# Properties

 Label 342.2.a.d.1.1 Level $342$ Weight $2$ Character 342.1 Self dual yes Analytic conductor $2.731$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$342 = 2 \cdot 3^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 342.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$2.73088374913$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 38) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 342.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} +1.00000 q^{4} +4.00000 q^{5} +3.00000 q^{7} -1.00000 q^{8} +O(q^{10})$$ $$q-1.00000 q^{2} +1.00000 q^{4} +4.00000 q^{5} +3.00000 q^{7} -1.00000 q^{8} -4.00000 q^{10} -2.00000 q^{11} -1.00000 q^{13} -3.00000 q^{14} +1.00000 q^{16} -3.00000 q^{17} -1.00000 q^{19} +4.00000 q^{20} +2.00000 q^{22} +1.00000 q^{23} +11.0000 q^{25} +1.00000 q^{26} +3.00000 q^{28} +5.00000 q^{29} -8.00000 q^{31} -1.00000 q^{32} +3.00000 q^{34} +12.0000 q^{35} -2.00000 q^{37} +1.00000 q^{38} -4.00000 q^{40} +8.00000 q^{41} +4.00000 q^{43} -2.00000 q^{44} -1.00000 q^{46} -8.00000 q^{47} +2.00000 q^{49} -11.0000 q^{50} -1.00000 q^{52} +1.00000 q^{53} -8.00000 q^{55} -3.00000 q^{56} -5.00000 q^{58} -15.0000 q^{59} +2.00000 q^{61} +8.00000 q^{62} +1.00000 q^{64} -4.00000 q^{65} +3.00000 q^{67} -3.00000 q^{68} -12.0000 q^{70} -2.00000 q^{71} +9.00000 q^{73} +2.00000 q^{74} -1.00000 q^{76} -6.00000 q^{77} -10.0000 q^{79} +4.00000 q^{80} -8.00000 q^{82} +6.00000 q^{83} -12.0000 q^{85} -4.00000 q^{86} +2.00000 q^{88} -3.00000 q^{91} +1.00000 q^{92} +8.00000 q^{94} -4.00000 q^{95} -2.00000 q^{97} -2.00000 q^{98} +O(q^{100})$$

## 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
$$7$$ 3.00000 1.13389 0.566947 0.823754i $$-0.308125\pi$$
0.566947 + 0.823754i $$0.308125\pi$$
$$8$$ −1.00000 −0.353553
$$9$$ 0 0
$$10$$ −4.00000 −1.26491
$$11$$ −2.00000 −0.603023 −0.301511 0.953463i $$-0.597491\pi$$
−0.301511 + 0.953463i $$0.597491\pi$$
$$12$$ 0 0
$$13$$ −1.00000 −0.277350 −0.138675 0.990338i $$-0.544284\pi$$
−0.138675 + 0.990338i $$0.544284\pi$$
$$14$$ −3.00000 −0.801784
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ −3.00000 −0.727607 −0.363803 0.931476i $$-0.618522\pi$$
−0.363803 + 0.931476i $$0.618522\pi$$
$$18$$ 0 0
$$19$$ −1.00000 −0.229416
$$20$$ 4.00000 0.894427
$$21$$ 0 0
$$22$$ 2.00000 0.426401
$$23$$ 1.00000 0.208514 0.104257 0.994550i $$-0.466753\pi$$
0.104257 + 0.994550i $$0.466753\pi$$
$$24$$ 0 0
$$25$$ 11.0000 2.20000
$$26$$ 1.00000 0.196116
$$27$$ 0 0
$$28$$ 3.00000 0.566947
$$29$$ 5.00000 0.928477 0.464238 0.885710i $$-0.346328\pi$$
0.464238 + 0.885710i $$0.346328\pi$$
$$30$$ 0 0
$$31$$ −8.00000 −1.43684 −0.718421 0.695608i $$-0.755135\pi$$
−0.718421 + 0.695608i $$0.755135\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ 0 0
$$34$$ 3.00000 0.514496
$$35$$ 12.0000 2.02837
$$36$$ 0 0
$$37$$ −2.00000 −0.328798 −0.164399 0.986394i $$-0.552568\pi$$
−0.164399 + 0.986394i $$0.552568\pi$$
$$38$$ 1.00000 0.162221
$$39$$ 0 0
$$40$$ −4.00000 −0.632456
$$41$$ 8.00000 1.24939 0.624695 0.780869i $$-0.285223\pi$$
0.624695 + 0.780869i $$0.285223\pi$$
$$42$$ 0 0
$$43$$ 4.00000 0.609994 0.304997 0.952353i $$-0.401344\pi$$
0.304997 + 0.952353i $$0.401344\pi$$
$$44$$ −2.00000 −0.301511
$$45$$ 0 0
$$46$$ −1.00000 −0.147442
$$47$$ −8.00000 −1.16692 −0.583460 0.812142i $$-0.698301\pi$$
−0.583460 + 0.812142i $$0.698301\pi$$
$$48$$ 0 0
$$49$$ 2.00000 0.285714
$$50$$ −11.0000 −1.55563
$$51$$ 0 0
$$52$$ −1.00000 −0.138675
$$53$$ 1.00000 0.137361 0.0686803 0.997639i $$-0.478121\pi$$
0.0686803 + 0.997639i $$0.478121\pi$$
$$54$$ 0 0
$$55$$ −8.00000 −1.07872
$$56$$ −3.00000 −0.400892
$$57$$ 0 0
$$58$$ −5.00000 −0.656532
$$59$$ −15.0000 −1.95283 −0.976417 0.215894i $$-0.930733\pi$$
−0.976417 + 0.215894i $$0.930733\pi$$
$$60$$ 0 0
$$61$$ 2.00000 0.256074 0.128037 0.991769i $$-0.459132\pi$$
0.128037 + 0.991769i $$0.459132\pi$$
$$62$$ 8.00000 1.01600
$$63$$ 0 0
$$64$$ 1.00000 0.125000
$$65$$ −4.00000 −0.496139
$$66$$ 0 0
$$67$$ 3.00000 0.366508 0.183254 0.983066i $$-0.441337\pi$$
0.183254 + 0.983066i $$0.441337\pi$$
$$68$$ −3.00000 −0.363803
$$69$$ 0 0
$$70$$ −12.0000 −1.43427
$$71$$ −2.00000 −0.237356 −0.118678 0.992933i $$-0.537866\pi$$
−0.118678 + 0.992933i $$0.537866\pi$$
$$72$$ 0 0
$$73$$ 9.00000 1.05337 0.526685 0.850060i $$-0.323435\pi$$
0.526685 + 0.850060i $$0.323435\pi$$
$$74$$ 2.00000 0.232495
$$75$$ 0 0
$$76$$ −1.00000 −0.114708
$$77$$ −6.00000 −0.683763
$$78$$ 0 0
$$79$$ −10.0000 −1.12509 −0.562544 0.826767i $$-0.690177\pi$$
−0.562544 + 0.826767i $$0.690177\pi$$
$$80$$ 4.00000 0.447214
$$81$$ 0 0
$$82$$ −8.00000 −0.883452
$$83$$ 6.00000 0.658586 0.329293 0.944228i $$-0.393190\pi$$
0.329293 + 0.944228i $$0.393190\pi$$
$$84$$ 0 0
$$85$$ −12.0000 −1.30158
$$86$$ −4.00000 −0.431331
$$87$$ 0 0
$$88$$ 2.00000 0.213201
$$89$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$90$$ 0 0
$$91$$ −3.00000 −0.314485
$$92$$ 1.00000 0.104257
$$93$$ 0 0
$$94$$ 8.00000 0.825137
$$95$$ −4.00000 −0.410391
$$96$$ 0 0
$$97$$ −2.00000 −0.203069 −0.101535 0.994832i $$-0.532375\pi$$
−0.101535 + 0.994832i $$0.532375\pi$$
$$98$$ −2.00000 −0.202031
$$99$$ 0 0
$$100$$ 11.0000 1.10000
$$101$$ −2.00000 −0.199007 −0.0995037 0.995037i $$-0.531726\pi$$
−0.0995037 + 0.995037i $$0.531726\pi$$
$$102$$ 0 0
$$103$$ −6.00000 −0.591198 −0.295599 0.955312i $$-0.595519\pi$$
−0.295599 + 0.955312i $$0.595519\pi$$
$$104$$ 1.00000 0.0980581
$$105$$ 0 0
$$106$$ −1.00000 −0.0971286
$$107$$ 7.00000 0.676716 0.338358 0.941018i $$-0.390129\pi$$
0.338358 + 0.941018i $$0.390129\pi$$
$$108$$ 0 0
$$109$$ −15.0000 −1.43674 −0.718370 0.695662i $$-0.755111\pi$$
−0.718370 + 0.695662i $$0.755111\pi$$
$$110$$ 8.00000 0.762770
$$111$$ 0 0
$$112$$ 3.00000 0.283473
$$113$$ −14.0000 −1.31701 −0.658505 0.752577i $$-0.728811\pi$$
−0.658505 + 0.752577i $$0.728811\pi$$
$$114$$ 0 0
$$115$$ 4.00000 0.373002
$$116$$ 5.00000 0.464238
$$117$$ 0 0
$$118$$ 15.0000 1.38086
$$119$$ −9.00000 −0.825029
$$120$$ 0 0
$$121$$ −7.00000 −0.636364
$$122$$ −2.00000 −0.181071
$$123$$ 0 0
$$124$$ −8.00000 −0.718421
$$125$$ 24.0000 2.14663
$$126$$ 0 0
$$127$$ 18.0000 1.59724 0.798621 0.601834i $$-0.205563\pi$$
0.798621 + 0.601834i $$0.205563\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ 0 0
$$130$$ 4.00000 0.350823
$$131$$ −12.0000 −1.04844 −0.524222 0.851581i $$-0.675644\pi$$
−0.524222 + 0.851581i $$0.675644\pi$$
$$132$$ 0 0
$$133$$ −3.00000 −0.260133
$$134$$ −3.00000 −0.259161
$$135$$ 0 0
$$136$$ 3.00000 0.257248
$$137$$ 17.0000 1.45241 0.726204 0.687479i $$-0.241283\pi$$
0.726204 + 0.687479i $$0.241283\pi$$
$$138$$ 0 0
$$139$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$140$$ 12.0000 1.01419
$$141$$ 0 0
$$142$$ 2.00000 0.167836
$$143$$ 2.00000 0.167248
$$144$$ 0 0
$$145$$ 20.0000 1.66091
$$146$$ −9.00000 −0.744845
$$147$$ 0 0
$$148$$ −2.00000 −0.164399
$$149$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$150$$ 0 0
$$151$$ 2.00000 0.162758 0.0813788 0.996683i $$-0.474068\pi$$
0.0813788 + 0.996683i $$0.474068\pi$$
$$152$$ 1.00000 0.0811107
$$153$$ 0 0
$$154$$ 6.00000 0.483494
$$155$$ −32.0000 −2.57030
$$156$$ 0 0
$$157$$ −2.00000 −0.159617 −0.0798087 0.996810i $$-0.525431\pi$$
−0.0798087 + 0.996810i $$0.525431\pi$$
$$158$$ 10.0000 0.795557
$$159$$ 0 0
$$160$$ −4.00000 −0.316228
$$161$$ 3.00000 0.236433
$$162$$ 0 0
$$163$$ −16.0000 −1.25322 −0.626608 0.779334i $$-0.715557\pi$$
−0.626608 + 0.779334i $$0.715557\pi$$
$$164$$ 8.00000 0.624695
$$165$$ 0 0
$$166$$ −6.00000 −0.465690
$$167$$ 12.0000 0.928588 0.464294 0.885681i $$-0.346308\pi$$
0.464294 + 0.885681i $$0.346308\pi$$
$$168$$ 0 0
$$169$$ −12.0000 −0.923077
$$170$$ 12.0000 0.920358
$$171$$ 0 0
$$172$$ 4.00000 0.304997
$$173$$ 6.00000 0.456172 0.228086 0.973641i $$-0.426753\pi$$
0.228086 + 0.973641i $$0.426753\pi$$
$$174$$ 0 0
$$175$$ 33.0000 2.49457
$$176$$ −2.00000 −0.150756
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$180$$ 0 0
$$181$$ 22.0000 1.63525 0.817624 0.575753i $$-0.195291\pi$$
0.817624 + 0.575753i $$0.195291\pi$$
$$182$$ 3.00000 0.222375
$$183$$ 0 0
$$184$$ −1.00000 −0.0737210
$$185$$ −8.00000 −0.588172
$$186$$ 0 0
$$187$$ 6.00000 0.438763
$$188$$ −8.00000 −0.583460
$$189$$ 0 0
$$190$$ 4.00000 0.290191
$$191$$ −7.00000 −0.506502 −0.253251 0.967401i $$-0.581500\pi$$
−0.253251 + 0.967401i $$0.581500\pi$$
$$192$$ 0 0
$$193$$ −6.00000 −0.431889 −0.215945 0.976406i $$-0.569283\pi$$
−0.215945 + 0.976406i $$0.569283\pi$$
$$194$$ 2.00000 0.143592
$$195$$ 0 0
$$196$$ 2.00000 0.142857
$$197$$ −8.00000 −0.569976 −0.284988 0.958531i $$-0.591990\pi$$
−0.284988 + 0.958531i $$0.591990\pi$$
$$198$$ 0 0
$$199$$ −25.0000 −1.77220 −0.886102 0.463491i $$-0.846597\pi$$
−0.886102 + 0.463491i $$0.846597\pi$$
$$200$$ −11.0000 −0.777817
$$201$$ 0 0
$$202$$ 2.00000 0.140720
$$203$$ 15.0000 1.05279
$$204$$ 0 0
$$205$$ 32.0000 2.23498
$$206$$ 6.00000 0.418040
$$207$$ 0 0
$$208$$ −1.00000 −0.0693375
$$209$$ 2.00000 0.138343
$$210$$ 0 0
$$211$$ 27.0000 1.85876 0.929378 0.369129i $$-0.120344\pi$$
0.929378 + 0.369129i $$0.120344\pi$$
$$212$$ 1.00000 0.0686803
$$213$$ 0 0
$$214$$ −7.00000 −0.478510
$$215$$ 16.0000 1.09119
$$216$$ 0 0
$$217$$ −24.0000 −1.62923
$$218$$ 15.0000 1.01593
$$219$$ 0 0
$$220$$ −8.00000 −0.539360
$$221$$ 3.00000 0.201802
$$222$$ 0 0
$$223$$ 14.0000 0.937509 0.468755 0.883328i $$-0.344703\pi$$
0.468755 + 0.883328i $$0.344703\pi$$
$$224$$ −3.00000 −0.200446
$$225$$ 0 0
$$226$$ 14.0000 0.931266
$$227$$ 17.0000 1.12833 0.564165 0.825662i $$-0.309198\pi$$
0.564165 + 0.825662i $$0.309198\pi$$
$$228$$ 0 0
$$229$$ −10.0000 −0.660819 −0.330409 0.943838i $$-0.607187\pi$$
−0.330409 + 0.943838i $$0.607187\pi$$
$$230$$ −4.00000 −0.263752
$$231$$ 0 0
$$232$$ −5.00000 −0.328266
$$233$$ 6.00000 0.393073 0.196537 0.980497i $$-0.437031\pi$$
0.196537 + 0.980497i $$0.437031\pi$$
$$234$$ 0 0
$$235$$ −32.0000 −2.08745
$$236$$ −15.0000 −0.976417
$$237$$ 0 0
$$238$$ 9.00000 0.583383
$$239$$ −15.0000 −0.970269 −0.485135 0.874439i $$-0.661229\pi$$
−0.485135 + 0.874439i $$0.661229\pi$$
$$240$$ 0 0
$$241$$ −8.00000 −0.515325 −0.257663 0.966235i $$-0.582952\pi$$
−0.257663 + 0.966235i $$0.582952\pi$$
$$242$$ 7.00000 0.449977
$$243$$ 0 0
$$244$$ 2.00000 0.128037
$$245$$ 8.00000 0.511101
$$246$$ 0 0
$$247$$ 1.00000 0.0636285
$$248$$ 8.00000 0.508001
$$249$$ 0 0
$$250$$ −24.0000 −1.51789
$$251$$ −2.00000 −0.126239 −0.0631194 0.998006i $$-0.520105\pi$$
−0.0631194 + 0.998006i $$0.520105\pi$$
$$252$$ 0 0
$$253$$ −2.00000 −0.125739
$$254$$ −18.0000 −1.12942
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ −8.00000 −0.499026 −0.249513 0.968371i $$-0.580271\pi$$
−0.249513 + 0.968371i $$0.580271\pi$$
$$258$$ 0 0
$$259$$ −6.00000 −0.372822
$$260$$ −4.00000 −0.248069
$$261$$ 0 0
$$262$$ 12.0000 0.741362
$$263$$ −24.0000 −1.47990 −0.739952 0.672660i $$-0.765152\pi$$
−0.739952 + 0.672660i $$0.765152\pi$$
$$264$$ 0 0
$$265$$ 4.00000 0.245718
$$266$$ 3.00000 0.183942
$$267$$ 0 0
$$268$$ 3.00000 0.183254
$$269$$ −30.0000 −1.82913 −0.914566 0.404436i $$-0.867468\pi$$
−0.914566 + 0.404436i $$0.867468\pi$$
$$270$$ 0 0
$$271$$ 7.00000 0.425220 0.212610 0.977137i $$-0.431804\pi$$
0.212610 + 0.977137i $$0.431804\pi$$
$$272$$ −3.00000 −0.181902
$$273$$ 0 0
$$274$$ −17.0000 −1.02701
$$275$$ −22.0000 −1.32665
$$276$$ 0 0
$$277$$ 28.0000 1.68236 0.841178 0.540758i $$-0.181862\pi$$
0.841178 + 0.540758i $$0.181862\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ −12.0000 −0.717137
$$281$$ 8.00000 0.477240 0.238620 0.971113i $$-0.423305\pi$$
0.238620 + 0.971113i $$0.423305\pi$$
$$282$$ 0 0
$$283$$ −6.00000 −0.356663 −0.178331 0.983970i $$-0.557070\pi$$
−0.178331 + 0.983970i $$0.557070\pi$$
$$284$$ −2.00000 −0.118678
$$285$$ 0 0
$$286$$ −2.00000 −0.118262
$$287$$ 24.0000 1.41668
$$288$$ 0 0
$$289$$ −8.00000 −0.470588
$$290$$ −20.0000 −1.17444
$$291$$ 0 0
$$292$$ 9.00000 0.526685
$$293$$ −9.00000 −0.525786 −0.262893 0.964825i $$-0.584677\pi$$
−0.262893 + 0.964825i $$0.584677\pi$$
$$294$$ 0 0
$$295$$ −60.0000 −3.49334
$$296$$ 2.00000 0.116248
$$297$$ 0 0
$$298$$ 0 0
$$299$$ −1.00000 −0.0578315
$$300$$ 0 0
$$301$$ 12.0000 0.691669
$$302$$ −2.00000 −0.115087
$$303$$ 0 0
$$304$$ −1.00000 −0.0573539
$$305$$ 8.00000 0.458079
$$306$$ 0 0
$$307$$ −12.0000 −0.684876 −0.342438 0.939540i $$-0.611253\pi$$
−0.342438 + 0.939540i $$0.611253\pi$$
$$308$$ −6.00000 −0.341882
$$309$$ 0 0
$$310$$ 32.0000 1.81748
$$311$$ −7.00000 −0.396934 −0.198467 0.980108i $$-0.563596\pi$$
−0.198467 + 0.980108i $$0.563596\pi$$
$$312$$ 0 0
$$313$$ 29.0000 1.63918 0.819588 0.572953i $$-0.194202\pi$$
0.819588 + 0.572953i $$0.194202\pi$$
$$314$$ 2.00000 0.112867
$$315$$ 0 0
$$316$$ −10.0000 −0.562544
$$317$$ 27.0000 1.51647 0.758236 0.651981i $$-0.226062\pi$$
0.758236 + 0.651981i $$0.226062\pi$$
$$318$$ 0 0
$$319$$ −10.0000 −0.559893
$$320$$ 4.00000 0.223607
$$321$$ 0 0
$$322$$ −3.00000 −0.167183
$$323$$ 3.00000 0.166924
$$324$$ 0 0
$$325$$ −11.0000 −0.610170
$$326$$ 16.0000 0.886158
$$327$$ 0 0
$$328$$ −8.00000 −0.441726
$$329$$ −24.0000 −1.32316
$$330$$ 0 0
$$331$$ 17.0000 0.934405 0.467202 0.884150i $$-0.345262\pi$$
0.467202 + 0.884150i $$0.345262\pi$$
$$332$$ 6.00000 0.329293
$$333$$ 0 0
$$334$$ −12.0000 −0.656611
$$335$$ 12.0000 0.655630
$$336$$ 0 0
$$337$$ −32.0000 −1.74315 −0.871576 0.490261i $$-0.836901\pi$$
−0.871576 + 0.490261i $$0.836901\pi$$
$$338$$ 12.0000 0.652714
$$339$$ 0 0
$$340$$ −12.0000 −0.650791
$$341$$ 16.0000 0.866449
$$342$$ 0 0
$$343$$ −15.0000 −0.809924
$$344$$ −4.00000 −0.215666
$$345$$ 0 0
$$346$$ −6.00000 −0.322562
$$347$$ 2.00000 0.107366 0.0536828 0.998558i $$-0.482904\pi$$
0.0536828 + 0.998558i $$0.482904\pi$$
$$348$$ 0 0
$$349$$ 10.0000 0.535288 0.267644 0.963518i $$-0.413755\pi$$
0.267644 + 0.963518i $$0.413755\pi$$
$$350$$ −33.0000 −1.76392
$$351$$ 0 0
$$352$$ 2.00000 0.106600
$$353$$ −9.00000 −0.479022 −0.239511 0.970894i $$-0.576987\pi$$
−0.239511 + 0.970894i $$0.576987\pi$$
$$354$$ 0 0
$$355$$ −8.00000 −0.424596
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 15.0000 0.791670 0.395835 0.918322i $$-0.370455\pi$$
0.395835 + 0.918322i $$0.370455\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ −22.0000 −1.15629
$$363$$ 0 0
$$364$$ −3.00000 −0.157243
$$365$$ 36.0000 1.88433
$$366$$ 0 0
$$367$$ 28.0000 1.46159 0.730794 0.682598i $$-0.239150\pi$$
0.730794 + 0.682598i $$0.239150\pi$$
$$368$$ 1.00000 0.0521286
$$369$$ 0 0
$$370$$ 8.00000 0.415900
$$371$$ 3.00000 0.155752
$$372$$ 0 0
$$373$$ 29.0000 1.50156 0.750782 0.660551i $$-0.229677\pi$$
0.750782 + 0.660551i $$0.229677\pi$$
$$374$$ −6.00000 −0.310253
$$375$$ 0 0
$$376$$ 8.00000 0.412568
$$377$$ −5.00000 −0.257513
$$378$$ 0 0
$$379$$ 15.0000 0.770498 0.385249 0.922813i $$-0.374116\pi$$
0.385249 + 0.922813i $$0.374116\pi$$
$$380$$ −4.00000 −0.205196
$$381$$ 0 0
$$382$$ 7.00000 0.358151
$$383$$ 26.0000 1.32854 0.664269 0.747494i $$-0.268743\pi$$
0.664269 + 0.747494i $$0.268743\pi$$
$$384$$ 0 0
$$385$$ −24.0000 −1.22315
$$386$$ 6.00000 0.305392
$$387$$ 0 0
$$388$$ −2.00000 −0.101535
$$389$$ 30.0000 1.52106 0.760530 0.649303i $$-0.224939\pi$$
0.760530 + 0.649303i $$0.224939\pi$$
$$390$$ 0 0
$$391$$ −3.00000 −0.151717
$$392$$ −2.00000 −0.101015
$$393$$ 0 0
$$394$$ 8.00000 0.403034
$$395$$ −40.0000 −2.01262
$$396$$ 0 0
$$397$$ 8.00000 0.401508 0.200754 0.979642i $$-0.435661\pi$$
0.200754 + 0.979642i $$0.435661\pi$$
$$398$$ 25.0000 1.25314
$$399$$ 0 0
$$400$$ 11.0000 0.550000
$$401$$ 8.00000 0.399501 0.199750 0.979847i $$-0.435987\pi$$
0.199750 + 0.979847i $$0.435987\pi$$
$$402$$ 0 0
$$403$$ 8.00000 0.398508
$$404$$ −2.00000 −0.0995037
$$405$$ 0 0
$$406$$ −15.0000 −0.744438
$$407$$ 4.00000 0.198273
$$408$$ 0 0
$$409$$ −20.0000 −0.988936 −0.494468 0.869196i $$-0.664637\pi$$
−0.494468 + 0.869196i $$0.664637\pi$$
$$410$$ −32.0000 −1.58037
$$411$$ 0 0
$$412$$ −6.00000 −0.295599
$$413$$ −45.0000 −2.21431
$$414$$ 0 0
$$415$$ 24.0000 1.17811
$$416$$ 1.00000 0.0490290
$$417$$ 0 0
$$418$$ −2.00000 −0.0978232
$$419$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$420$$ 0 0
$$421$$ −13.0000 −0.633581 −0.316791 0.948495i $$-0.602605\pi$$
−0.316791 + 0.948495i $$0.602605\pi$$
$$422$$ −27.0000 −1.31434
$$423$$ 0 0
$$424$$ −1.00000 −0.0485643
$$425$$ −33.0000 −1.60074
$$426$$ 0 0
$$427$$ 6.00000 0.290360
$$428$$ 7.00000 0.338358
$$429$$ 0 0
$$430$$ −16.0000 −0.771589
$$431$$ 18.0000 0.867029 0.433515 0.901146i $$-0.357273\pi$$
0.433515 + 0.901146i $$0.357273\pi$$
$$432$$ 0 0
$$433$$ 14.0000 0.672797 0.336399 0.941720i $$-0.390791\pi$$
0.336399 + 0.941720i $$0.390791\pi$$
$$434$$ 24.0000 1.15204
$$435$$ 0 0
$$436$$ −15.0000 −0.718370
$$437$$ −1.00000 −0.0478365
$$438$$ 0 0
$$439$$ 20.0000 0.954548 0.477274 0.878755i $$-0.341625\pi$$
0.477274 + 0.878755i $$0.341625\pi$$
$$440$$ 8.00000 0.381385
$$441$$ 0 0
$$442$$ −3.00000 −0.142695
$$443$$ 26.0000 1.23530 0.617649 0.786454i $$-0.288085\pi$$
0.617649 + 0.786454i $$0.288085\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ −14.0000 −0.662919
$$447$$ 0 0
$$448$$ 3.00000 0.141737
$$449$$ −10.0000 −0.471929 −0.235965 0.971762i $$-0.575825\pi$$
−0.235965 + 0.971762i $$0.575825\pi$$
$$450$$ 0 0
$$451$$ −16.0000 −0.753411
$$452$$ −14.0000 −0.658505
$$453$$ 0 0
$$454$$ −17.0000 −0.797850
$$455$$ −12.0000 −0.562569
$$456$$ 0 0
$$457$$ −7.00000 −0.327446 −0.163723 0.986506i $$-0.552350\pi$$
−0.163723 + 0.986506i $$0.552350\pi$$
$$458$$ 10.0000 0.467269
$$459$$ 0 0
$$460$$ 4.00000 0.186501
$$461$$ 28.0000 1.30409 0.652045 0.758180i $$-0.273911\pi$$
0.652045 + 0.758180i $$0.273911\pi$$
$$462$$ 0 0
$$463$$ 4.00000 0.185896 0.0929479 0.995671i $$-0.470371\pi$$
0.0929479 + 0.995671i $$0.470371\pi$$
$$464$$ 5.00000 0.232119
$$465$$ 0 0
$$466$$ −6.00000 −0.277945
$$467$$ 2.00000 0.0925490 0.0462745 0.998929i $$-0.485265\pi$$
0.0462745 + 0.998929i $$0.485265\pi$$
$$468$$ 0 0
$$469$$ 9.00000 0.415581
$$470$$ 32.0000 1.47605
$$471$$ 0 0
$$472$$ 15.0000 0.690431
$$473$$ −8.00000 −0.367840
$$474$$ 0 0
$$475$$ −11.0000 −0.504715
$$476$$ −9.00000 −0.412514
$$477$$ 0 0
$$478$$ 15.0000 0.686084
$$479$$ 20.0000 0.913823 0.456912 0.889512i $$-0.348956\pi$$
0.456912 + 0.889512i $$0.348956\pi$$
$$480$$ 0 0
$$481$$ 2.00000 0.0911922
$$482$$ 8.00000 0.364390
$$483$$ 0 0
$$484$$ −7.00000 −0.318182
$$485$$ −8.00000 −0.363261
$$486$$ 0 0
$$487$$ −2.00000 −0.0906287 −0.0453143 0.998973i $$-0.514429\pi$$
−0.0453143 + 0.998973i $$0.514429\pi$$
$$488$$ −2.00000 −0.0905357
$$489$$ 0 0
$$490$$ −8.00000 −0.361403
$$491$$ 28.0000 1.26362 0.631811 0.775122i $$-0.282312\pi$$
0.631811 + 0.775122i $$0.282312\pi$$
$$492$$ 0 0
$$493$$ −15.0000 −0.675566
$$494$$ −1.00000 −0.0449921
$$495$$ 0 0
$$496$$ −8.00000 −0.359211
$$497$$ −6.00000 −0.269137
$$498$$ 0 0
$$499$$ 40.0000 1.79065 0.895323 0.445418i $$-0.146945\pi$$
0.895323 + 0.445418i $$0.146945\pi$$
$$500$$ 24.0000 1.07331
$$501$$ 0 0
$$502$$ 2.00000 0.0892644
$$503$$ −39.0000 −1.73892 −0.869462 0.494000i $$-0.835534\pi$$
−0.869462 + 0.494000i $$0.835534\pi$$
$$504$$ 0 0
$$505$$ −8.00000 −0.355995
$$506$$ 2.00000 0.0889108
$$507$$ 0 0
$$508$$ 18.0000 0.798621
$$509$$ 30.0000 1.32973 0.664863 0.746965i $$-0.268490\pi$$
0.664863 + 0.746965i $$0.268490\pi$$
$$510$$ 0 0
$$511$$ 27.0000 1.19441
$$512$$ −1.00000 −0.0441942
$$513$$ 0 0
$$514$$ 8.00000 0.352865
$$515$$ −24.0000 −1.05757
$$516$$ 0 0
$$517$$ 16.0000 0.703679
$$518$$ 6.00000 0.263625
$$519$$ 0 0
$$520$$ 4.00000 0.175412
$$521$$ 28.0000 1.22670 0.613351 0.789810i $$-0.289821\pi$$
0.613351 + 0.789810i $$0.289821\pi$$
$$522$$ 0 0
$$523$$ 29.0000 1.26808 0.634041 0.773300i $$-0.281395\pi$$
0.634041 + 0.773300i $$0.281395\pi$$
$$524$$ −12.0000 −0.524222
$$525$$ 0 0
$$526$$ 24.0000 1.04645
$$527$$ 24.0000 1.04546
$$528$$ 0 0
$$529$$ −22.0000 −0.956522
$$530$$ −4.00000 −0.173749
$$531$$ 0 0
$$532$$ −3.00000 −0.130066
$$533$$ −8.00000 −0.346518
$$534$$ 0 0
$$535$$ 28.0000 1.21055
$$536$$ −3.00000 −0.129580
$$537$$ 0 0
$$538$$ 30.0000 1.29339
$$539$$ −4.00000 −0.172292
$$540$$ 0 0
$$541$$ 2.00000 0.0859867 0.0429934 0.999075i $$-0.486311\pi$$
0.0429934 + 0.999075i $$0.486311\pi$$
$$542$$ −7.00000 −0.300676
$$543$$ 0 0
$$544$$ 3.00000 0.128624
$$545$$ −60.0000 −2.57012
$$546$$ 0 0
$$547$$ 28.0000 1.19719 0.598597 0.801050i $$-0.295725\pi$$
0.598597 + 0.801050i $$0.295725\pi$$
$$548$$ 17.0000 0.726204
$$549$$ 0 0
$$550$$ 22.0000 0.938083
$$551$$ −5.00000 −0.213007
$$552$$ 0 0
$$553$$ −30.0000 −1.27573
$$554$$ −28.0000 −1.18961
$$555$$ 0 0
$$556$$ 0 0
$$557$$ −28.0000 −1.18640 −0.593199 0.805056i $$-0.702135\pi$$
−0.593199 + 0.805056i $$0.702135\pi$$
$$558$$ 0 0
$$559$$ −4.00000 −0.169182
$$560$$ 12.0000 0.507093
$$561$$ 0 0
$$562$$ −8.00000 −0.337460
$$563$$ 36.0000 1.51722 0.758610 0.651546i $$-0.225879\pi$$
0.758610 + 0.651546i $$0.225879\pi$$
$$564$$ 0 0
$$565$$ −56.0000 −2.35594
$$566$$ 6.00000 0.252199
$$567$$ 0 0
$$568$$ 2.00000 0.0839181
$$569$$ −40.0000 −1.67689 −0.838444 0.544988i $$-0.816534\pi$$
−0.838444 + 0.544988i $$0.816534\pi$$
$$570$$ 0 0
$$571$$ −28.0000 −1.17176 −0.585882 0.810397i $$-0.699252\pi$$
−0.585882 + 0.810397i $$0.699252\pi$$
$$572$$ 2.00000 0.0836242
$$573$$ 0 0
$$574$$ −24.0000 −1.00174
$$575$$ 11.0000 0.458732
$$576$$ 0 0
$$577$$ −37.0000 −1.54033 −0.770165 0.637845i $$-0.779826\pi$$
−0.770165 + 0.637845i $$0.779826\pi$$
$$578$$ 8.00000 0.332756
$$579$$ 0 0
$$580$$ 20.0000 0.830455
$$581$$ 18.0000 0.746766
$$582$$ 0 0
$$583$$ −2.00000 −0.0828315
$$584$$ −9.00000 −0.372423
$$585$$ 0 0
$$586$$ 9.00000 0.371787
$$587$$ 12.0000 0.495293 0.247647 0.968850i $$-0.420343\pi$$
0.247647 + 0.968850i $$0.420343\pi$$
$$588$$ 0 0
$$589$$ 8.00000 0.329634
$$590$$ 60.0000 2.47016
$$591$$ 0 0
$$592$$ −2.00000 −0.0821995
$$593$$ −34.0000 −1.39621 −0.698106 0.715994i $$-0.745974\pi$$
−0.698106 + 0.715994i $$0.745974\pi$$
$$594$$ 0 0
$$595$$ −36.0000 −1.47586
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 1.00000 0.0408930
$$599$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$600$$ 0 0
$$601$$ −8.00000 −0.326327 −0.163163 0.986599i $$-0.552170\pi$$
−0.163163 + 0.986599i $$0.552170\pi$$
$$602$$ −12.0000 −0.489083
$$603$$ 0 0
$$604$$ 2.00000 0.0813788
$$605$$ −28.0000 −1.13836
$$606$$ 0 0
$$607$$ −22.0000 −0.892952 −0.446476 0.894795i $$-0.647321\pi$$
−0.446476 + 0.894795i $$0.647321\pi$$
$$608$$ 1.00000 0.0405554
$$609$$ 0 0
$$610$$ −8.00000 −0.323911
$$611$$ 8.00000 0.323645
$$612$$ 0 0
$$613$$ 34.0000 1.37325 0.686624 0.727013i $$-0.259092\pi$$
0.686624 + 0.727013i $$0.259092\pi$$
$$614$$ 12.0000 0.484281
$$615$$ 0 0
$$616$$ 6.00000 0.241747
$$617$$ −18.0000 −0.724653 −0.362326 0.932051i $$-0.618017\pi$$
−0.362326 + 0.932051i $$0.618017\pi$$
$$618$$ 0 0
$$619$$ 10.0000 0.401934 0.200967 0.979598i $$-0.435592\pi$$
0.200967 + 0.979598i $$0.435592\pi$$
$$620$$ −32.0000 −1.28515
$$621$$ 0 0
$$622$$ 7.00000 0.280674
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 41.0000 1.64000
$$626$$ −29.0000 −1.15907
$$627$$ 0 0
$$628$$ −2.00000 −0.0798087
$$629$$ 6.00000 0.239236
$$630$$ 0 0
$$631$$ 32.0000 1.27390 0.636950 0.770905i $$-0.280196\pi$$
0.636950 + 0.770905i $$0.280196\pi$$
$$632$$ 10.0000 0.397779
$$633$$ 0 0
$$634$$ −27.0000 −1.07231
$$635$$ 72.0000 2.85723
$$636$$ 0 0
$$637$$ −2.00000 −0.0792429
$$638$$ 10.0000 0.395904
$$639$$ 0 0
$$640$$ −4.00000 −0.158114
$$641$$ −42.0000 −1.65890 −0.829450 0.558581i $$-0.811346\pi$$
−0.829450 + 0.558581i $$0.811346\pi$$
$$642$$ 0 0
$$643$$ −26.0000 −1.02534 −0.512670 0.858586i $$-0.671344\pi$$
−0.512670 + 0.858586i $$0.671344\pi$$
$$644$$ 3.00000 0.118217
$$645$$ 0 0
$$646$$ −3.00000 −0.118033
$$647$$ −23.0000 −0.904223 −0.452112 0.891961i $$-0.649329\pi$$
−0.452112 + 0.891961i $$0.649329\pi$$
$$648$$ 0 0
$$649$$ 30.0000 1.17760
$$650$$ 11.0000 0.431455
$$651$$ 0 0
$$652$$ −16.0000 −0.626608
$$653$$ 36.0000 1.40879 0.704394 0.709809i $$-0.251219\pi$$
0.704394 + 0.709809i $$0.251219\pi$$
$$654$$ 0 0
$$655$$ −48.0000 −1.87552
$$656$$ 8.00000 0.312348
$$657$$ 0 0
$$658$$ 24.0000 0.935617
$$659$$ −5.00000 −0.194772 −0.0973862 0.995247i $$-0.531048\pi$$
−0.0973862 + 0.995247i $$0.531048\pi$$
$$660$$ 0 0
$$661$$ −23.0000 −0.894596 −0.447298 0.894385i $$-0.647614\pi$$
−0.447298 + 0.894385i $$0.647614\pi$$
$$662$$ −17.0000 −0.660724
$$663$$ 0 0
$$664$$ −6.00000 −0.232845
$$665$$ −12.0000 −0.465340
$$666$$ 0 0
$$667$$ 5.00000 0.193601
$$668$$ 12.0000 0.464294
$$669$$ 0 0
$$670$$ −12.0000 −0.463600
$$671$$ −4.00000 −0.154418
$$672$$ 0 0
$$673$$ 44.0000 1.69608 0.848038 0.529936i $$-0.177784\pi$$
0.848038 + 0.529936i $$0.177784\pi$$
$$674$$ 32.0000 1.23259
$$675$$ 0 0
$$676$$ −12.0000 −0.461538
$$677$$ −13.0000 −0.499631 −0.249815 0.968294i $$-0.580370\pi$$
−0.249815 + 0.968294i $$0.580370\pi$$
$$678$$ 0 0
$$679$$ −6.00000 −0.230259
$$680$$ 12.0000 0.460179
$$681$$ 0 0
$$682$$ −16.0000 −0.612672
$$683$$ −4.00000 −0.153056 −0.0765279 0.997067i $$-0.524383\pi$$
−0.0765279 + 0.997067i $$0.524383\pi$$
$$684$$ 0 0
$$685$$ 68.0000 2.59815
$$686$$ 15.0000 0.572703
$$687$$ 0 0
$$688$$ 4.00000 0.152499
$$689$$ −1.00000 −0.0380970
$$690$$ 0 0
$$691$$ 42.0000 1.59776 0.798878 0.601494i $$-0.205427\pi$$
0.798878 + 0.601494i $$0.205427\pi$$
$$692$$ 6.00000 0.228086
$$693$$ 0 0
$$694$$ −2.00000 −0.0759190
$$695$$ 0 0
$$696$$ 0 0
$$697$$ −24.0000 −0.909065
$$698$$ −10.0000 −0.378506
$$699$$ 0 0
$$700$$ 33.0000 1.24728
$$701$$ 28.0000 1.05755 0.528773 0.848763i $$-0.322652\pi$$
0.528773 + 0.848763i $$0.322652\pi$$
$$702$$ 0 0
$$703$$ 2.00000 0.0754314
$$704$$ −2.00000 −0.0753778
$$705$$ 0 0
$$706$$ 9.00000 0.338719
$$707$$ −6.00000 −0.225653
$$708$$ 0 0
$$709$$ −30.0000 −1.12667 −0.563337 0.826227i $$-0.690483\pi$$
−0.563337 + 0.826227i $$0.690483\pi$$
$$710$$ 8.00000 0.300235
$$711$$ 0 0
$$712$$ 0 0
$$713$$ −8.00000 −0.299602
$$714$$ 0 0
$$715$$ 8.00000 0.299183
$$716$$ 0 0
$$717$$ 0 0
$$718$$ −15.0000 −0.559795
$$719$$ 5.00000 0.186469 0.0932343 0.995644i $$-0.470279\pi$$
0.0932343 + 0.995644i $$0.470279\pi$$
$$720$$ 0 0
$$721$$ −18.0000 −0.670355
$$722$$ −1.00000 −0.0372161
$$723$$ 0 0
$$724$$ 22.0000 0.817624
$$725$$ 55.0000 2.04265
$$726$$ 0 0
$$727$$ −17.0000 −0.630495 −0.315248 0.949009i $$-0.602088\pi$$
−0.315248 + 0.949009i $$0.602088\pi$$
$$728$$ 3.00000 0.111187
$$729$$ 0 0
$$730$$ −36.0000 −1.33242
$$731$$ −12.0000 −0.443836
$$732$$ 0 0
$$733$$ −36.0000 −1.32969 −0.664845 0.746981i $$-0.731502\pi$$
−0.664845 + 0.746981i $$0.731502\pi$$
$$734$$ −28.0000 −1.03350
$$735$$ 0 0
$$736$$ −1.00000 −0.0368605
$$737$$ −6.00000 −0.221013
$$738$$ 0 0
$$739$$ −40.0000 −1.47142 −0.735712 0.677295i $$-0.763152\pi$$
−0.735712 + 0.677295i $$0.763152\pi$$
$$740$$ −8.00000 −0.294086
$$741$$ 0 0
$$742$$ −3.00000 −0.110133
$$743$$ 16.0000 0.586983 0.293492 0.955962i $$-0.405183\pi$$
0.293492 + 0.955962i $$0.405183\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ −29.0000 −1.06177
$$747$$ 0 0
$$748$$ 6.00000 0.219382
$$749$$ 21.0000 0.767323
$$750$$ 0 0
$$751$$ 32.0000 1.16770 0.583848 0.811863i $$-0.301546\pi$$
0.583848 + 0.811863i $$0.301546\pi$$
$$752$$ −8.00000 −0.291730
$$753$$ 0 0
$$754$$ 5.00000 0.182089
$$755$$ 8.00000 0.291150
$$756$$ 0 0
$$757$$ −2.00000 −0.0726912 −0.0363456 0.999339i $$-0.511572\pi$$
−0.0363456 + 0.999339i $$0.511572\pi$$
$$758$$ −15.0000 −0.544825
$$759$$ 0 0
$$760$$ 4.00000 0.145095
$$761$$ −27.0000 −0.978749 −0.489375 0.872074i $$-0.662775\pi$$
−0.489375 + 0.872074i $$0.662775\pi$$
$$762$$ 0 0
$$763$$ −45.0000 −1.62911
$$764$$ −7.00000 −0.253251
$$765$$ 0 0
$$766$$ −26.0000 −0.939418
$$767$$ 15.0000 0.541619
$$768$$ 0 0
$$769$$ −35.0000 −1.26213 −0.631066 0.775729i $$-0.717382\pi$$
−0.631066 + 0.775729i $$0.717382\pi$$
$$770$$ 24.0000 0.864900
$$771$$ 0 0
$$772$$ −6.00000 −0.215945
$$773$$ −9.00000 −0.323708 −0.161854 0.986815i $$-0.551747\pi$$
−0.161854 + 0.986815i $$0.551747\pi$$
$$774$$ 0 0
$$775$$ −88.0000 −3.16105
$$776$$ 2.00000 0.0717958
$$777$$ 0 0
$$778$$ −30.0000 −1.07555
$$779$$ −8.00000 −0.286630
$$780$$ 0 0
$$781$$ 4.00000 0.143131
$$782$$ 3.00000 0.107280
$$783$$ 0 0
$$784$$ 2.00000 0.0714286
$$785$$ −8.00000 −0.285532
$$786$$ 0 0
$$787$$ −17.0000 −0.605985 −0.302992 0.952993i $$-0.597986\pi$$
−0.302992 + 0.952993i $$0.597986\pi$$
$$788$$ −8.00000 −0.284988
$$789$$ 0 0
$$790$$ 40.0000 1.42314
$$791$$ −42.0000 −1.49335
$$792$$ 0 0
$$793$$ −2.00000 −0.0710221
$$794$$ −8.00000 −0.283909
$$795$$ 0 0
$$796$$ −25.0000 −0.886102
$$797$$ −3.00000 −0.106265 −0.0531327 0.998587i $$-0.516921\pi$$
−0.0531327 + 0.998587i $$0.516921\pi$$
$$798$$ 0 0
$$799$$ 24.0000 0.849059
$$800$$ −11.0000 −0.388909
$$801$$ 0 0
$$802$$ −8.00000 −0.282490
$$803$$ −18.0000 −0.635206
$$804$$ 0 0
$$805$$ 12.0000 0.422944
$$806$$ −8.00000 −0.281788
$$807$$ 0 0
$$808$$ 2.00000 0.0703598
$$809$$ 15.0000 0.527372 0.263686 0.964609i $$-0.415062\pi$$
0.263686 + 0.964609i $$0.415062\pi$$
$$810$$ 0 0
$$811$$ −3.00000 −0.105344 −0.0526721 0.998612i $$-0.516774\pi$$
−0.0526721 + 0.998612i $$0.516774\pi$$
$$812$$ 15.0000 0.526397
$$813$$ 0 0
$$814$$ −4.00000 −0.140200
$$815$$ −64.0000 −2.24182
$$816$$ 0 0
$$817$$ −4.00000 −0.139942
$$818$$ 20.0000 0.699284
$$819$$ 0 0
$$820$$ 32.0000 1.11749
$$821$$ −12.0000 −0.418803 −0.209401 0.977830i $$-0.567152\pi$$
−0.209401 + 0.977830i $$0.567152\pi$$
$$822$$ 0 0
$$823$$ 29.0000 1.01088 0.505438 0.862863i $$-0.331331\pi$$
0.505438 + 0.862863i $$0.331331\pi$$
$$824$$ 6.00000 0.209020
$$825$$ 0 0
$$826$$ 45.0000 1.56575
$$827$$ −23.0000 −0.799788 −0.399894 0.916561i $$-0.630953\pi$$
−0.399894 + 0.916561i $$0.630953\pi$$
$$828$$ 0 0
$$829$$ −15.0000 −0.520972 −0.260486 0.965478i $$-0.583883\pi$$
−0.260486 + 0.965478i $$0.583883\pi$$
$$830$$ −24.0000 −0.833052
$$831$$ 0 0
$$832$$ −1.00000 −0.0346688
$$833$$ −6.00000 −0.207888
$$834$$ 0 0
$$835$$ 48.0000 1.66111
$$836$$ 2.00000 0.0691714
$$837$$ 0 0
$$838$$ 0 0
$$839$$ −20.0000 −0.690477 −0.345238 0.938515i $$-0.612202\pi$$
−0.345238 + 0.938515i $$0.612202\pi$$
$$840$$ 0 0
$$841$$ −4.00000 −0.137931
$$842$$ 13.0000 0.448010
$$843$$ 0 0
$$844$$ 27.0000 0.929378
$$845$$ −48.0000 −1.65125
$$846$$ 0 0
$$847$$ −21.0000 −0.721569
$$848$$ 1.00000 0.0343401
$$849$$ 0 0
$$850$$ 33.0000 1.13189
$$851$$ −2.00000 −0.0685591
$$852$$ 0 0
$$853$$ −6.00000 −0.205436 −0.102718 0.994711i $$-0.532754\pi$$
−0.102718 + 0.994711i $$0.532754\pi$$
$$854$$ −6.00000 −0.205316
$$855$$ 0 0
$$856$$ −7.00000 −0.239255
$$857$$ 12.0000 0.409912 0.204956 0.978771i $$-0.434295\pi$$
0.204956 + 0.978771i $$0.434295\pi$$
$$858$$ 0 0
$$859$$ −50.0000 −1.70598 −0.852989 0.521929i $$-0.825213\pi$$
−0.852989 + 0.521929i $$0.825213\pi$$
$$860$$ 16.0000 0.545595
$$861$$ 0 0
$$862$$ −18.0000 −0.613082
$$863$$ −54.0000 −1.83818 −0.919091 0.394046i $$-0.871075\pi$$
−0.919091 + 0.394046i $$0.871075\pi$$
$$864$$ 0 0
$$865$$ 24.0000 0.816024
$$866$$ −14.0000 −0.475739
$$867$$ 0 0
$$868$$ −24.0000 −0.814613
$$869$$ 20.0000 0.678454
$$870$$ 0 0
$$871$$ −3.00000 −0.101651
$$872$$ 15.0000 0.507964
$$873$$ 0 0
$$874$$ 1.00000 0.0338255
$$875$$ 72.0000 2.43404
$$876$$ 0 0
$$877$$ 13.0000 0.438979 0.219489 0.975615i $$-0.429561\pi$$
0.219489 + 0.975615i $$0.429561\pi$$
$$878$$ −20.0000 −0.674967
$$879$$ 0 0
$$880$$ −8.00000 −0.269680
$$881$$ 18.0000 0.606435 0.303218 0.952921i $$-0.401939\pi$$
0.303218 + 0.952921i $$0.401939\pi$$
$$882$$ 0 0
$$883$$ 34.0000 1.14419 0.572096 0.820187i $$-0.306131\pi$$
0.572096 + 0.820187i $$0.306131\pi$$
$$884$$ 3.00000 0.100901
$$885$$ 0 0
$$886$$ −26.0000 −0.873487
$$887$$ 2.00000 0.0671534 0.0335767 0.999436i $$-0.489310\pi$$
0.0335767 + 0.999436i $$0.489310\pi$$
$$888$$ 0 0
$$889$$ 54.0000 1.81110
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 14.0000 0.468755
$$893$$ 8.00000 0.267710
$$894$$ 0 0
$$895$$ 0 0
$$896$$ −3.00000 −0.100223
$$897$$ 0 0
$$898$$ 10.0000 0.333704
$$899$$ −40.0000 −1.33407
$$900$$ 0 0
$$901$$ −3.00000 −0.0999445
$$902$$ 16.0000 0.532742
$$903$$ 0 0
$$904$$ 14.0000 0.465633
$$905$$ 88.0000 2.92522
$$906$$ 0 0
$$907$$ 53.0000 1.75984 0.879918 0.475125i $$-0.157597\pi$$
0.879918 + 0.475125i $$0.157597\pi$$
$$908$$ 17.0000 0.564165
$$909$$ 0 0
$$910$$ 12.0000 0.397796
$$911$$ −12.0000 −0.397578 −0.198789 0.980042i $$-0.563701\pi$$
−0.198789 + 0.980042i $$0.563701\pi$$
$$912$$ 0 0
$$913$$ −12.0000 −0.397142
$$914$$ 7.00000 0.231539
$$915$$ 0 0
$$916$$ −10.0000 −0.330409
$$917$$ −36.0000 −1.18882
$$918$$ 0 0
$$919$$ 5.00000 0.164935 0.0824674 0.996594i $$-0.473720\pi$$
0.0824674 + 0.996594i $$0.473720\pi$$
$$920$$ −4.00000 −0.131876
$$921$$ 0 0
$$922$$ −28.0000 −0.922131
$$923$$ 2.00000 0.0658308
$$924$$ 0 0
$$925$$ −22.0000 −0.723356
$$926$$ −4.00000 −0.131448
$$927$$ 0 0
$$928$$ −5.00000 −0.164133
$$929$$ 55.0000 1.80449 0.902246 0.431222i $$-0.141918\pi$$
0.902246 + 0.431222i $$0.141918\pi$$
$$930$$ 0 0
$$931$$ −2.00000 −0.0655474
$$932$$ 6.00000 0.196537
$$933$$ 0 0
$$934$$ −2.00000 −0.0654420
$$935$$ 24.0000 0.784884
$$936$$ 0 0
$$937$$ −7.00000 −0.228680 −0.114340 0.993442i $$-0.536475\pi$$
−0.114340 + 0.993442i $$0.536475\pi$$
$$938$$ −9.00000 −0.293860
$$939$$ 0 0
$$940$$ −32.0000 −1.04372
$$941$$ −7.00000 −0.228193 −0.114097 0.993470i $$-0.536397\pi$$
−0.114097 + 0.993470i $$0.536397\pi$$
$$942$$ 0 0
$$943$$ 8.00000 0.260516
$$944$$ −15.0000 −0.488208
$$945$$ 0 0
$$946$$ 8.00000 0.260102
$$947$$ 12.0000 0.389948 0.194974 0.980808i $$-0.437538\pi$$
0.194974 + 0.980808i $$0.437538\pi$$
$$948$$ 0 0
$$949$$ −9.00000 −0.292152
$$950$$ 11.0000 0.356887
$$951$$ 0 0
$$952$$ 9.00000 0.291692
$$953$$ 46.0000 1.49009 0.745043 0.667016i $$-0.232429\pi$$
0.745043 + 0.667016i $$0.232429\pi$$
$$954$$ 0 0
$$955$$ −28.0000 −0.906059
$$956$$ −15.0000 −0.485135
$$957$$ 0 0
$$958$$ −20.0000 −0.646171
$$959$$ 51.0000 1.64688
$$960$$ 0 0
$$961$$ 33.0000 1.06452
$$962$$ −2.00000 −0.0644826
$$963$$ 0 0
$$964$$ −8.00000 −0.257663
$$965$$ −24.0000 −0.772587
$$966$$ 0 0
$$967$$ 48.0000 1.54358 0.771788 0.635880i $$-0.219363\pi$$
0.771788 + 0.635880i $$0.219363\pi$$
$$968$$ 7.00000 0.224989
$$969$$ 0 0
$$970$$ 8.00000 0.256865
$$971$$ 28.0000 0.898563 0.449281 0.893390i $$-0.351680\pi$$
0.449281 + 0.893390i $$0.351680\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 2.00000 0.0640841
$$975$$ 0 0
$$976$$ 2.00000 0.0640184
$$977$$ −8.00000 −0.255943 −0.127971 0.991778i $$-0.540847\pi$$
−0.127971 + 0.991778i $$0.540847\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ 8.00000 0.255551
$$981$$ 0 0
$$982$$ −28.0000 −0.893516
$$983$$ 6.00000 0.191370 0.0956851 0.995412i $$-0.469496\pi$$
0.0956851 + 0.995412i $$0.469496\pi$$
$$984$$ 0 0
$$985$$ −32.0000 −1.01960
$$986$$ 15.0000 0.477697
$$987$$ 0 0
$$988$$ 1.00000 0.0318142
$$989$$ 4.00000 0.127193
$$990$$ 0 0
$$991$$ −8.00000 −0.254128 −0.127064 0.991894i $$-0.540555\pi$$
−0.127064 + 0.991894i $$0.540555\pi$$
$$992$$ 8.00000 0.254000
$$993$$ 0 0
$$994$$ 6.00000 0.190308
$$995$$ −100.000 −3.17021
$$996$$ 0 0
$$997$$ 28.0000 0.886769 0.443384 0.896332i $$-0.353778\pi$$
0.443384 + 0.896332i $$0.353778\pi$$
$$998$$ −40.0000 −1.26618
$$999$$ 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 342.2.a.d.1.1 1
3.2 odd 2 38.2.a.b.1.1 1
4.3 odd 2 2736.2.a.w.1.1 1
5.4 even 2 8550.2.a.u.1.1 1
12.11 even 2 304.2.a.d.1.1 1
15.2 even 4 950.2.b.c.799.2 2
15.8 even 4 950.2.b.c.799.1 2
15.14 odd 2 950.2.a.b.1.1 1
19.18 odd 2 6498.2.a.y.1.1 1
21.20 even 2 1862.2.a.f.1.1 1
24.5 odd 2 1216.2.a.n.1.1 1
24.11 even 2 1216.2.a.g.1.1 1
33.32 even 2 4598.2.a.a.1.1 1
39.38 odd 2 6422.2.a.b.1.1 1
57.2 even 18 722.2.e.d.99.1 6
57.5 odd 18 722.2.e.c.595.1 6
57.8 even 6 722.2.c.f.653.1 2
57.11 odd 6 722.2.c.d.653.1 2
57.14 even 18 722.2.e.d.595.1 6
57.17 odd 18 722.2.e.c.99.1 6
57.23 odd 18 722.2.e.c.415.1 6
57.26 odd 6 722.2.c.d.429.1 2
57.29 even 18 722.2.e.d.423.1 6
57.32 even 18 722.2.e.d.245.1 6
57.35 odd 18 722.2.e.c.389.1 6
57.41 even 18 722.2.e.d.389.1 6
57.44 odd 18 722.2.e.c.245.1 6
57.47 odd 18 722.2.e.c.423.1 6
57.50 even 6 722.2.c.f.429.1 2
57.53 even 18 722.2.e.d.415.1 6
57.56 even 2 722.2.a.b.1.1 1
60.59 even 2 7600.2.a.h.1.1 1
228.227 odd 2 5776.2.a.d.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
38.2.a.b.1.1 1 3.2 odd 2
304.2.a.d.1.1 1 12.11 even 2
342.2.a.d.1.1 1 1.1 even 1 trivial
722.2.a.b.1.1 1 57.56 even 2
722.2.c.d.429.1 2 57.26 odd 6
722.2.c.d.653.1 2 57.11 odd 6
722.2.c.f.429.1 2 57.50 even 6
722.2.c.f.653.1 2 57.8 even 6
722.2.e.c.99.1 6 57.17 odd 18
722.2.e.c.245.1 6 57.44 odd 18
722.2.e.c.389.1 6 57.35 odd 18
722.2.e.c.415.1 6 57.23 odd 18
722.2.e.c.423.1 6 57.47 odd 18
722.2.e.c.595.1 6 57.5 odd 18
722.2.e.d.99.1 6 57.2 even 18
722.2.e.d.245.1 6 57.32 even 18
722.2.e.d.389.1 6 57.41 even 18
722.2.e.d.415.1 6 57.53 even 18
722.2.e.d.423.1 6 57.29 even 18
722.2.e.d.595.1 6 57.14 even 18
950.2.a.b.1.1 1 15.14 odd 2
950.2.b.c.799.1 2 15.8 even 4
950.2.b.c.799.2 2 15.2 even 4
1216.2.a.g.1.1 1 24.11 even 2
1216.2.a.n.1.1 1 24.5 odd 2
1862.2.a.f.1.1 1 21.20 even 2
2736.2.a.w.1.1 1 4.3 odd 2
4598.2.a.a.1.1 1 33.32 even 2
5776.2.a.d.1.1 1 228.227 odd 2
6422.2.a.b.1.1 1 39.38 odd 2
6498.2.a.y.1.1 1 19.18 odd 2
7600.2.a.h.1.1 1 60.59 even 2
8550.2.a.u.1.1 1 5.4 even 2