# Properties

 Label 2394.2.a.f.1.1 Level $2394$ Weight $2$ Character 2394.1 Self dual yes Analytic conductor $19.116$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2394 = 2 \cdot 3^{2} \cdot 7 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2394.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$19.1161862439$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 798) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 2394.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} +1.00000 q^{4} +4.00000 q^{5} -1.00000 q^{7} -1.00000 q^{8} +O(q^{10})$$ $$q-1.00000 q^{2} +1.00000 q^{4} +4.00000 q^{5} -1.00000 q^{7} -1.00000 q^{8} -4.00000 q^{10} +6.00000 q^{11} -4.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} +4.00000 q^{17} -1.00000 q^{19} +4.00000 q^{20} -6.00000 q^{22} -2.00000 q^{23} +11.0000 q^{25} +4.00000 q^{26} -1.00000 q^{28} -2.00000 q^{29} +4.00000 q^{31} -1.00000 q^{32} -4.00000 q^{34} -4.00000 q^{35} +2.00000 q^{37} +1.00000 q^{38} -4.00000 q^{40} -6.00000 q^{41} +6.00000 q^{44} +2.00000 q^{46} +8.00000 q^{47} +1.00000 q^{49} -11.0000 q^{50} -4.00000 q^{52} +14.0000 q^{53} +24.0000 q^{55} +1.00000 q^{56} +2.00000 q^{58} +4.00000 q^{59} -10.0000 q^{61} -4.00000 q^{62} +1.00000 q^{64} -16.0000 q^{65} +10.0000 q^{67} +4.00000 q^{68} +4.00000 q^{70} +4.00000 q^{71} -14.0000 q^{73} -2.00000 q^{74} -1.00000 q^{76} -6.00000 q^{77} +2.00000 q^{79} +4.00000 q^{80} +6.00000 q^{82} +16.0000 q^{85} -6.00000 q^{88} -6.00000 q^{89} +4.00000 q^{91} -2.00000 q^{92} -8.00000 q^{94} -4.00000 q^{95} -1.00000 q^{98} +O(q^{100})$$

## 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
$$5$$ 4.00000 1.78885 0.894427 0.447214i $$-0.147584\pi$$
0.894427 + 0.447214i $$0.147584\pi$$
$$6$$ 0 0
$$7$$ −1.00000 −0.377964
$$8$$ −1.00000 −0.353553
$$9$$ 0 0
$$10$$ −4.00000 −1.26491
$$11$$ 6.00000 1.80907 0.904534 0.426401i $$-0.140219\pi$$
0.904534 + 0.426401i $$0.140219\pi$$
$$12$$ 0 0
$$13$$ −4.00000 −1.10940 −0.554700 0.832050i $$-0.687167\pi$$
−0.554700 + 0.832050i $$0.687167\pi$$
$$14$$ 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
$$19$$ −1.00000 −0.229416
$$20$$ 4.00000 0.894427
$$21$$ 0 0
$$22$$ −6.00000 −1.27920
$$23$$ −2.00000 −0.417029 −0.208514 0.978019i $$-0.566863\pi$$
−0.208514 + 0.978019i $$0.566863\pi$$
$$24$$ 0 0
$$25$$ 11.0000 2.20000
$$26$$ 4.00000 0.784465
$$27$$ 0 0
$$28$$ −1.00000 −0.188982
$$29$$ −2.00000 −0.371391 −0.185695 0.982607i $$-0.559454\pi$$
−0.185695 + 0.982607i $$0.559454\pi$$
$$30$$ 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
$$33$$ 0 0
$$34$$ −4.00000 −0.685994
$$35$$ −4.00000 −0.676123
$$36$$ 0 0
$$37$$ 2.00000 0.328798 0.164399 0.986394i $$-0.447432\pi$$
0.164399 + 0.986394i $$0.447432\pi$$
$$38$$ 1.00000 0.162221
$$39$$ 0 0
$$40$$ −4.00000 −0.632456
$$41$$ −6.00000 −0.937043 −0.468521 0.883452i $$-0.655213\pi$$
−0.468521 + 0.883452i $$0.655213\pi$$
$$42$$ 0 0
$$43$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$44$$ 6.00000 0.904534
$$45$$ 0 0
$$46$$ 2.00000 0.294884
$$47$$ 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
$$50$$ −11.0000 −1.55563
$$51$$ 0 0
$$52$$ −4.00000 −0.554700
$$53$$ 14.0000 1.92305 0.961524 0.274721i $$-0.0885855\pi$$
0.961524 + 0.274721i $$0.0885855\pi$$
$$54$$ 0 0
$$55$$ 24.0000 3.23616
$$56$$ 1.00000 0.133631
$$57$$ 0 0
$$58$$ 2.00000 0.262613
$$59$$ 4.00000 0.520756 0.260378 0.965507i $$-0.416153\pi$$
0.260378 + 0.965507i $$0.416153\pi$$
$$60$$ 0 0
$$61$$ −10.0000 −1.28037 −0.640184 0.768221i $$-0.721142\pi$$
−0.640184 + 0.768221i $$0.721142\pi$$
$$62$$ −4.00000 −0.508001
$$63$$ 0 0
$$64$$ 1.00000 0.125000
$$65$$ −16.0000 −1.98456
$$66$$ 0 0
$$67$$ 10.0000 1.22169 0.610847 0.791748i $$-0.290829\pi$$
0.610847 + 0.791748i $$0.290829\pi$$
$$68$$ 4.00000 0.485071
$$69$$ 0 0
$$70$$ 4.00000 0.478091
$$71$$ 4.00000 0.474713 0.237356 0.971423i $$-0.423719\pi$$
0.237356 + 0.971423i $$0.423719\pi$$
$$72$$ 0 0
$$73$$ −14.0000 −1.63858 −0.819288 0.573382i $$-0.805631\pi$$
−0.819288 + 0.573382i $$0.805631\pi$$
$$74$$ −2.00000 −0.232495
$$75$$ 0 0
$$76$$ −1.00000 −0.114708
$$77$$ −6.00000 −0.683763
$$78$$ 0 0
$$79$$ 2.00000 0.225018 0.112509 0.993651i $$-0.464111\pi$$
0.112509 + 0.993651i $$0.464111\pi$$
$$80$$ 4.00000 0.447214
$$81$$ 0 0
$$82$$ 6.00000 0.662589
$$83$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$84$$ 0 0
$$85$$ 16.0000 1.73544
$$86$$ 0 0
$$87$$ 0 0
$$88$$ −6.00000 −0.639602
$$89$$ −6.00000 −0.635999 −0.317999 0.948091i $$-0.603011\pi$$
−0.317999 + 0.948091i $$0.603011\pi$$
$$90$$ 0 0
$$91$$ 4.00000 0.419314
$$92$$ −2.00000 −0.208514
$$93$$ 0 0
$$94$$ −8.00000 −0.825137
$$95$$ −4.00000 −0.410391
$$96$$ 0 0
$$97$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$98$$ −1.00000 −0.101015
$$99$$ 0 0
$$100$$ 11.0000 1.10000
$$101$$ −4.00000 −0.398015 −0.199007 0.979998i $$-0.563772\pi$$
−0.199007 + 0.979998i $$0.563772\pi$$
$$102$$ 0 0
$$103$$ −4.00000 −0.394132 −0.197066 0.980390i $$-0.563141\pi$$
−0.197066 + 0.980390i $$0.563141\pi$$
$$104$$ 4.00000 0.392232
$$105$$ 0 0
$$106$$ −14.0000 −1.35980
$$107$$ −12.0000 −1.16008 −0.580042 0.814587i $$-0.696964\pi$$
−0.580042 + 0.814587i $$0.696964\pi$$
$$108$$ 0 0
$$109$$ −6.00000 −0.574696 −0.287348 0.957826i $$-0.592774\pi$$
−0.287348 + 0.957826i $$0.592774\pi$$
$$110$$ −24.0000 −2.28831
$$111$$ 0 0
$$112$$ −1.00000 −0.0944911
$$113$$ −18.0000 −1.69330 −0.846649 0.532152i $$-0.821383\pi$$
−0.846649 + 0.532152i $$0.821383\pi$$
$$114$$ 0 0
$$115$$ −8.00000 −0.746004
$$116$$ −2.00000 −0.185695
$$117$$ 0 0
$$118$$ −4.00000 −0.368230
$$119$$ −4.00000 −0.366679
$$120$$ 0 0
$$121$$ 25.0000 2.27273
$$122$$ 10.0000 0.905357
$$123$$ 0 0
$$124$$ 4.00000 0.359211
$$125$$ 24.0000 2.14663
$$126$$ 0 0
$$127$$ −2.00000 −0.177471 −0.0887357 0.996055i $$-0.528283\pi$$
−0.0887357 + 0.996055i $$0.528283\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ 0 0
$$130$$ 16.0000 1.40329
$$131$$ 20.0000 1.74741 0.873704 0.486458i $$-0.161711\pi$$
0.873704 + 0.486458i $$0.161711\pi$$
$$132$$ 0 0
$$133$$ 1.00000 0.0867110
$$134$$ −10.0000 −0.863868
$$135$$ 0 0
$$136$$ −4.00000 −0.342997
$$137$$ 18.0000 1.53784 0.768922 0.639343i $$-0.220793\pi$$
0.768922 + 0.639343i $$0.220793\pi$$
$$138$$ 0 0
$$139$$ 20.0000 1.69638 0.848189 0.529694i $$-0.177693\pi$$
0.848189 + 0.529694i $$0.177693\pi$$
$$140$$ −4.00000 −0.338062
$$141$$ 0 0
$$142$$ −4.00000 −0.335673
$$143$$ −24.0000 −2.00698
$$144$$ 0 0
$$145$$ −8.00000 −0.664364
$$146$$ 14.0000 1.15865
$$147$$ 0 0
$$148$$ 2.00000 0.164399
$$149$$ 18.0000 1.47462 0.737309 0.675556i $$-0.236096\pi$$
0.737309 + 0.675556i $$0.236096\pi$$
$$150$$ 0 0
$$151$$ −14.0000 −1.13930 −0.569652 0.821886i $$-0.692922\pi$$
−0.569652 + 0.821886i $$0.692922\pi$$
$$152$$ 1.00000 0.0811107
$$153$$ 0 0
$$154$$ 6.00000 0.483494
$$155$$ 16.0000 1.28515
$$156$$ 0 0
$$157$$ −22.0000 −1.75579 −0.877896 0.478852i $$-0.841053\pi$$
−0.877896 + 0.478852i $$0.841053\pi$$
$$158$$ −2.00000 −0.159111
$$159$$ 0 0
$$160$$ −4.00000 −0.316228
$$161$$ 2.00000 0.157622
$$162$$ 0 0
$$163$$ −24.0000 −1.87983 −0.939913 0.341415i $$-0.889094\pi$$
−0.939913 + 0.341415i $$0.889094\pi$$
$$164$$ −6.00000 −0.468521
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −8.00000 −0.619059 −0.309529 0.950890i $$-0.600171\pi$$
−0.309529 + 0.950890i $$0.600171\pi$$
$$168$$ 0 0
$$169$$ 3.00000 0.230769
$$170$$ −16.0000 −1.22714
$$171$$ 0 0
$$172$$ 0 0
$$173$$ −10.0000 −0.760286 −0.380143 0.924928i $$-0.624125\pi$$
−0.380143 + 0.924928i $$0.624125\pi$$
$$174$$ 0 0
$$175$$ −11.0000 −0.831522
$$176$$ 6.00000 0.452267
$$177$$ 0 0
$$178$$ 6.00000 0.449719
$$179$$ −4.00000 −0.298974 −0.149487 0.988764i $$-0.547762\pi$$
−0.149487 + 0.988764i $$0.547762\pi$$
$$180$$ 0 0
$$181$$ 16.0000 1.18927 0.594635 0.803996i $$-0.297296\pi$$
0.594635 + 0.803996i $$0.297296\pi$$
$$182$$ −4.00000 −0.296500
$$183$$ 0 0
$$184$$ 2.00000 0.147442
$$185$$ 8.00000 0.588172
$$186$$ 0 0
$$187$$ 24.0000 1.75505
$$188$$ 8.00000 0.583460
$$189$$ 0 0
$$190$$ 4.00000 0.290191
$$191$$ 14.0000 1.01300 0.506502 0.862239i $$-0.330938\pi$$
0.506502 + 0.862239i $$0.330938\pi$$
$$192$$ 0 0
$$193$$ 26.0000 1.87152 0.935760 0.352636i $$-0.114715\pi$$
0.935760 + 0.352636i $$0.114715\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 1.00000 0.0714286
$$197$$ −22.0000 −1.56744 −0.783718 0.621117i $$-0.786679\pi$$
−0.783718 + 0.621117i $$0.786679\pi$$
$$198$$ 0 0
$$199$$ 8.00000 0.567105 0.283552 0.958957i $$-0.408487\pi$$
0.283552 + 0.958957i $$0.408487\pi$$
$$200$$ −11.0000 −0.777817
$$201$$ 0 0
$$202$$ 4.00000 0.281439
$$203$$ 2.00000 0.140372
$$204$$ 0 0
$$205$$ −24.0000 −1.67623
$$206$$ 4.00000 0.278693
$$207$$ 0 0
$$208$$ −4.00000 −0.277350
$$209$$ −6.00000 −0.415029
$$210$$ 0 0
$$211$$ 10.0000 0.688428 0.344214 0.938891i $$-0.388145\pi$$
0.344214 + 0.938891i $$0.388145\pi$$
$$212$$ 14.0000 0.961524
$$213$$ 0 0
$$214$$ 12.0000 0.820303
$$215$$ 0 0
$$216$$ 0 0
$$217$$ −4.00000 −0.271538
$$218$$ 6.00000 0.406371
$$219$$ 0 0
$$220$$ 24.0000 1.61808
$$221$$ −16.0000 −1.07628
$$222$$ 0 0
$$223$$ −16.0000 −1.07144 −0.535720 0.844396i $$-0.679960\pi$$
−0.535720 + 0.844396i $$0.679960\pi$$
$$224$$ 1.00000 0.0668153
$$225$$ 0 0
$$226$$ 18.0000 1.19734
$$227$$ 20.0000 1.32745 0.663723 0.747978i $$-0.268975\pi$$
0.663723 + 0.747978i $$0.268975\pi$$
$$228$$ 0 0
$$229$$ −2.00000 −0.132164 −0.0660819 0.997814i $$-0.521050\pi$$
−0.0660819 + 0.997814i $$0.521050\pi$$
$$230$$ 8.00000 0.527504
$$231$$ 0 0
$$232$$ 2.00000 0.131306
$$233$$ −6.00000 −0.393073 −0.196537 0.980497i $$-0.562969\pi$$
−0.196537 + 0.980497i $$0.562969\pi$$
$$234$$ 0 0
$$235$$ 32.0000 2.08745
$$236$$ 4.00000 0.260378
$$237$$ 0 0
$$238$$ 4.00000 0.259281
$$239$$ 22.0000 1.42306 0.711531 0.702655i $$-0.248002\pi$$
0.711531 + 0.702655i $$0.248002\pi$$
$$240$$ 0 0
$$241$$ 28.0000 1.80364 0.901819 0.432113i $$-0.142232\pi$$
0.901819 + 0.432113i $$0.142232\pi$$
$$242$$ −25.0000 −1.60706
$$243$$ 0 0
$$244$$ −10.0000 −0.640184
$$245$$ 4.00000 0.255551
$$246$$ 0 0
$$247$$ 4.00000 0.254514
$$248$$ −4.00000 −0.254000
$$249$$ 0 0
$$250$$ −24.0000 −1.51789
$$251$$ 20.0000 1.26239 0.631194 0.775625i $$-0.282565\pi$$
0.631194 + 0.775625i $$0.282565\pi$$
$$252$$ 0 0
$$253$$ −12.0000 −0.754434
$$254$$ 2.00000 0.125491
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 6.00000 0.374270 0.187135 0.982334i $$-0.440080\pi$$
0.187135 + 0.982334i $$0.440080\pi$$
$$258$$ 0 0
$$259$$ −2.00000 −0.124274
$$260$$ −16.0000 −0.992278
$$261$$ 0 0
$$262$$ −20.0000 −1.23560
$$263$$ 2.00000 0.123325 0.0616626 0.998097i $$-0.480360\pi$$
0.0616626 + 0.998097i $$0.480360\pi$$
$$264$$ 0 0
$$265$$ 56.0000 3.44005
$$266$$ −1.00000 −0.0613139
$$267$$ 0 0
$$268$$ 10.0000 0.610847
$$269$$ 6.00000 0.365826 0.182913 0.983129i $$-0.441447\pi$$
0.182913 + 0.983129i $$0.441447\pi$$
$$270$$ 0 0
$$271$$ 8.00000 0.485965 0.242983 0.970031i $$-0.421874\pi$$
0.242983 + 0.970031i $$0.421874\pi$$
$$272$$ 4.00000 0.242536
$$273$$ 0 0
$$274$$ −18.0000 −1.08742
$$275$$ 66.0000 3.97995
$$276$$ 0 0
$$277$$ −10.0000 −0.600842 −0.300421 0.953807i $$-0.597127\pi$$
−0.300421 + 0.953807i $$0.597127\pi$$
$$278$$ −20.0000 −1.19952
$$279$$ 0 0
$$280$$ 4.00000 0.239046
$$281$$ −10.0000 −0.596550 −0.298275 0.954480i $$-0.596411\pi$$
−0.298275 + 0.954480i $$0.596411\pi$$
$$282$$ 0 0
$$283$$ −4.00000 −0.237775 −0.118888 0.992908i $$-0.537933\pi$$
−0.118888 + 0.992908i $$0.537933\pi$$
$$284$$ 4.00000 0.237356
$$285$$ 0 0
$$286$$ 24.0000 1.41915
$$287$$ 6.00000 0.354169
$$288$$ 0 0
$$289$$ −1.00000 −0.0588235
$$290$$ 8.00000 0.469776
$$291$$ 0 0
$$292$$ −14.0000 −0.819288
$$293$$ −18.0000 −1.05157 −0.525786 0.850617i $$-0.676229\pi$$
−0.525786 + 0.850617i $$0.676229\pi$$
$$294$$ 0 0
$$295$$ 16.0000 0.931556
$$296$$ −2.00000 −0.116248
$$297$$ 0 0
$$298$$ −18.0000 −1.04271
$$299$$ 8.00000 0.462652
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 14.0000 0.805609
$$303$$ 0 0
$$304$$ −1.00000 −0.0573539
$$305$$ −40.0000 −2.29039
$$306$$ 0 0
$$307$$ −8.00000 −0.456584 −0.228292 0.973593i $$-0.573314\pi$$
−0.228292 + 0.973593i $$0.573314\pi$$
$$308$$ −6.00000 −0.341882
$$309$$ 0 0
$$310$$ −16.0000 −0.908739
$$311$$ 4.00000 0.226819 0.113410 0.993548i $$-0.463823\pi$$
0.113410 + 0.993548i $$0.463823\pi$$
$$312$$ 0 0
$$313$$ −14.0000 −0.791327 −0.395663 0.918396i $$-0.629485\pi$$
−0.395663 + 0.918396i $$0.629485\pi$$
$$314$$ 22.0000 1.24153
$$315$$ 0 0
$$316$$ 2.00000 0.112509
$$317$$ 18.0000 1.01098 0.505490 0.862832i $$-0.331312\pi$$
0.505490 + 0.862832i $$0.331312\pi$$
$$318$$ 0 0
$$319$$ −12.0000 −0.671871
$$320$$ 4.00000 0.223607
$$321$$ 0 0
$$322$$ −2.00000 −0.111456
$$323$$ −4.00000 −0.222566
$$324$$ 0 0
$$325$$ −44.0000 −2.44068
$$326$$ 24.0000 1.32924
$$327$$ 0 0
$$328$$ 6.00000 0.331295
$$329$$ −8.00000 −0.441054
$$330$$ 0 0
$$331$$ −10.0000 −0.549650 −0.274825 0.961494i $$-0.588620\pi$$
−0.274825 + 0.961494i $$0.588620\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 8.00000 0.437741
$$335$$ 40.0000 2.18543
$$336$$ 0 0
$$337$$ 6.00000 0.326841 0.163420 0.986557i $$-0.447747\pi$$
0.163420 + 0.986557i $$0.447747\pi$$
$$338$$ −3.00000 −0.163178
$$339$$ 0 0
$$340$$ 16.0000 0.867722
$$341$$ 24.0000 1.29967
$$342$$ 0 0
$$343$$ −1.00000 −0.0539949
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 10.0000 0.537603
$$347$$ −6.00000 −0.322097 −0.161048 0.986947i $$-0.551488\pi$$
−0.161048 + 0.986947i $$0.551488\pi$$
$$348$$ 0 0
$$349$$ −30.0000 −1.60586 −0.802932 0.596071i $$-0.796728\pi$$
−0.802932 + 0.596071i $$0.796728\pi$$
$$350$$ 11.0000 0.587975
$$351$$ 0 0
$$352$$ −6.00000 −0.319801
$$353$$ −4.00000 −0.212899 −0.106449 0.994318i $$-0.533948\pi$$
−0.106449 + 0.994318i $$0.533948\pi$$
$$354$$ 0 0
$$355$$ 16.0000 0.849192
$$356$$ −6.00000 −0.317999
$$357$$ 0 0
$$358$$ 4.00000 0.211407
$$359$$ 34.0000 1.79445 0.897226 0.441572i $$-0.145579\pi$$
0.897226 + 0.441572i $$0.145579\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ −16.0000 −0.840941
$$363$$ 0 0
$$364$$ 4.00000 0.209657
$$365$$ −56.0000 −2.93117
$$366$$ 0 0
$$367$$ −32.0000 −1.67039 −0.835193 0.549957i $$-0.814644\pi$$
−0.835193 + 0.549957i $$0.814644\pi$$
$$368$$ −2.00000 −0.104257
$$369$$ 0 0
$$370$$ −8.00000 −0.415900
$$371$$ −14.0000 −0.726844
$$372$$ 0 0
$$373$$ 14.0000 0.724893 0.362446 0.932005i $$-0.381942\pi$$
0.362446 + 0.932005i $$0.381942\pi$$
$$374$$ −24.0000 −1.24101
$$375$$ 0 0
$$376$$ −8.00000 −0.412568
$$377$$ 8.00000 0.412021
$$378$$ 0 0
$$379$$ −34.0000 −1.74646 −0.873231 0.487306i $$-0.837980\pi$$
−0.873231 + 0.487306i $$0.837980\pi$$
$$380$$ −4.00000 −0.205196
$$381$$ 0 0
$$382$$ −14.0000 −0.716302
$$383$$ 24.0000 1.22634 0.613171 0.789950i $$-0.289894\pi$$
0.613171 + 0.789950i $$0.289894\pi$$
$$384$$ 0 0
$$385$$ −24.0000 −1.22315
$$386$$ −26.0000 −1.32337
$$387$$ 0 0
$$388$$ 0 0
$$389$$ −34.0000 −1.72387 −0.861934 0.507020i $$-0.830747\pi$$
−0.861934 + 0.507020i $$0.830747\pi$$
$$390$$ 0 0
$$391$$ −8.00000 −0.404577
$$392$$ −1.00000 −0.0505076
$$393$$ 0 0
$$394$$ 22.0000 1.10834
$$395$$ 8.00000 0.402524
$$396$$ 0 0
$$397$$ 10.0000 0.501886 0.250943 0.968002i $$-0.419259\pi$$
0.250943 + 0.968002i $$0.419259\pi$$
$$398$$ −8.00000 −0.401004
$$399$$ 0 0
$$400$$ 11.0000 0.550000
$$401$$ −30.0000 −1.49813 −0.749064 0.662497i $$-0.769497\pi$$
−0.749064 + 0.662497i $$0.769497\pi$$
$$402$$ 0 0
$$403$$ −16.0000 −0.797017
$$404$$ −4.00000 −0.199007
$$405$$ 0 0
$$406$$ −2.00000 −0.0992583
$$407$$ 12.0000 0.594818
$$408$$ 0 0
$$409$$ 24.0000 1.18672 0.593362 0.804936i $$-0.297800\pi$$
0.593362 + 0.804936i $$0.297800\pi$$
$$410$$ 24.0000 1.18528
$$411$$ 0 0
$$412$$ −4.00000 −0.197066
$$413$$ −4.00000 −0.196827
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 4.00000 0.196116
$$417$$ 0 0
$$418$$ 6.00000 0.293470
$$419$$ 12.0000 0.586238 0.293119 0.956076i $$-0.405307\pi$$
0.293119 + 0.956076i $$0.405307\pi$$
$$420$$ 0 0
$$421$$ −2.00000 −0.0974740 −0.0487370 0.998812i $$-0.515520\pi$$
−0.0487370 + 0.998812i $$0.515520\pi$$
$$422$$ −10.0000 −0.486792
$$423$$ 0 0
$$424$$ −14.0000 −0.679900
$$425$$ 44.0000 2.13431
$$426$$ 0 0
$$427$$ 10.0000 0.483934
$$428$$ −12.0000 −0.580042
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −12.0000 −0.578020 −0.289010 0.957326i $$-0.593326\pi$$
−0.289010 + 0.957326i $$0.593326\pi$$
$$432$$ 0 0
$$433$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$434$$ 4.00000 0.192006
$$435$$ 0 0
$$436$$ −6.00000 −0.287348
$$437$$ 2.00000 0.0956730
$$438$$ 0 0
$$439$$ 20.0000 0.954548 0.477274 0.878755i $$-0.341625\pi$$
0.477274 + 0.878755i $$0.341625\pi$$
$$440$$ −24.0000 −1.14416
$$441$$ 0 0
$$442$$ 16.0000 0.761042
$$443$$ −18.0000 −0.855206 −0.427603 0.903967i $$-0.640642\pi$$
−0.427603 + 0.903967i $$0.640642\pi$$
$$444$$ 0 0
$$445$$ −24.0000 −1.13771
$$446$$ 16.0000 0.757622
$$447$$ 0 0
$$448$$ −1.00000 −0.0472456
$$449$$ 30.0000 1.41579 0.707894 0.706319i $$-0.249646\pi$$
0.707894 + 0.706319i $$0.249646\pi$$
$$450$$ 0 0
$$451$$ −36.0000 −1.69517
$$452$$ −18.0000 −0.846649
$$453$$ 0 0
$$454$$ −20.0000 −0.938647
$$455$$ 16.0000 0.750092
$$456$$ 0 0
$$457$$ −10.0000 −0.467780 −0.233890 0.972263i $$-0.575146\pi$$
−0.233890 + 0.972263i $$0.575146\pi$$
$$458$$ 2.00000 0.0934539
$$459$$ 0 0
$$460$$ −8.00000 −0.373002
$$461$$ −36.0000 −1.67669 −0.838344 0.545142i $$-0.816476\pi$$
−0.838344 + 0.545142i $$0.816476\pi$$
$$462$$ 0 0
$$463$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$464$$ −2.00000 −0.0928477
$$465$$ 0 0
$$466$$ 6.00000 0.277945
$$467$$ 8.00000 0.370196 0.185098 0.982720i $$-0.440740\pi$$
0.185098 + 0.982720i $$0.440740\pi$$
$$468$$ 0 0
$$469$$ −10.0000 −0.461757
$$470$$ −32.0000 −1.47605
$$471$$ 0 0
$$472$$ −4.00000 −0.184115
$$473$$ 0 0
$$474$$ 0 0
$$475$$ −11.0000 −0.504715
$$476$$ −4.00000 −0.183340
$$477$$ 0 0
$$478$$ −22.0000 −1.00626
$$479$$ −4.00000 −0.182765 −0.0913823 0.995816i $$-0.529129\pi$$
−0.0913823 + 0.995816i $$0.529129\pi$$
$$480$$ 0 0
$$481$$ −8.00000 −0.364769
$$482$$ −28.0000 −1.27537
$$483$$ 0 0
$$484$$ 25.0000 1.13636
$$485$$ 0 0
$$486$$ 0 0
$$487$$ −34.0000 −1.54069 −0.770344 0.637629i $$-0.779915\pi$$
−0.770344 + 0.637629i $$0.779915\pi$$
$$488$$ 10.0000 0.452679
$$489$$ 0 0
$$490$$ −4.00000 −0.180702
$$491$$ −30.0000 −1.35388 −0.676941 0.736038i $$-0.736695\pi$$
−0.676941 + 0.736038i $$0.736695\pi$$
$$492$$ 0 0
$$493$$ −8.00000 −0.360302
$$494$$ −4.00000 −0.179969
$$495$$ 0 0
$$496$$ 4.00000 0.179605
$$497$$ −4.00000 −0.179425
$$498$$ 0 0
$$499$$ 24.0000 1.07439 0.537194 0.843459i $$-0.319484\pi$$
0.537194 + 0.843459i $$0.319484\pi$$
$$500$$ 24.0000 1.07331
$$501$$ 0 0
$$502$$ −20.0000 −0.892644
$$503$$ −28.0000 −1.24846 −0.624229 0.781241i $$-0.714587\pi$$
−0.624229 + 0.781241i $$0.714587\pi$$
$$504$$ 0 0
$$505$$ −16.0000 −0.711991
$$506$$ 12.0000 0.533465
$$507$$ 0 0
$$508$$ −2.00000 −0.0887357
$$509$$ −42.0000 −1.86162 −0.930809 0.365507i $$-0.880896\pi$$
−0.930809 + 0.365507i $$0.880896\pi$$
$$510$$ 0 0
$$511$$ 14.0000 0.619324
$$512$$ −1.00000 −0.0441942
$$513$$ 0 0
$$514$$ −6.00000 −0.264649
$$515$$ −16.0000 −0.705044
$$516$$ 0 0
$$517$$ 48.0000 2.11104
$$518$$ 2.00000 0.0878750
$$519$$ 0 0
$$520$$ 16.0000 0.701646
$$521$$ −38.0000 −1.66481 −0.832405 0.554168i $$-0.813037\pi$$
−0.832405 + 0.554168i $$0.813037\pi$$
$$522$$ 0 0
$$523$$ 16.0000 0.699631 0.349816 0.936819i $$-0.386244\pi$$
0.349816 + 0.936819i $$0.386244\pi$$
$$524$$ 20.0000 0.873704
$$525$$ 0 0
$$526$$ −2.00000 −0.0872041
$$527$$ 16.0000 0.696971
$$528$$ 0 0
$$529$$ −19.0000 −0.826087
$$530$$ −56.0000 −2.43248
$$531$$ 0 0
$$532$$ 1.00000 0.0433555
$$533$$ 24.0000 1.03956
$$534$$ 0 0
$$535$$ −48.0000 −2.07522
$$536$$ −10.0000 −0.431934
$$537$$ 0 0
$$538$$ −6.00000 −0.258678
$$539$$ 6.00000 0.258438
$$540$$ 0 0
$$541$$ −14.0000 −0.601907 −0.300954 0.953639i $$-0.597305\pi$$
−0.300954 + 0.953639i $$0.597305\pi$$
$$542$$ −8.00000 −0.343629
$$543$$ 0 0
$$544$$ −4.00000 −0.171499
$$545$$ −24.0000 −1.02805
$$546$$ 0 0
$$547$$ 10.0000 0.427569 0.213785 0.976881i $$-0.431421\pi$$
0.213785 + 0.976881i $$0.431421\pi$$
$$548$$ 18.0000 0.768922
$$549$$ 0 0
$$550$$ −66.0000 −2.81425
$$551$$ 2.00000 0.0852029
$$552$$ 0 0
$$553$$ −2.00000 −0.0850487
$$554$$ 10.0000 0.424859
$$555$$ 0 0
$$556$$ 20.0000 0.848189
$$557$$ 10.0000 0.423714 0.211857 0.977301i $$-0.432049\pi$$
0.211857 + 0.977301i $$0.432049\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ −4.00000 −0.169031
$$561$$ 0 0
$$562$$ 10.0000 0.421825
$$563$$ −44.0000 −1.85438 −0.927189 0.374593i $$-0.877783\pi$$
−0.927189 + 0.374593i $$0.877783\pi$$
$$564$$ 0 0
$$565$$ −72.0000 −3.02906
$$566$$ 4.00000 0.168133
$$567$$ 0 0
$$568$$ −4.00000 −0.167836
$$569$$ 10.0000 0.419222 0.209611 0.977785i $$-0.432780\pi$$
0.209611 + 0.977785i $$0.432780\pi$$
$$570$$ 0 0
$$571$$ −4.00000 −0.167395 −0.0836974 0.996491i $$-0.526673\pi$$
−0.0836974 + 0.996491i $$0.526673\pi$$
$$572$$ −24.0000 −1.00349
$$573$$ 0 0
$$574$$ −6.00000 −0.250435
$$575$$ −22.0000 −0.917463
$$576$$ 0 0
$$577$$ −38.0000 −1.58196 −0.790980 0.611842i $$-0.790429\pi$$
−0.790980 + 0.611842i $$0.790429\pi$$
$$578$$ 1.00000 0.0415945
$$579$$ 0 0
$$580$$ −8.00000 −0.332182
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 84.0000 3.47892
$$584$$ 14.0000 0.579324
$$585$$ 0 0
$$586$$ 18.0000 0.743573
$$587$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$588$$ 0 0
$$589$$ −4.00000 −0.164817
$$590$$ −16.0000 −0.658710
$$591$$ 0 0
$$592$$ 2.00000 0.0821995
$$593$$ −16.0000 −0.657041 −0.328521 0.944497i $$-0.606550\pi$$
−0.328521 + 0.944497i $$0.606550\pi$$
$$594$$ 0 0
$$595$$ −16.0000 −0.655936
$$596$$ 18.0000 0.737309
$$597$$ 0 0
$$598$$ −8.00000 −0.327144
$$599$$ 28.0000 1.14405 0.572024 0.820237i $$-0.306158\pi$$
0.572024 + 0.820237i $$0.306158\pi$$
$$600$$ 0 0
$$601$$ −8.00000 −0.326327 −0.163163 0.986599i $$-0.552170\pi$$
−0.163163 + 0.986599i $$0.552170\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ −14.0000 −0.569652
$$605$$ 100.000 4.06558
$$606$$ 0 0
$$607$$ −24.0000 −0.974130 −0.487065 0.873366i $$-0.661933\pi$$
−0.487065 + 0.873366i $$0.661933\pi$$
$$608$$ 1.00000 0.0405554
$$609$$ 0 0
$$610$$ 40.0000 1.61955
$$611$$ −32.0000 −1.29458
$$612$$ 0 0
$$613$$ −34.0000 −1.37325 −0.686624 0.727013i $$-0.740908\pi$$
−0.686624 + 0.727013i $$0.740908\pi$$
$$614$$ 8.00000 0.322854
$$615$$ 0 0
$$616$$ 6.00000 0.241747
$$617$$ 2.00000 0.0805170 0.0402585 0.999189i $$-0.487182\pi$$
0.0402585 + 0.999189i $$0.487182\pi$$
$$618$$ 0 0
$$619$$ 12.0000 0.482321 0.241160 0.970485i $$-0.422472\pi$$
0.241160 + 0.970485i $$0.422472\pi$$
$$620$$ 16.0000 0.642575
$$621$$ 0 0
$$622$$ −4.00000 −0.160385
$$623$$ 6.00000 0.240385
$$624$$ 0 0
$$625$$ 41.0000 1.64000
$$626$$ 14.0000 0.559553
$$627$$ 0 0
$$628$$ −22.0000 −0.877896
$$629$$ 8.00000 0.318981
$$630$$ 0 0
$$631$$ 12.0000 0.477712 0.238856 0.971055i $$-0.423228\pi$$
0.238856 + 0.971055i $$0.423228\pi$$
$$632$$ −2.00000 −0.0795557
$$633$$ 0 0
$$634$$ −18.0000 −0.714871
$$635$$ −8.00000 −0.317470
$$636$$ 0 0
$$637$$ −4.00000 −0.158486
$$638$$ 12.0000 0.475085
$$639$$ 0 0
$$640$$ −4.00000 −0.158114
$$641$$ 22.0000 0.868948 0.434474 0.900684i $$-0.356934\pi$$
0.434474 + 0.900684i $$0.356934\pi$$
$$642$$ 0 0
$$643$$ −36.0000 −1.41970 −0.709851 0.704352i $$-0.751238\pi$$
−0.709851 + 0.704352i $$0.751238\pi$$
$$644$$ 2.00000 0.0788110
$$645$$ 0 0
$$646$$ 4.00000 0.157378
$$647$$ −12.0000 −0.471769 −0.235884 0.971781i $$-0.575799\pi$$
−0.235884 + 0.971781i $$0.575799\pi$$
$$648$$ 0 0
$$649$$ 24.0000 0.942082
$$650$$ 44.0000 1.72582
$$651$$ 0 0
$$652$$ −24.0000 −0.939913
$$653$$ 14.0000 0.547862 0.273931 0.961749i $$-0.411676\pi$$
0.273931 + 0.961749i $$0.411676\pi$$
$$654$$ 0 0
$$655$$ 80.0000 3.12586
$$656$$ −6.00000 −0.234261
$$657$$ 0 0
$$658$$ 8.00000 0.311872
$$659$$ 36.0000 1.40236 0.701180 0.712984i $$-0.252657\pi$$
0.701180 + 0.712984i $$0.252657\pi$$
$$660$$ 0 0
$$661$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$662$$ 10.0000 0.388661
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 4.00000 0.155113
$$666$$ 0 0
$$667$$ 4.00000 0.154881
$$668$$ −8.00000 −0.309529
$$669$$ 0 0
$$670$$ −40.0000 −1.54533
$$671$$ −60.0000 −2.31627
$$672$$ 0 0
$$673$$ 22.0000 0.848038 0.424019 0.905653i $$-0.360619\pi$$
0.424019 + 0.905653i $$0.360619\pi$$
$$674$$ −6.00000 −0.231111
$$675$$ 0 0
$$676$$ 3.00000 0.115385
$$677$$ −6.00000 −0.230599 −0.115299 0.993331i $$-0.536783\pi$$
−0.115299 + 0.993331i $$0.536783\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ −16.0000 −0.613572
$$681$$ 0 0
$$682$$ −24.0000 −0.919007
$$683$$ −8.00000 −0.306111 −0.153056 0.988218i $$-0.548911\pi$$
−0.153056 + 0.988218i $$0.548911\pi$$
$$684$$ 0 0
$$685$$ 72.0000 2.75098
$$686$$ 1.00000 0.0381802
$$687$$ 0 0
$$688$$ 0 0
$$689$$ −56.0000 −2.13343
$$690$$ 0 0
$$691$$ −4.00000 −0.152167 −0.0760836 0.997101i $$-0.524242\pi$$
−0.0760836 + 0.997101i $$0.524242\pi$$
$$692$$ −10.0000 −0.380143
$$693$$ 0 0
$$694$$ 6.00000 0.227757
$$695$$ 80.0000 3.03457
$$696$$ 0 0
$$697$$ −24.0000 −0.909065
$$698$$ 30.0000 1.13552
$$699$$ 0 0
$$700$$ −11.0000 −0.415761
$$701$$ 6.00000 0.226617 0.113308 0.993560i $$-0.463855\pi$$
0.113308 + 0.993560i $$0.463855\pi$$
$$702$$ 0 0
$$703$$ −2.00000 −0.0754314
$$704$$ 6.00000 0.226134
$$705$$ 0 0
$$706$$ 4.00000 0.150542
$$707$$ 4.00000 0.150435
$$708$$ 0 0
$$709$$ 10.0000 0.375558 0.187779 0.982211i $$-0.439871\pi$$
0.187779 + 0.982211i $$0.439871\pi$$
$$710$$ −16.0000 −0.600469
$$711$$ 0 0
$$712$$ 6.00000 0.224860
$$713$$ −8.00000 −0.299602
$$714$$ 0 0
$$715$$ −96.0000 −3.59020
$$716$$ −4.00000 −0.149487
$$717$$ 0 0
$$718$$ −34.0000 −1.26887
$$719$$ −12.0000 −0.447524 −0.223762 0.974644i $$-0.571834\pi$$
−0.223762 + 0.974644i $$0.571834\pi$$
$$720$$ 0 0
$$721$$ 4.00000 0.148968
$$722$$ −1.00000 −0.0372161
$$723$$ 0 0
$$724$$ 16.0000 0.594635
$$725$$ −22.0000 −0.817059
$$726$$ 0 0
$$727$$ −32.0000 −1.18681 −0.593407 0.804902i $$-0.702218\pi$$
−0.593407 + 0.804902i $$0.702218\pi$$
$$728$$ −4.00000 −0.148250
$$729$$ 0 0
$$730$$ 56.0000 2.07265
$$731$$ 0 0
$$732$$ 0 0
$$733$$ −30.0000 −1.10808 −0.554038 0.832492i $$-0.686914\pi$$
−0.554038 + 0.832492i $$0.686914\pi$$
$$734$$ 32.0000 1.18114
$$735$$ 0 0
$$736$$ 2.00000 0.0737210
$$737$$ 60.0000 2.21013
$$738$$ 0 0
$$739$$ 4.00000 0.147142 0.0735712 0.997290i $$-0.476560\pi$$
0.0735712 + 0.997290i $$0.476560\pi$$
$$740$$ 8.00000 0.294086
$$741$$ 0 0
$$742$$ 14.0000 0.513956
$$743$$ 32.0000 1.17397 0.586983 0.809599i $$-0.300316\pi$$
0.586983 + 0.809599i $$0.300316\pi$$
$$744$$ 0 0
$$745$$ 72.0000 2.63788
$$746$$ −14.0000 −0.512576
$$747$$ 0 0
$$748$$ 24.0000 0.877527
$$749$$ 12.0000 0.438470
$$750$$ 0 0
$$751$$ 22.0000 0.802791 0.401396 0.915905i $$-0.368525\pi$$
0.401396 + 0.915905i $$0.368525\pi$$
$$752$$ 8.00000 0.291730
$$753$$ 0 0
$$754$$ −8.00000 −0.291343
$$755$$ −56.0000 −2.03805
$$756$$ 0 0
$$757$$ 46.0000 1.67190 0.835949 0.548807i $$-0.184918\pi$$
0.835949 + 0.548807i $$0.184918\pi$$
$$758$$ 34.0000 1.23494
$$759$$ 0 0
$$760$$ 4.00000 0.145095
$$761$$ −16.0000 −0.580000 −0.290000 0.957027i $$-0.593655\pi$$
−0.290000 + 0.957027i $$0.593655\pi$$
$$762$$ 0 0
$$763$$ 6.00000 0.217215
$$764$$ 14.0000 0.506502
$$765$$ 0 0
$$766$$ −24.0000 −0.867155
$$767$$ −16.0000 −0.577727
$$768$$ 0 0
$$769$$ 18.0000 0.649097 0.324548 0.945869i $$-0.394788\pi$$
0.324548 + 0.945869i $$0.394788\pi$$
$$770$$ 24.0000 0.864900
$$771$$ 0 0
$$772$$ 26.0000 0.935760
$$773$$ −42.0000 −1.51064 −0.755318 0.655359i $$-0.772517\pi$$
−0.755318 + 0.655359i $$0.772517\pi$$
$$774$$ 0 0
$$775$$ 44.0000 1.58053
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 34.0000 1.21896
$$779$$ 6.00000 0.214972
$$780$$ 0 0
$$781$$ 24.0000 0.858788
$$782$$ 8.00000 0.286079
$$783$$ 0 0
$$784$$ 1.00000 0.0357143
$$785$$ −88.0000 −3.14085
$$786$$ 0 0
$$787$$ 32.0000 1.14068 0.570338 0.821410i $$-0.306812\pi$$
0.570338 + 0.821410i $$0.306812\pi$$
$$788$$ −22.0000 −0.783718
$$789$$ 0 0
$$790$$ −8.00000 −0.284627
$$791$$ 18.0000 0.640006
$$792$$ 0 0
$$793$$ 40.0000 1.42044
$$794$$ −10.0000 −0.354887
$$795$$ 0 0
$$796$$ 8.00000 0.283552
$$797$$ −2.00000 −0.0708436 −0.0354218 0.999372i $$-0.511277\pi$$
−0.0354218 + 0.999372i $$0.511277\pi$$
$$798$$ 0 0
$$799$$ 32.0000 1.13208
$$800$$ −11.0000 −0.388909
$$801$$ 0 0
$$802$$ 30.0000 1.05934
$$803$$ −84.0000 −2.96430
$$804$$ 0 0
$$805$$ 8.00000 0.281963
$$806$$ 16.0000 0.563576
$$807$$ 0 0
$$808$$ 4.00000 0.140720
$$809$$ −10.0000 −0.351581 −0.175791 0.984428i $$-0.556248\pi$$
−0.175791 + 0.984428i $$0.556248\pi$$
$$810$$ 0 0
$$811$$ −40.0000 −1.40459 −0.702295 0.711886i $$-0.747841\pi$$
−0.702295 + 0.711886i $$0.747841\pi$$
$$812$$ 2.00000 0.0701862
$$813$$ 0 0
$$814$$ −12.0000 −0.420600
$$815$$ −96.0000 −3.36273
$$816$$ 0 0
$$817$$ 0 0
$$818$$ −24.0000 −0.839140
$$819$$ 0 0
$$820$$ −24.0000 −0.838116
$$821$$ −30.0000 −1.04701 −0.523504 0.852023i $$-0.675375\pi$$
−0.523504 + 0.852023i $$0.675375\pi$$
$$822$$ 0 0
$$823$$ 24.0000 0.836587 0.418294 0.908312i $$-0.362628\pi$$
0.418294 + 0.908312i $$0.362628\pi$$
$$824$$ 4.00000 0.139347
$$825$$ 0 0
$$826$$ 4.00000 0.139178
$$827$$ 28.0000 0.973655 0.486828 0.873498i $$-0.338154\pi$$
0.486828 + 0.873498i $$0.338154\pi$$
$$828$$ 0 0
$$829$$ −44.0000 −1.52818 −0.764092 0.645108i $$-0.776812\pi$$
−0.764092 + 0.645108i $$0.776812\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ −4.00000 −0.138675
$$833$$ 4.00000 0.138592
$$834$$ 0 0
$$835$$ −32.0000 −1.10741
$$836$$ −6.00000 −0.207514
$$837$$ 0 0
$$838$$ −12.0000 −0.414533
$$839$$ −48.0000 −1.65714 −0.828572 0.559883i $$-0.810846\pi$$
−0.828572 + 0.559883i $$0.810846\pi$$
$$840$$ 0 0
$$841$$ −25.0000 −0.862069
$$842$$ 2.00000 0.0689246
$$843$$ 0 0
$$844$$ 10.0000 0.344214
$$845$$ 12.0000 0.412813
$$846$$ 0 0
$$847$$ −25.0000 −0.859010
$$848$$ 14.0000 0.480762
$$849$$ 0 0
$$850$$ −44.0000 −1.50919
$$851$$ −4.00000 −0.137118
$$852$$ 0 0
$$853$$ 46.0000 1.57501 0.787505 0.616308i $$-0.211372\pi$$
0.787505 + 0.616308i $$0.211372\pi$$
$$854$$ −10.0000 −0.342193
$$855$$ 0 0
$$856$$ 12.0000 0.410152
$$857$$ 18.0000 0.614868 0.307434 0.951569i $$-0.400530\pi$$
0.307434 + 0.951569i $$0.400530\pi$$
$$858$$ 0 0
$$859$$ 4.00000 0.136478 0.0682391 0.997669i $$-0.478262\pi$$
0.0682391 + 0.997669i $$0.478262\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 12.0000 0.408722
$$863$$ −48.0000 −1.63394 −0.816970 0.576681i $$-0.804348\pi$$
−0.816970 + 0.576681i $$0.804348\pi$$
$$864$$ 0 0
$$865$$ −40.0000 −1.36004
$$866$$ 0 0
$$867$$ 0 0
$$868$$ −4.00000 −0.135769
$$869$$ 12.0000 0.407072
$$870$$ 0 0
$$871$$ −40.0000 −1.35535
$$872$$ 6.00000 0.203186
$$873$$ 0 0
$$874$$ −2.00000 −0.0676510
$$875$$ −24.0000 −0.811348
$$876$$ 0 0
$$877$$ −54.0000 −1.82345 −0.911725 0.410801i $$-0.865249\pi$$
−0.911725 + 0.410801i $$0.865249\pi$$
$$878$$ −20.0000 −0.674967
$$879$$ 0 0
$$880$$ 24.0000 0.809040
$$881$$ −12.0000 −0.404290 −0.202145 0.979356i $$-0.564791\pi$$
−0.202145 + 0.979356i $$0.564791\pi$$
$$882$$ 0 0
$$883$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$884$$ −16.0000 −0.538138
$$885$$ 0 0
$$886$$ 18.0000 0.604722
$$887$$ 24.0000 0.805841 0.402921 0.915235i $$-0.367995\pi$$
0.402921 + 0.915235i $$0.367995\pi$$
$$888$$ 0 0
$$889$$ 2.00000 0.0670778
$$890$$ 24.0000 0.804482
$$891$$ 0 0
$$892$$ −16.0000 −0.535720
$$893$$ −8.00000 −0.267710
$$894$$ 0 0
$$895$$ −16.0000 −0.534821
$$896$$ 1.00000 0.0334077
$$897$$ 0 0
$$898$$ −30.0000 −1.00111
$$899$$ −8.00000 −0.266815
$$900$$ 0 0
$$901$$ 56.0000 1.86563
$$902$$ 36.0000 1.19867
$$903$$ 0 0
$$904$$ 18.0000 0.598671
$$905$$ 64.0000 2.12743
$$906$$ 0 0
$$907$$ −26.0000 −0.863316 −0.431658 0.902037i $$-0.642071\pi$$
−0.431658 + 0.902037i $$0.642071\pi$$
$$908$$ 20.0000 0.663723
$$909$$ 0 0
$$910$$ −16.0000 −0.530395
$$911$$ −52.0000 −1.72284 −0.861418 0.507896i $$-0.830423\pi$$
−0.861418 + 0.507896i $$0.830423\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ 10.0000 0.330771
$$915$$ 0 0
$$916$$ −2.00000 −0.0660819
$$917$$ −20.0000 −0.660458
$$918$$ 0 0
$$919$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$920$$ 8.00000 0.263752
$$921$$ 0 0
$$922$$ 36.0000 1.18560
$$923$$ −16.0000 −0.526646
$$924$$ 0 0
$$925$$ 22.0000 0.723356
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 2.00000 0.0656532
$$929$$ 52.0000 1.70606 0.853032 0.521858i $$-0.174761\pi$$
0.853032 + 0.521858i $$0.174761\pi$$
$$930$$ 0 0
$$931$$ −1.00000 −0.0327737
$$932$$ −6.00000 −0.196537
$$933$$ 0 0
$$934$$ −8.00000 −0.261768
$$935$$ 96.0000 3.13954
$$936$$ 0 0
$$937$$ 38.0000 1.24141 0.620703 0.784046i $$-0.286847\pi$$
0.620703 + 0.784046i $$0.286847\pi$$
$$938$$ 10.0000 0.326512
$$939$$ 0 0
$$940$$ 32.0000 1.04372
$$941$$ −2.00000 −0.0651981 −0.0325991 0.999469i $$-0.510378\pi$$
−0.0325991 + 0.999469i $$0.510378\pi$$
$$942$$ 0 0
$$943$$ 12.0000 0.390774
$$944$$ 4.00000 0.130189
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −30.0000 −0.974869 −0.487435 0.873160i $$-0.662067\pi$$
−0.487435 + 0.873160i $$0.662067\pi$$
$$948$$ 0 0
$$949$$ 56.0000 1.81784
$$950$$ 11.0000 0.356887
$$951$$ 0 0
$$952$$ 4.00000 0.129641
$$953$$ 22.0000 0.712650 0.356325 0.934362i $$-0.384030\pi$$
0.356325 + 0.934362i $$0.384030\pi$$
$$954$$ 0 0
$$955$$ 56.0000 1.81212
$$956$$ 22.0000 0.711531
$$957$$ 0 0
$$958$$ 4.00000 0.129234
$$959$$ −18.0000 −0.581250
$$960$$ 0 0
$$961$$ −15.0000 −0.483871
$$962$$ 8.00000 0.257930
$$963$$ 0 0
$$964$$ 28.0000 0.901819
$$965$$ 104.000 3.34788
$$966$$ 0 0
$$967$$ 4.00000 0.128631 0.0643157 0.997930i $$-0.479514\pi$$
0.0643157 + 0.997930i $$0.479514\pi$$
$$968$$ −25.0000 −0.803530
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 12.0000 0.385098 0.192549 0.981287i $$-0.438325\pi$$
0.192549 + 0.981287i $$0.438325\pi$$
$$972$$ 0 0
$$973$$ −20.0000 −0.641171
$$974$$ 34.0000 1.08943
$$975$$ 0 0
$$976$$ −10.0000 −0.320092
$$977$$ 18.0000 0.575871 0.287936 0.957650i $$-0.407031\pi$$
0.287936 + 0.957650i $$0.407031\pi$$
$$978$$ 0 0
$$979$$ −36.0000 −1.15056
$$980$$ 4.00000 0.127775
$$981$$ 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
$$985$$ −88.0000 −2.80391
$$986$$ 8.00000 0.254772
$$987$$ 0 0
$$988$$ 4.00000 0.127257
$$989$$ 0 0
$$990$$ 0 0
$$991$$ −30.0000 −0.952981 −0.476491 0.879180i $$-0.658091\pi$$
−0.476491 + 0.879180i $$0.658091\pi$$
$$992$$ −4.00000 −0.127000
$$993$$ 0 0
$$994$$ 4.00000 0.126872
$$995$$ 32.0000 1.01447
$$996$$ 0 0
$$997$$ −14.0000 −0.443384 −0.221692 0.975117i $$-0.571158\pi$$
−0.221692 + 0.975117i $$0.571158\pi$$
$$998$$ −24.0000 −0.759707
$$999$$ 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 2394.2.a.f.1.1 1
3.2 odd 2 798.2.a.h.1.1 1
12.11 even 2 6384.2.a.c.1.1 1
21.20 even 2 5586.2.a.x.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
798.2.a.h.1.1 1 3.2 odd 2
2394.2.a.f.1.1 1 1.1 even 1 trivial
5586.2.a.x.1.1 1 21.20 even 2
6384.2.a.c.1.1 1 12.11 even 2