$$ 8.00000 0.530979 0.265489 0.964114i $$-0.414466\pi$$
0.265489 + 0.964114i $$0.414466\pi$$
$$228$$ 0 0
$$229$$ 4.00000 0.264327 0.132164 0.991228i $$-0.457808\pi$$
0.132164 + 0.991228i $$0.457808\pi$$
$$230$$ 8.00000 0.527504
$$231$$ 0 0
$$232$$ 10.0000 0.656532
$$233$$ −6.00000 −0.393073 −0.196537 0.980497i $$-0.562969\pi$$
−0.196537 + 0.980497i $$0.562969\pi$$
$$234$$ 0 0
$$235$$ 24.0000 1.56559
$$236$$ −4.00000 −0.260378
$$237$$ 0 0
$$238$$ 4.00000 0.259281
$$239$$ −16.0000 −1.03495 −0.517477 0.855697i $$-0.673129\pi$$
−0.517477 + 0.855697i $$0.673129\pi$$
$$240$$ 0 0
$$241$$ 20.0000 1.28831 0.644157 0.764894i $$-0.277208\pi$$
0.644157 + 0.764894i $$0.277208\pi$$
$$242$$ −11.0000 −0.707107
$$243$$ 0 0
$$244$$ 2.00000 0.128037
$$245$$ −6.00000 −0.383326
$$246$$ 0 0
$$247$$ 0 0
$$248$$ −10.0000 −0.635001
$$249$$ 0 0
$$250$$ −12.0000 −0.758947
$$251$$ −28.0000 −1.76734 −0.883672 0.468106i $$-0.844936\pi$$
−0.883672 + 0.468106i $$0.844936\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ −8.00000 −0.501965
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 18.0000 1.12281 0.561405 0.827541i $$-0.310261\pi$$
0.561405 + 0.827541i $$0.310261\pi$$
$$258$$ 0 0
$$259$$ −16.0000 −0.994192
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 8.00000 0.494242
$$263$$ −24.0000 −1.47990 −0.739952 0.672660i $$-0.765152\pi$$
−0.739952 + 0.672660i $$0.765152\pi$$
$$264$$ 0 0
$$265$$ 12.0000 0.737154
$$266$$ −12.0000 −0.735767
$$267$$ 0 0
$$268$$ 2.00000 0.122169
$$269$$ 10.0000 0.609711 0.304855 0.952399i $$-0.401392\pi$$
0.304855 + 0.952399i $$0.401392\pi$$
$$270$$ 0 0
$$271$$ −10.0000 −0.607457 −0.303728 0.952759i $$-0.598232\pi$$
−0.303728 + 0.952759i $$0.598232\pi$$
$$272$$ −2.00000 −0.121268
$$273$$ 0 0
$$274$$ 2.00000 0.120824
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 2.00000 0.120168 0.0600842 0.998193i $$-0.480863\pi$$
0.0600842 + 0.998193i $$0.480863\pi$$
$$278$$ −20.0000 −1.19952
$$279$$ 0 0
$$280$$ −4.00000 −0.239046
$$281$$ 10.0000 0.596550 0.298275 0.954480i $$-0.403589\pi$$
0.298275 + 0.954480i $$0.403589\pi$$
$$282$$ 0 0
$$283$$ −4.00000 −0.237775 −0.118888 0.992908i $$-0.537933\pi$$
−0.118888 + 0.992908i $$0.537933\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −20.0000 −1.18056
$$288$$ 0 0
$$289$$ −13.0000 −0.764706
$$290$$ 20.0000 1.17444
$$291$$ 0 0
$$292$$ −4.00000 −0.234082
$$293$$ 14.0000 0.817889 0.408944 0.912559i $$-0.365897\pi$$
0.408944 + 0.912559i $$0.365897\pi$$
$$294$$ 0 0
$$295$$ −8.00000 −0.465778
$$296$$ 8.00000 0.464991
$$297$$ 0 0
$$298$$ 14.0000 0.810998
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 8.00000 0.461112
$$302$$ 10.0000 0.575435
$$303$$ 0 0
$$304$$ 6.00000 0.344124
$$305$$ 4.00000 0.229039
$$306$$ 0 0
$$307$$ −2.00000 −0.114146 −0.0570730 0.998370i $$-0.518177\pi$$
−0.0570730 + 0.998370i $$0.518177\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ −20.0000 −1.13592
$$311$$ −28.0000 −1.58773 −0.793867 0.608091i $$-0.791935\pi$$
−0.793867 + 0.608091i $$0.791935\pi$$
$$312$$ 0 0
$$313$$ −26.0000 −1.46961 −0.734803 0.678280i $$-0.762726\pi$$
−0.734803 + 0.678280i $$0.762726\pi$$
$$314$$ −2.00000 −0.112867
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 18.0000 1.01098 0.505490 0.862832i $$-0.331312\pi$$
0.505490 + 0.862832i $$0.331312\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 2.00000 0.111803
$$321$$ 0 0
$$322$$ −8.00000 −0.445823
$$323$$ −12.0000 −0.667698
$$324$$ 0 0
$$325$$ 0 0
$$326$$ −14.0000 −0.775388
$$327$$ 0 0
$$328$$ 10.0000 0.552158
$$329$$ −24.0000 −1.32316
$$330$$ 0 0
$$331$$ 10.0000 0.549650 0.274825 0.961494i $$-0.411380\pi$$
0.274825 + 0.961494i $$0.411380\pi$$
$$332$$ −4.00000 −0.219529
$$333$$ 0 0
$$334$$ 12.0000 0.656611
$$335$$ 4.00000 0.218543
$$336$$ 0 0
$$337$$ 2.00000 0.108947 0.0544735 0.998515i $$-0.482652\pi$$
0.0544735 + 0.998515i $$0.482652\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ −4.00000 −0.216930
$$341$$ 0 0
$$342$$ 0 0
$$343$$ 20.0000 1.07990
$$344$$ −4.00000 −0.215666
$$345$$ 0 0
$$346$$ −6.00000 −0.322562
$$347$$ 12.0000 0.644194 0.322097 0.946707i $$-0.395612\pi$$
0.322097 + 0.946707i $$0.395612\pi$$
$$348$$ 0 0
$$349$$ −16.0000 −0.856460 −0.428230 0.903670i $$-0.640863\pi$$
−0.428230 + 0.903670i $$0.640863\pi$$
$$350$$ 2.00000 0.106904
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 26.0000 1.38384 0.691920 0.721974i $$-0.256765\pi$$
0.691920 + 0.721974i $$0.256765\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 6.00000 0.317999
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −4.00000 −0.211112 −0.105556 0.994413i $$-0.533662\pi$$
−0.105556 + 0.994413i $$0.533662\pi$$
$$360$$ 0 0
$$361$$ 17.0000 0.894737
$$362$$ −22.0000 −1.15629
$$363$$ 0 0
$$364$$ 0 0
$$365$$ −8.00000 −0.418739
$$366$$ 0 0
$$367$$ 8.00000 0.417597 0.208798 0.977959i $$-0.433045\pi$$
0.208798 + 0.977959i $$0.433045\pi$$
$$368$$ 4.00000 0.208514
$$369$$ 0 0
$$370$$ 16.0000 0.831800
$$371$$ −12.0000 −0.623009
$$372$$ 0 0
$$373$$ −6.00000 −0.310668 −0.155334 0.987862i $$-0.549645\pi$$
−0.155334 + 0.987862i $$0.549645\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 12.0000 0.618853
$$377$$ 0 0
$$378$$ 0 0
$$379$$ −34.0000 −1.74646 −0.873231 0.487306i $$-0.837980\pi$$
−0.873231 + 0.487306i $$0.837980\pi$$
$$380$$ 12.0000 0.615587
$$381$$ 0 0
$$382$$ −12.0000 −0.613973
$$383$$ −4.00000 −0.204390 −0.102195 0.994764i $$-0.532587\pi$$
−0.102195 + 0.994764i $$0.532587\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 16.0000 0.814379
$$387$$ 0 0
$$388$$ 12.0000 0.609208
$$389$$ 30.0000 1.52106 0.760530 0.649303i $$-0.224939\pi$$
0.760530 + 0.649303i $$0.224939\pi$$
$$390$$ 0 0
$$391$$ −8.00000 −0.404577
$$392$$ −3.00000 −0.151523
$$393$$ 0 0
$$394$$ −22.0000 −1.10834
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 8.00000 0.401508 0.200754 0.979642i $$-0.435661\pi$$
0.200754 + 0.979642i $$0.435661\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ −1.00000 −0.0500000
$$401$$ 30.0000 1.49813 0.749064 0.662497i $$-0.230503\pi$$
0.749064 + 0.662497i $$0.230503\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 2.00000 0.0995037
$$405$$ 0 0
$$406$$ −20.0000 −0.992583
$$407$$ 0 0
$$408$$ 0 0
$$409$$ −4.00000 −0.197787 −0.0988936 0.995098i $$-0.531530\pi$$
−0.0988936 + 0.995098i $$0.531530\pi$$
$$410$$ 20.0000 0.987730
$$411$$ 0 0
$$412$$ 16.0000 0.788263
$$413$$ 8.00000 0.393654
$$414$$ 0 0
$$415$$ −8.00000 −0.392705
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ −40.0000 −1.95413 −0.977064 0.212946i $$-0.931694\pi$$
−0.977064 + 0.212946i $$0.931694\pi$$
$$420$$ 0 0
$$421$$ −20.0000 −0.974740 −0.487370 0.873195i $$-0.662044\pi$$
−0.487370 + 0.873195i $$0.662044\pi$$
$$422$$ 12.0000 0.584151
$$423$$ 0 0
$$424$$ 6.00000 0.291386
$$425$$ 2.00000 0.0970143
$$426$$ 0 0
$$427$$ −4.00000 −0.193574
$$428$$ −8.00000 −0.386695
$$429$$ 0 0
$$430$$ −8.00000 −0.385794
$$431$$ −20.0000 −0.963366 −0.481683 0.876346i $$-0.659974\pi$$
−0.481683 + 0.876346i $$0.659974\pi$$
$$432$$ 0 0
$$433$$ 26.0000 1.24948 0.624740 0.780833i $$-0.285205\pi$$
0.624740 + 0.780833i $$0.285205\pi$$
$$434$$ 20.0000 0.960031
$$435$$ 0 0
$$436$$ −4.00000 −0.191565
$$437$$ 24.0000 1.14808
$$438$$ 0 0
$$439$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 16.0000 0.760183 0.380091 0.924949i $$-0.375893\pi$$
0.380091 + 0.924949i $$0.375893\pi$$
$$444$$ 0 0
$$445$$ 12.0000 0.568855
$$446$$ 14.0000 0.662919
$$447$$ 0 0
$$448$$ −2.00000 −0.0944911
$$449$$ 6.00000 0.283158 0.141579 0.989927i $$-0.454782\pi$$
0.141579 + 0.989927i $$0.454782\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ −14.0000 −0.658505
$$453$$ 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 $$0.727286\pi$$
$$458$$ 4.00000 0.186908
$$459$$ 0 0
$$460$$ 8.00000 0.373002
$$461$$ −30.0000 −1.39724 −0.698620 0.715493i $$-0.746202\pi$$
−0.698620 + 0.715493i $$0.746202\pi$$
$$462$$ 0 0
$$463$$ 6.00000 0.278844 0.139422 0.990233i $$-0.455476\pi$$
0.139422 + 0.990233i $$0.455476\pi$$
$$464$$ 10.0000 0.464238
$$465$$ 0 0
$$466$$ −6.00000 −0.277945
$$467$$ −12.0000 −0.555294 −0.277647 0.960683i $$-0.589555\pi$$
−0.277647 + 0.960683i $$0.589555\pi$$
$$468$$ 0 0
$$469$$ −4.00000 −0.184703
$$470$$ 24.0000 1.10704
$$471$$ 0 0
$$472$$ −4.00000 −0.184115
$$473$$ 0 0
$$474$$ 0 0
$$475$$ −6.00000 −0.275299
$$476$$ 4.00000 0.183340
$$477$$ 0 0
$$478$$ −16.0000 −0.731823
$$479$$ −24.0000 −1.09659 −0.548294 0.836286i $$-0.684723\pi$$
−0.548294 + 0.836286i $$0.684723\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 20.0000 0.910975
$$483$$ 0 0
$$484$$ −11.0000 −0.500000
$$485$$ 24.0000 1.08978
$$486$$ 0 0
$$487$$ −18.0000 −0.815658 −0.407829 0.913058i $$-0.633714\pi$$
−0.407829 + 0.913058i $$0.633714\pi$$
$$488$$ 2.00000 0.0905357
$$489$$ 0 0
$$490$$ −6.00000 −0.271052
$$491$$ −28.0000 −1.26362 −0.631811 0.775122i $$-0.717688\pi$$
−0.631811 + 0.775122i $$0.717688\pi$$
$$492$$ 0 0
$$493$$ −20.0000 −0.900755
$$494$$ 0 0
$$495$$ 0 0
$$496$$ −10.0000 −0.449013
$$497$$ 0 0
$$498$$ 0 0
$$499$$ −14.0000 −0.626726 −0.313363 0.949633i $$-0.601456\pi$$
−0.313363 + 0.949633i $$0.601456\pi$$
$$500$$ −12.0000 −0.536656
$$501$$ 0 0
$$502$$ −28.0000 −1.24970
$$503$$ −24.0000 −1.07011 −0.535054 0.844818i $$-0.679709\pi$$
−0.535054 + 0.844818i $$0.679709\pi$$
$$504$$ 0 0
$$505$$ 4.00000 0.177998
$$506$$ 0 0
$$507$$ 0 0
$$508$$ −8.00000 −0.354943
$$509$$ −6.00000 −0.265945 −0.132973 0.991120i $$-0.542452\pi$$
−0.132973 + 0.991120i $$0.542452\pi$$
$$510$$ 0 0
$$511$$ 8.00000 0.353899
$$512$$ 1.00000 0.0441942
$$513$$ 0 0
$$514$$ 18.0000 0.793946
$$515$$ 32.0000 1.41009
$$516$$ 0 0
$$517$$ 0 0
$$518$$ −16.0000 −0.703000
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 18.0000 0.788594 0.394297 0.918983i $$-0.370988\pi$$
0.394297 + 0.918983i $$0.370988\pi$$
$$522$$ 0 0
$$523$$ 4.00000 0.174908 0.0874539 0.996169i $$-0.472127\pi$$
0.0874539 + 0.996169i $$0.472127\pi$$
$$524$$ 8.00000 0.349482
$$525$$ 0 0
$$526$$ −24.0000 −1.04645
$$527$$ 20.0000 0.871214
$$528$$ 0 0
$$529$$ −7.00000 −0.304348
$$530$$ 12.0000 0.521247
$$531$$ 0 0
$$532$$ −12.0000 −0.520266
$$533$$ 0 0
$$534$$ 0 0
$$535$$ −16.0000 −0.691740
$$536$$ 2.00000 0.0863868
$$537$$ 0 0
$$538$$ 10.0000 0.431131
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −20.0000 −0.859867 −0.429934 0.902861i $$-0.641463\pi$$
−0.429934 + 0.902861i $$0.641463\pi$$
$$542$$ −10.0000 −0.429537
$$543$$ 0 0
$$544$$ −2.00000 −0.0857493
$$545$$ −8.00000 −0.342682
$$546$$ 0 0
$$547$$ 28.0000 1.19719 0.598597 0.801050i $$-0.295725\pi$$
0.598597 + 0.801050i $$0.295725\pi$$
$$548$$ 2.00000 0.0854358
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 60.0000 2.55609
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 2.00000 0.0849719
$$555$$ 0 0
$$556$$ −20.0000 −0.848189
$$557$$ −18.0000 −0.762684 −0.381342 0.924434i $$-0.624538\pi$$
−0.381342 + 0.924434i $$0.624538\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ −4.00000 −0.169031
$$561$$ 0 0
$$562$$ 10.0000 0.421825
$$563$$ −16.0000 −0.674320 −0.337160 0.941447i $$-0.609466\pi$$
−0.337160 + 0.941447i $$0.609466\pi$$
$$564$$ 0 0
$$565$$ −28.0000 −1.17797
$$566$$ −4.00000 −0.168133
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 10.0000 0.419222 0.209611 0.977785i $$-0.432780\pi$$
0.209611 + 0.977785i $$0.432780\pi$$
$$570$$ 0 0
$$571$$ 28.0000 1.17176 0.585882 0.810397i $$-0.300748\pi$$
0.585882 + 0.810397i $$0.300748\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ −20.0000 −0.834784
$$575$$ −4.00000 −0.166812
$$576$$ 0 0
$$577$$ −8.00000 −0.333044 −0.166522 0.986038i $$-0.553254\pi$$
−0.166522 + 0.986038i $$0.553254\pi$$
$$578$$ −13.0000 −0.540729
$$579$$ 0 0
$$580$$ 20.0000 0.830455
$$581$$ 8.00000 0.331896
$$582$$ 0 0
$$583$$ 0 0
$$584$$ −4.00000 −0.165521
$$585$$ 0 0
$$586$$ 14.0000 0.578335
$$587$$ 28.0000 1.15568 0.577842 0.816149i $$-0.303895\pi$$
0.577842 + 0.816149i $$0.303895\pi$$
$$588$$ 0 0
$$589$$ −60.0000 −2.47226
$$590$$ −8.00000 −0.329355
$$591$$ 0 0
$$592$$ 8.00000 0.328798
$$593$$ −26.0000 −1.06769 −0.533846 0.845582i $$-0.679254\pi$$
−0.533846 + 0.845582i $$0.679254\pi$$
$$594$$ 0 0
$$595$$ 8.00000 0.327968
$$596$$ 14.0000 0.573462
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$600$$ 0 0
$$601$$ 22.0000 0.897399 0.448699 0.893683i $$-0.351887\pi$$
0.448699 + 0.893683i $$0.351887\pi$$
$$602$$ 8.00000 0.326056
$$603$$ 0 0
$$604$$ 10.0000 0.406894
$$605$$ −22.0000 −0.894427
$$606$$ 0 0
$$607$$ −32.0000 −1.29884 −0.649420 0.760430i $$-0.724988\pi$$
−0.649420 + 0.760430i $$0.724988\pi$$
$$608$$ 6.00000 0.243332
$$609$$ 0 0
$$610$$ 4.00000 0.161955
$$611$$ 0 0
$$612$$ 0 0
$$613$$ −16.0000 −0.646234 −0.323117 0.946359i $$-0.604731\pi$$
−0.323117 + 0.946359i $$0.604731\pi$$
$$614$$ −2.00000 −0.0807134
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −22.0000 −0.885687 −0.442843 0.896599i $$-0.646030\pi$$
−0.442843 + 0.896599i $$0.646030\pi$$
$$618$$ 0 0
$$619$$ −26.0000 −1.04503 −0.522514 0.852631i $$-0.675006\pi$$
−0.522514 + 0.852631i $$0.675006\pi$$
$$620$$ −20.0000 −0.803219
$$621$$ 0 0
$$622$$ −28.0000 −1.12270
$$623$$ −12.0000 −0.480770
$$624$$ 0 0
$$625$$ −19.0000 −0.760000
$$626$$ −26.0000 −1.03917
$$627$$ 0 0
$$628$$ −2.00000 −0.0798087
$$629$$ −16.0000 −0.637962
$$630$$ 0 0
$$631$$ 10.0000 0.398094 0.199047 0.979990i $$-0.436215\pi$$
0.199047 + 0.979990i $$0.436215\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 18.0000 0.714871
$$635$$ −16.0000 −0.634941
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 2.00000 0.0790569
$$641$$ −18.0000 −0.710957 −0.355479 0.934684i $$-0.615682\pi$$
−0.355479 + 0.934684i $$0.615682\pi$$
$$642$$ 0 0
$$643$$ −6.00000 −0.236617 −0.118308 0.992977i $$-0.537747\pi$$
−0.118308 + 0.992977i $$0.537747\pi$$
$$644$$ −8.00000 −0.315244
$$645$$ 0 0
$$646$$ −12.0000 −0.472134
$$647$$ −32.0000 −1.25805 −0.629025 0.777385i $$-0.716546\pi$$
−0.629025 + 0.777385i $$0.716546\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 0 0
$$652$$ −14.0000 −0.548282
$$653$$ 26.0000 1.01746 0.508729 0.860927i $$-0.330115\pi$$
0.508729 + 0.860927i $$0.330115\pi$$
$$654$$ 0 0
$$655$$ 16.0000 0.625172
$$656$$ 10.0000 0.390434
$$657$$ 0 0
$$658$$ −24.0000 −0.935617
$$659$$ −20.0000 −0.779089 −0.389545 0.921008i $$-0.627368\pi$$
−0.389545 + 0.921008i $$0.627368\pi$$
$$660$$ 0 0
$$661$$ −40.0000 −1.55582 −0.777910 0.628376i $$-0.783720\pi$$
−0.777910 + 0.628376i $$0.783720\pi$$
$$662$$ 10.0000 0.388661
$$663$$ 0 0
$$664$$ −4.00000 −0.155230
$$665$$ −24.0000 −0.930680
$$666$$ 0 0
$$667$$ 40.0000 1.54881
$$668$$ 12.0000 0.464294
$$669$$ 0 0
$$670$$ 4.00000 0.154533
$$671$$ 0 0
$$672$$ 0 0
$$673$$ 6.00000 0.231283 0.115642 0.993291i $$-0.463108\pi$$
0.115642 + 0.993291i $$0.463108\pi$$
$$674$$ 2.00000 0.0770371
$$675$$ 0 0
$$676$$ 0 0
$$677$$ −18.0000 −0.691796 −0.345898 0.938272i $$-0.612426\pi$$
−0.345898 + 0.938272i $$0.612426\pi$$
$$678$$ 0 0
$$679$$ −24.0000 −0.921035
$$680$$ −4.00000 −0.153393
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 24.0000 0.918334 0.459167 0.888350i $$-0.348148\pi$$
0.459167 + 0.888350i $$0.348148\pi$$
$$684$$ 0 0
$$685$$ 4.00000 0.152832
$$686$$ 20.0000 0.763604
$$687$$ 0 0
$$688$$ −4.00000 −0.152499
$$689$$ 0 0
$$690$$ 0 0
$$691$$ 10.0000 0.380418 0.190209 0.981744i $$-0.439083\pi$$
0.190209 + 0.981744i $$0.439083\pi$$
$$692$$ −6.00000 −0.228086
$$693$$ 0 0
$$694$$ 12.0000 0.455514
$$695$$ −40.0000 −1.51729
$$696$$ 0 0
$$697$$ −20.0000 −0.757554
$$698$$ −16.0000 −0.605609
$$699$$ 0 0
$$700$$ 2.00000 0.0755929
$$701$$ 22.0000 0.830929 0.415464 0.909610i $$-0.363619\pi$$
0.415464 + 0.909610i $$0.363619\pi$$
$$702$$ 0 0
$$703$$ 48.0000 1.81035
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 26.0000 0.978523
$$707$$ −4.00000 −0.150435
$$708$$ 0 0
$$709$$ −36.0000 −1.35201 −0.676004 0.736898i $$-0.736290\pi$$
−0.676004 + 0.736898i $$0.736290\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 6.00000 0.224860
$$713$$ −40.0000 −1.49801
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 0 0
$$718$$ −4.00000 −0.149279
$$719$$ 20.0000 0.745874 0.372937 0.927857i $$-0.378351\pi$$
0.372937 + 0.927857i $$0.378351\pi$$
$$720$$ 0 0
$$721$$ −32.0000 −1.19174
$$722$$ 17.0000 0.632674
$$723$$ 0 0
$$724$$ −22.0000 −0.817624
$$725$$ −10.0000 −0.371391
$$726$$ 0 0
$$727$$ −8.00000 −0.296704 −0.148352 0.988935i $$-0.547397\pi$$
−0.148352 + 0.988935i $$0.547397\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ −8.00000 −0.296093
$$731$$ 8.00000 0.295891
$$732$$ 0 0
$$733$$ 44.0000 1.62518 0.812589 0.582838i $$-0.198058\pi$$
0.812589 + 0.582838i $$0.198058\pi$$
$$734$$ 8.00000 0.295285
$$735$$ 0 0
$$736$$ 4.00000 0.147442
$$737$$ 0 0
$$738$$ 0 0
$$739$$ −26.0000 −0.956425 −0.478213 0.878244i $$-0.658715\pi$$
−0.478213 + 0.878244i $$0.658715\pi$$
$$740$$ 16.0000 0.588172
$$741$$ 0 0
$$742$$ −12.0000 −0.440534
$$743$$ 16.0000 0.586983 0.293492 0.955962i $$-0.405183\pi$$
0.293492 + 0.955962i $$0.405183\pi$$
$$744$$ 0 0
$$745$$ 28.0000 1.02584
$$746$$ −6.00000 −0.219676
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 16.0000 0.584627
$$750$$ 0 0
$$751$$ −32.0000 −1.16770 −0.583848 0.811863i $$-0.698454\pi$$
−0.583848 + 0.811863i $$0.698454\pi$$
$$752$$ 12.0000 0.437595
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 20.0000 0.727875
$$756$$ 0 0
$$757$$ −22.0000 −0.799604 −0.399802 0.916602i $$-0.630921\pi$$
−0.399802 + 0.916602i $$0.630921\pi$$
$$758$$ −34.0000 −1.23494
$$759$$ 0 0
$$760$$ 12.0000 0.435286
$$761$$ −30.0000 −1.08750 −0.543750 0.839248i $$-0.682996\pi$$
−0.543750 + 0.839248i $$0.682996\pi$$
$$762$$ 0 0
$$763$$ 8.00000 0.289619
$$764$$ −12.0000 −0.434145
$$765$$ 0 0
$$766$$ −4.00000 −0.144526
$$767$$ 0 0
$$768$$ 0 0
$$769$$ −24.0000 −0.865462 −0.432731 0.901523i $$-0.642450\pi$$
−0.432731 + 0.901523i $$0.642450\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 16.0000 0.575853
$$773$$ 6.00000 0.215805 0.107903 0.994161i $$-0.465587\pi$$
0.107903 + 0.994161i $$0.465587\pi$$
$$774$$ 0 0
$$775$$ 10.0000 0.359211
$$776$$ 12.0000 0.430775
$$777$$ 0 0
$$778$$ 30.0000 1.07555
$$779$$ 60.0000 2.14972
$$780$$ 0 0
$$781$$ 0 0
$$782$$ −8.00000 −0.286079
$$783$$ 0 0
$$784$$ −3.00000 −0.107143
$$785$$ −4.00000 −0.142766
$$786$$ 0 0
$$787$$ 38.0000 1.35455 0.677277 0.735728i $$-0.263160\pi$$
0.677277 + 0.735728i $$0.263160\pi$$
$$788$$ −22.0000 −0.783718
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 28.0000 0.995565
$$792$$ 0 0
$$793$$ 0 0
$$794$$ 8.00000 0.283909
$$795$$ 0 0
$$796$$ 0 0
$$797$$ −2.00000 −0.0708436 −0.0354218 0.999372i $$-0.511277\pi$$
−0.0354218 + 0.999372i $$0.511277\pi$$
$$798$$ 0 0
$$799$$ −24.0000 −0.849059
$$800$$ −1.00000 −0.0353553
$$801$$ 0 0
$$802$$ 30.0000 1.05934
$$803$$ 0 0
$$804$$ 0 0
$$805$$ −16.0000 −0.563926
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 2.00000 0.0703598
$$809$$ 50.0000 1.75791 0.878953 0.476908i $$-0.158243\pi$$
0.878953 + 0.476908i $$0.158243\pi$$
$$810$$ 0 0
$$811$$ −10.0000 −0.351147 −0.175574 0.984466i $$-0.556178\pi$$
−0.175574 + 0.984466i $$0.556178\pi$$
$$812$$ −20.0000 −0.701862
$$813$$ 0 0
$$814$$ 0 0
$$815$$ −28.0000 −0.980797
$$816$$ 0 0
$$817$$ −24.0000 −0.839654
$$818$$ −4.00000 −0.139857
$$819$$ 0 0
$$820$$ 20.0000 0.698430
$$821$$ 30.0000 1.04701 0.523504 0.852023i $$-0.324625\pi$$
0.523504 + 0.852023i $$0.324625\pi$$
$$822$$ 0 0
$$823$$ −24.0000 −0.836587 −0.418294 0.908312i $$-0.637372\pi$$
−0.418294 + 0.908312i $$0.637372\pi$$
$$824$$ 16.0000 0.557386
$$825$$ 0 0
$$826$$ 8.00000 0.278356
$$827$$ −48.0000 −1.66912 −0.834562 0.550914i $$-0.814279\pi$$
−0.834562 + 0.550914i $$0.814279\pi$$
$$828$$ 0 0
$$829$$ 30.0000 1.04194 0.520972 0.853574i $$-0.325570\pi$$
0.520972 + 0.853574i $$0.325570\pi$$
$$830$$ −8.00000 −0.277684
$$831$$ 0 0
$$832$$ 0 0
$$833$$ 6.00000 0.207888
$$834$$ 0 0
$$835$$ 24.0000 0.830554
$$836$$ 0 0
$$837$$ 0 0
$$838$$ −40.0000 −1.38178
$$839$$ 16.0000 0.552381 0.276191 0.961103i $$-0.410928\pi$$
0.276191 + 0.961103i $$0.410928\pi$$
$$840$$ 0 0
$$841$$ 71.0000 2.44828
$$842$$ −20.0000 −0.689246
$$843$$ 0 0
$$844$$ 12.0000 0.413057
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 22.0000 0.755929
$$848$$ 6.00000 0.206041
$$849$$ 0 0
$$850$$ 2.00000 0.0685994
$$851$$ 32.0000 1.09695
$$852$$ 0 0
$$853$$ 56.0000 1.91740 0.958702 0.284413i $$-0.0917988\pi$$
0.958702 + 0.284413i $$0.0917988\pi$$
$$854$$ −4.00000 −0.136877
$$855$$ 0 0
$$856$$ −8.00000 −0.273434
$$857$$ −22.0000 −0.751506 −0.375753 0.926720i $$-0.622616\pi$$
−0.375753 + 0.926720i $$0.622616\pi$$
$$858$$ 0 0
$$859$$ −20.0000 −0.682391 −0.341196 0.939992i $$-0.610832\pi$$
−0.341196 + 0.939992i $$0.610832\pi$$
$$860$$ −8.00000 −0.272798
$$861$$ 0 0
$$862$$ −20.0000 −0.681203
$$863$$ −44.0000 −1.49778 −0.748889 0.662696i $$-0.769412\pi$$
−0.748889 + 0.662696i $$0.769412\pi$$
$$864$$ 0 0
$$865$$ −12.0000 −0.408012
$$866$$ 26.0000 0.883516
$$867$$ 0 0
$$868$$ 20.0000 0.678844
$$869$$ 0 0
$$870$$ 0 0
$$871$$ 0 0
$$872$$ −4.00000 −0.135457
$$873$$ 0 0
$$874$$ 24.0000 0.811812
$$875$$ 24.0000 0.811348
$$876$$ 0 0
$$877$$ −8.00000 −0.270141 −0.135070 0.990836i $$-0.543126\pi$$
−0.135070 + 0.990836i $$0.543126\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 42.0000 1.41502 0.707508 0.706705i $$-0.249819\pi$$
0.707508 + 0.706705i $$0.249819\pi$$
$$882$$ 0 0
$$883$$ 36.0000 1.21150 0.605748 0.795656i $$-0.292874\pi$$
0.605748 + 0.795656i $$0.292874\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 16.0000 0.537531
$$887$$ 12.0000 0.402921 0.201460 0.979497i $$-0.435431\pi$$
0.201460 + 0.979497i $$0.435431\pi$$
$$888$$ 0 0
$$889$$ 16.0000 0.536623
$$890$$ 12.0000 0.402241
$$891$$ 0 0
$$892$$ 14.0000 0.468755
$$893$$ 72.0000 2.40939
$$894$$ 0 0
$$895$$ 0 0
$$896$$ −2.00000 −0.0668153
$$897$$ 0 0
$$898$$ 6.00000 0.200223
$$899$$ −100.000 −3.33519
$$900$$ 0 0
$$901$$ −12.0000 −0.399778
$$902$$ 0 0
$$903$$ 0 0
$$904$$ −14.0000 −0.465633
$$905$$ −44.0000 −1.46261
$$906$$ 0 0
$$907$$ −28.0000 −0.929725 −0.464862 0.885383i $$-0.653896\pi$$
−0.464862 + 0.885383i $$0.653896\pi$$
$$908$$ 8.00000 0.265489
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −12.0000 −0.397578 −0.198789 0.980042i $$-0.563701\pi$$
−0.198789 + 0.980042i $$0.563701\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ −28.0000 −0.926158
$$915$$ 0 0
$$916$$ 4.00000 0.132164
$$917$$ −16.0000 −0.528367
$$918$$ 0 0
$$919$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$920$$ 8.00000 0.263752
$$921$$ 0 0
$$922$$ −30.0000 −0.987997
$$923$$ 0 0
$$924$$ 0 0
$$925$$ −8.00000 −0.263038
$$926$$ 6.00000 0.197172
$$927$$ 0 0
$$928$$ 10.0000 0.328266
$$929$$ −6.00000 −0.196854 −0.0984268 0.995144i $$-0.531381\pi$$
−0.0984268 + 0.995144i $$0.531381\pi$$
$$930$$ 0 0
$$931$$ −18.0000 −0.589926
$$932$$ −6.00000 −0.196537
$$933$$ 0 0
$$934$$ −12.0000 −0.392652
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −2.00000 −0.0653372 −0.0326686 0.999466i $$-0.510401\pi$$
−0.0326686 + 0.999466i $$0.510401\pi$$
$$938$$ −4.00000 −0.130605
$$939$$ 0 0
$$940$$ 24.0000 0.782794
$$941$$ −10.0000 −0.325991 −0.162995 0.986627i $$-0.552116\pi$$
−0.162995 + 0.986627i $$0.552116\pi$$
$$942$$ 0 0
$$943$$ 40.0000 1.30258
$$944$$ −4.00000 −0.130189
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 52.0000 1.68977 0.844886 0.534946i $$-0.179668\pi$$
0.844886 + 0.534946i $$0.179668\pi$$
$$948$$ 0 0
$$949$$ 0 0
$$950$$ −6.00000 −0.194666
$$951$$ 0 0
$$952$$ 4.00000 0.129641
$$953$$ −6.00000 −0.194359 −0.0971795 0.995267i $$-0.530982\pi$$
−0.0971795 + 0.995267i $$0.530982\pi$$
$$954$$ 0 0
$$955$$ −24.0000 −0.776622
$$956$$ −16.0000 −0.517477
$$957$$ 0 0
$$958$$ −24.0000 −0.775405
$$959$$ −4.00000 −0.129167
$$960$$ 0 0
$$961$$ 69.0000 2.22581
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 20.0000 0.644157
$$965$$ 32.0000 1.03012
$$966$$ 0 0
$$967$$ 22.0000 0.707472 0.353736 0.935345i $$-0.384911\pi$$
0.353736 + 0.935345i $$0.384911\pi$$
$$968$$ −11.0000 −0.353553
$$969$$ 0 0
$$970$$ 24.0000 0.770594
$$971$$ −12.0000 −0.385098 −0.192549 0.981287i $$-0.561675\pi$$
−0.192549 + 0.981287i $$0.561675\pi$$
$$972$$ 0 0
$$973$$ 40.0000 1.28234
$$974$$ −18.0000 −0.576757
$$975$$ 0 0
$$976$$ 2.00000 0.0640184
$$977$$ −42.0000 −1.34370 −0.671850 0.740688i $$-0.734500\pi$$
−0.671850 + 0.740688i $$0.734500\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ −6.00000 −0.191663
$$981$$ 0 0
$$982$$ −28.0000 −0.893516
$$983$$ 24.0000 0.765481 0.382741 0.923856i $$-0.374980\pi$$
0.382741 + 0.923856i $$0.374980\pi$$
$$984$$ 0 0
$$985$$ −44.0000 −1.40196
$$986$$ −20.0000 −0.636930
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −16.0000 −0.508770
$$990$$ 0 0
$$991$$ −8.00000 −0.254128 −0.127064 0.991894i $$-0.540555\pi$$
−0.127064 + 0.991894i $$0.540555\pi$$
$$992$$ −10.0000 −0.317500
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ −42.0000 −1.33015 −0.665077 0.746775i $$-0.731601\pi$$
−0.665077 + 0.746775i $$0.731601\pi$$
$$998$$ −14.0000 −0.443162
$$999$$ 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 3042.2.a.n.1.1 1
3.2 odd 2 1014.2.a.b.1.1 1
12.11 even 2 8112.2.a.g.1.1 1
13.5 odd 4 234.2.b.a.181.1 2
13.8 odd 4 234.2.b.a.181.2 2
13.12 even 2 3042.2.a.c.1.1 1
39.2 even 12 1014.2.i.c.823.1 4
39.5 even 4 78.2.b.a.25.2 yes 2
39.8 even 4 78.2.b.a.25.1 2
39.11 even 12 1014.2.i.c.823.2 4
39.17 odd 6 1014.2.e.b.991.1 2
39.20 even 12 1014.2.i.c.361.1 4
39.23 odd 6 1014.2.e.b.529.1 2
39.29 odd 6 1014.2.e.e.529.1 2
39.32 even 12 1014.2.i.c.361.2 4
39.35 odd 6 1014.2.e.e.991.1 2
39.38 odd 2 1014.2.a.g.1.1 1
52.31 even 4 1872.2.c.b.1585.1 2
52.47 even 4 1872.2.c.b.1585.2 2
156.47 odd 4 624.2.c.a.337.1 2
156.83 odd 4 624.2.c.a.337.2 2
156.155 even 2 8112.2.a.j.1.1 1
195.8 odd 4 1950.2.f.d.649.2 2
195.44 even 4 1950.2.b.c.1351.1 2
195.47 odd 4 1950.2.f.g.649.1 2
195.83 odd 4 1950.2.f.g.649.2 2
195.122 odd 4 1950.2.f.d.649.1 2
195.164 even 4 1950.2.b.c.1351.2 2
273.83 odd 4 3822.2.c.d.883.2 2
273.125 odd 4 3822.2.c.d.883.1 2
312.5 even 4 2496.2.c.f.961.1 2
312.83 odd 4 2496.2.c.m.961.1 2
312.125 even 4 2496.2.c.f.961.2 2
312.203 odd 4 2496.2.c.m.961.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
78.2.b.a.25.1 2 39.8 even 4
78.2.b.a.25.2 yes 2 39.5 even 4
234.2.b.a.181.1 2 13.5 odd 4
234.2.b.a.181.2 2 13.8 odd 4
624.2.c.a.337.1 2 156.47 odd 4
624.2.c.a.337.2 2 156.83 odd 4
1014.2.a.b.1.1 1 3.2 odd 2
1014.2.a.g.1.1 1 39.38 odd 2
1014.2.e.b.529.1 2 39.23 odd 6
1014.2.e.b.991.1 2 39.17 odd 6
1014.2.e.e.529.1 2 39.29 odd 6
1014.2.e.e.991.1 2 39.35 odd 6
1014.2.i.c.361.1 4 39.20 even 12
1014.2.i.c.361.2 4 39.32 even 12
1014.2.i.c.823.1 4 39.2 even 12
1014.2.i.c.823.2 4 39.11 even 12
1872.2.c.b.1585.1 2 52.31 even 4
1872.2.c.b.1585.2 2 52.47 even 4
1950.2.b.c.1351.1 2 195.44 even 4
1950.2.b.c.1351.2 2 195.164 even 4
1950.2.f.d.649.1 2 195.122 odd 4
1950.2.f.d.649.2 2 195.8 odd 4
1950.2.f.g.649.1 2 195.47 odd 4
1950.2.f.g.649.2 2 195.83 odd 4
2496.2.c.f.961.1 2 312.5 even 4
2496.2.c.f.961.2 2 312.125 even 4
2496.2.c.m.961.1 2 312.83 odd 4
2496.2.c.m.961.2 2 312.203 odd 4
3042.2.a.c.1.1 1 13.12 even 2
3042.2.a.n.1.1 1 1.1 even 1 trivial
3822.2.c.d.883.1 2 273.125 odd 4
3822.2.c.d.883.2 2 273.83 odd 4
8112.2.a.g.1.1 1 12.11 even 2
8112.2.a.j.1.1 1 156.155 even 2