# Properties

 Label 1078.2.a.d.1.1 Level $1078$ Weight $2$ Character 1078.1 Self dual yes Analytic conductor $8.608$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1078 = 2 \cdot 7^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1078.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$8.60787333789$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 154) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 1078.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} +1.00000 q^{4} +4.00000 q^{5} -1.00000 q^{8} -3.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} +1.00000 q^{4} +4.00000 q^{5} -1.00000 q^{8} -3.00000 q^{9} -4.00000 q^{10} -1.00000 q^{11} -2.00000 q^{13} +1.00000 q^{16} +4.00000 q^{17} +3.00000 q^{18} +6.00000 q^{19} +4.00000 q^{20} +1.00000 q^{22} +4.00000 q^{23} +11.0000 q^{25} +2.00000 q^{26} -2.00000 q^{29} +2.00000 q^{31} -1.00000 q^{32} -4.00000 q^{34} -3.00000 q^{36} +10.0000 q^{37} -6.00000 q^{38} -4.00000 q^{40} -4.00000 q^{41} -8.00000 q^{43} -1.00000 q^{44} -12.0000 q^{45} -4.00000 q^{46} -2.00000 q^{47} -11.0000 q^{50} -2.00000 q^{52} +6.00000 q^{53} -4.00000 q^{55} +2.00000 q^{58} +12.0000 q^{59} +14.0000 q^{61} -2.00000 q^{62} +1.00000 q^{64} -8.00000 q^{65} -12.0000 q^{67} +4.00000 q^{68} -8.00000 q^{71} +3.00000 q^{72} -4.00000 q^{73} -10.0000 q^{74} +6.00000 q^{76} +4.00000 q^{80} +9.00000 q^{81} +4.00000 q^{82} +6.00000 q^{83} +16.0000 q^{85} +8.00000 q^{86} +1.00000 q^{88} +6.00000 q^{89} +12.0000 q^{90} +4.00000 q^{92} +2.00000 q^{94} +24.0000 q^{95} +14.0000 q^{97} +3.00000 q^{99} +O(q^{100})$$

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.00000 −0.707107
$$3$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$4$$ 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
$$7$$ 0 0
$$8$$ −1.00000 −0.353553
$$9$$ −3.00000 −1.00000
$$10$$ −4.00000 −1.26491
$$11$$ −1.00000 −0.301511
$$12$$ 0 0
$$13$$ −2.00000 −0.554700 −0.277350 0.960769i $$-0.589456\pi$$
−0.277350 + 0.960769i $$0.589456\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 4.00000 0.970143 0.485071 0.874475i $$-0.338794\pi$$
0.485071 + 0.874475i $$0.338794\pi$$
$$18$$ 3.00000 0.707107
$$19$$ 6.00000 1.37649 0.688247 0.725476i $$-0.258380\pi$$
0.688247 + 0.725476i $$0.258380\pi$$
$$20$$ 4.00000 0.894427
$$21$$ 0 0
$$22$$ 1.00000 0.213201
$$23$$ 4.00000 0.834058 0.417029 0.908893i $$-0.363071\pi$$
0.417029 + 0.908893i $$0.363071\pi$$
$$24$$ 0 0
$$25$$ 11.0000 2.20000
$$26$$ 2.00000 0.392232
$$27$$ 0 0
$$28$$ 0 0
$$29$$ −2.00000 −0.371391 −0.185695 0.982607i $$-0.559454\pi$$
−0.185695 + 0.982607i $$0.559454\pi$$
$$30$$ 0 0
$$31$$ 2.00000 0.359211 0.179605 0.983739i $$-0.442518\pi$$
0.179605 + 0.983739i $$0.442518\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ 0 0
$$34$$ −4.00000 −0.685994
$$35$$ 0 0
$$36$$ −3.00000 −0.500000
$$37$$ 10.0000 1.64399 0.821995 0.569495i $$-0.192861\pi$$
0.821995 + 0.569495i $$0.192861\pi$$
$$38$$ −6.00000 −0.973329
$$39$$ 0 0
$$40$$ −4.00000 −0.632456
$$41$$ −4.00000 −0.624695 −0.312348 0.949968i $$-0.601115\pi$$
−0.312348 + 0.949968i $$0.601115\pi$$
$$42$$ 0 0
$$43$$ −8.00000 −1.21999 −0.609994 0.792406i $$-0.708828\pi$$
−0.609994 + 0.792406i $$0.708828\pi$$
$$44$$ −1.00000 −0.150756
$$45$$ −12.0000 −1.78885
$$46$$ −4.00000 −0.589768
$$47$$ −2.00000 −0.291730 −0.145865 0.989305i $$-0.546597\pi$$
−0.145865 + 0.989305i $$0.546597\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ −11.0000 −1.55563
$$51$$ 0 0
$$52$$ −2.00000 −0.277350
$$53$$ 6.00000 0.824163 0.412082 0.911147i $$-0.364802\pi$$
0.412082 + 0.911147i $$0.364802\pi$$
$$54$$ 0 0
$$55$$ −4.00000 −0.539360
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 2.00000 0.262613
$$59$$ 12.0000 1.56227 0.781133 0.624364i $$-0.214642\pi$$
0.781133 + 0.624364i $$0.214642\pi$$
$$60$$ 0 0
$$61$$ 14.0000 1.79252 0.896258 0.443533i $$-0.146275\pi$$
0.896258 + 0.443533i $$0.146275\pi$$
$$62$$ −2.00000 −0.254000
$$63$$ 0 0
$$64$$ 1.00000 0.125000
$$65$$ −8.00000 −0.992278
$$66$$ 0 0
$$67$$ −12.0000 −1.46603 −0.733017 0.680211i $$-0.761888\pi$$
−0.733017 + 0.680211i $$0.761888\pi$$
$$68$$ 4.00000 0.485071
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −8.00000 −0.949425 −0.474713 0.880141i $$-0.657448\pi$$
−0.474713 + 0.880141i $$0.657448\pi$$
$$72$$ 3.00000 0.353553
$$73$$ −4.00000 −0.468165 −0.234082 0.972217i $$-0.575209\pi$$
−0.234082 + 0.972217i $$0.575209\pi$$
$$74$$ −10.0000 −1.16248
$$75$$ 0 0
$$76$$ 6.00000 0.688247
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$80$$ 4.00000 0.447214
$$81$$ 9.00000 1.00000
$$82$$ 4.00000 0.441726
$$83$$ 6.00000 0.658586 0.329293 0.944228i $$-0.393190\pi$$
0.329293 + 0.944228i $$0.393190\pi$$
$$84$$ 0 0
$$85$$ 16.0000 1.73544
$$86$$ 8.00000 0.862662
$$87$$ 0 0
$$88$$ 1.00000 0.106600
$$89$$ 6.00000 0.635999 0.317999 0.948091i $$-0.396989\pi$$
0.317999 + 0.948091i $$0.396989\pi$$
$$90$$ 12.0000 1.26491
$$91$$ 0 0
$$92$$ 4.00000 0.417029
$$93$$ 0 0
$$94$$ 2.00000 0.206284
$$95$$ 24.0000 2.46235
$$96$$ 0 0
$$97$$ 14.0000 1.42148 0.710742 0.703452i $$-0.248359\pi$$
0.710742 + 0.703452i $$0.248359\pi$$
$$98$$ 0 0
$$99$$ 3.00000 0.301511
$$100$$ 11.0000 1.10000
$$101$$ −6.00000 −0.597022 −0.298511 0.954406i $$-0.596490\pi$$
−0.298511 + 0.954406i $$0.596490\pi$$
$$102$$ 0 0
$$103$$ −18.0000 −1.77359 −0.886796 0.462160i $$-0.847074\pi$$
−0.886796 + 0.462160i $$0.847074\pi$$
$$104$$ 2.00000 0.196116
$$105$$ 0 0
$$106$$ −6.00000 −0.582772
$$107$$ −16.0000 −1.54678 −0.773389 0.633932i $$-0.781440\pi$$
−0.773389 + 0.633932i $$0.781440\pi$$
$$108$$ 0 0
$$109$$ −14.0000 −1.34096 −0.670478 0.741929i $$-0.733911\pi$$
−0.670478 + 0.741929i $$0.733911\pi$$
$$110$$ 4.00000 0.381385
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 14.0000 1.31701 0.658505 0.752577i $$-0.271189\pi$$
0.658505 + 0.752577i $$0.271189\pi$$
$$114$$ 0 0
$$115$$ 16.0000 1.49201
$$116$$ −2.00000 −0.185695
$$117$$ 6.00000 0.554700
$$118$$ −12.0000 −1.10469
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ −14.0000 −1.26750
$$123$$ 0 0
$$124$$ 2.00000 0.179605
$$125$$ 24.0000 2.14663
$$126$$ 0 0
$$127$$ 8.00000 0.709885 0.354943 0.934888i $$-0.384500\pi$$
0.354943 + 0.934888i $$0.384500\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ 0 0
$$130$$ 8.00000 0.701646
$$131$$ −6.00000 −0.524222 −0.262111 0.965038i $$-0.584419\pi$$
−0.262111 + 0.965038i $$0.584419\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 12.0000 1.03664
$$135$$ 0 0
$$136$$ −4.00000 −0.342997
$$137$$ 6.00000 0.512615 0.256307 0.966595i $$-0.417494\pi$$
0.256307 + 0.966595i $$0.417494\pi$$
$$138$$ 0 0
$$139$$ −14.0000 −1.18746 −0.593732 0.804663i $$-0.702346\pi$$
−0.593732 + 0.804663i $$0.702346\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 8.00000 0.671345
$$143$$ 2.00000 0.167248
$$144$$ −3.00000 −0.250000
$$145$$ −8.00000 −0.664364
$$146$$ 4.00000 0.331042
$$147$$ 0 0
$$148$$ 10.0000 0.821995
$$149$$ 2.00000 0.163846 0.0819232 0.996639i $$-0.473894\pi$$
0.0819232 + 0.996639i $$0.473894\pi$$
$$150$$ 0 0
$$151$$ −24.0000 −1.95309 −0.976546 0.215308i $$-0.930924\pi$$
−0.976546 + 0.215308i $$0.930924\pi$$
$$152$$ −6.00000 −0.486664
$$153$$ −12.0000 −0.970143
$$154$$ 0 0
$$155$$ 8.00000 0.642575
$$156$$ 0 0
$$157$$ 8.00000 0.638470 0.319235 0.947676i $$-0.396574\pi$$
0.319235 + 0.947676i $$0.396574\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ −4.00000 −0.316228
$$161$$ 0 0
$$162$$ −9.00000 −0.707107
$$163$$ 4.00000 0.313304 0.156652 0.987654i $$-0.449930\pi$$
0.156652 + 0.987654i $$0.449930\pi$$
$$164$$ −4.00000 −0.312348
$$165$$ 0 0
$$166$$ −6.00000 −0.465690
$$167$$ −4.00000 −0.309529 −0.154765 0.987951i $$-0.549462\pi$$
−0.154765 + 0.987951i $$0.549462\pi$$
$$168$$ 0 0
$$169$$ −9.00000 −0.692308
$$170$$ −16.0000 −1.22714
$$171$$ −18.0000 −1.37649
$$172$$ −8.00000 −0.609994
$$173$$ 14.0000 1.06440 0.532200 0.846619i $$-0.321365\pi$$
0.532200 + 0.846619i $$0.321365\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ −1.00000 −0.0753778
$$177$$ 0 0
$$178$$ −6.00000 −0.449719
$$179$$ 4.00000 0.298974 0.149487 0.988764i $$-0.452238\pi$$
0.149487 + 0.988764i $$0.452238\pi$$
$$180$$ −12.0000 −0.894427
$$181$$ −20.0000 −1.48659 −0.743294 0.668965i $$-0.766738\pi$$
−0.743294 + 0.668965i $$0.766738\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ −4.00000 −0.294884
$$185$$ 40.0000 2.94086
$$186$$ 0 0
$$187$$ −4.00000 −0.292509
$$188$$ −2.00000 −0.145865
$$189$$ 0 0
$$190$$ −24.0000 −1.74114
$$191$$ −4.00000 −0.289430 −0.144715 0.989473i $$-0.546227\pi$$
−0.144715 + 0.989473i $$0.546227\pi$$
$$192$$ 0 0
$$193$$ 2.00000 0.143963 0.0719816 0.997406i $$-0.477068\pi$$
0.0719816 + 0.997406i $$0.477068\pi$$
$$194$$ −14.0000 −1.00514
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 6.00000 0.427482 0.213741 0.976890i $$-0.431435\pi$$
0.213741 + 0.976890i $$0.431435\pi$$
$$198$$ −3.00000 −0.213201
$$199$$ 14.0000 0.992434 0.496217 0.868199i $$-0.334722\pi$$
0.496217 + 0.868199i $$0.334722\pi$$
$$200$$ −11.0000 −0.777817
$$201$$ 0 0
$$202$$ 6.00000 0.422159
$$203$$ 0 0
$$204$$ 0 0
$$205$$ −16.0000 −1.11749
$$206$$ 18.0000 1.25412
$$207$$ −12.0000 −0.834058
$$208$$ −2.00000 −0.138675
$$209$$ −6.00000 −0.415029
$$210$$ 0 0
$$211$$ −8.00000 −0.550743 −0.275371 0.961338i $$-0.588801\pi$$
−0.275371 + 0.961338i $$0.588801\pi$$
$$212$$ 6.00000 0.412082
$$213$$ 0 0
$$214$$ 16.0000 1.09374
$$215$$ −32.0000 −2.18238
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 14.0000 0.948200
$$219$$ 0 0
$$220$$ −4.00000 −0.269680
$$221$$ −8.00000 −0.538138
$$222$$ 0 0
$$223$$ 2.00000 0.133930 0.0669650 0.997755i $$-0.478668\pi$$
0.0669650 + 0.997755i $$0.478668\pi$$
$$224$$ 0 0
$$225$$ −33.0000 −2.20000
$$226$$ −14.0000 −0.931266
$$227$$ 2.00000 0.132745 0.0663723 0.997795i $$-0.478857\pi$$
0.0663723 + 0.997795i $$0.478857\pi$$
$$228$$ 0 0
$$229$$ −20.0000 −1.32164 −0.660819 0.750546i $$-0.729791\pi$$
−0.660819 + 0.750546i $$0.729791\pi$$
$$230$$ −16.0000 −1.05501
$$231$$ 0 0
$$232$$ 2.00000 0.131306
$$233$$ 30.0000 1.96537 0.982683 0.185296i $$-0.0593245\pi$$
0.982683 + 0.185296i $$0.0593245\pi$$
$$234$$ −6.00000 −0.392232
$$235$$ −8.00000 −0.521862
$$236$$ 12.0000 0.781133
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 16.0000 1.03495 0.517477 0.855697i $$-0.326871\pi$$
0.517477 + 0.855697i $$0.326871\pi$$
$$240$$ 0 0
$$241$$ −12.0000 −0.772988 −0.386494 0.922292i $$-0.626314\pi$$
−0.386494 + 0.922292i $$0.626314\pi$$
$$242$$ −1.00000 −0.0642824
$$243$$ 0 0
$$244$$ 14.0000 0.896258
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −12.0000 −0.763542
$$248$$ −2.00000 −0.127000
$$249$$ 0 0
$$250$$ −24.0000 −1.51789
$$251$$ −12.0000 −0.757433 −0.378717 0.925513i $$-0.623635\pi$$
−0.378717 + 0.925513i $$0.623635\pi$$
$$252$$ 0 0
$$253$$ −4.00000 −0.251478
$$254$$ −8.00000 −0.501965
$$255$$ 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
$$259$$ 0 0
$$260$$ −8.00000 −0.496139
$$261$$ 6.00000 0.371391
$$262$$ 6.00000 0.370681
$$263$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$264$$ 0 0
$$265$$ 24.0000 1.47431
$$266$$ 0 0
$$267$$ 0 0
$$268$$ −12.0000 −0.733017
$$269$$ −12.0000 −0.731653 −0.365826 0.930683i $$-0.619214\pi$$
−0.365826 + 0.930683i $$0.619214\pi$$
$$270$$ 0 0
$$271$$ 20.0000 1.21491 0.607457 0.794353i $$-0.292190\pi$$
0.607457 + 0.794353i $$0.292190\pi$$
$$272$$ 4.00000 0.242536
$$273$$ 0 0
$$274$$ −6.00000 −0.362473
$$275$$ −11.0000 −0.663325
$$276$$ 0 0
$$277$$ −30.0000 −1.80253 −0.901263 0.433273i $$-0.857359\pi$$
−0.901263 + 0.433273i $$0.857359\pi$$
$$278$$ 14.0000 0.839664
$$279$$ −6.00000 −0.359211
$$280$$ 0 0
$$281$$ −10.0000 −0.596550 −0.298275 0.954480i $$-0.596411\pi$$
−0.298275 + 0.954480i $$0.596411\pi$$
$$282$$ 0 0
$$283$$ −6.00000 −0.356663 −0.178331 0.983970i $$-0.557070\pi$$
−0.178331 + 0.983970i $$0.557070\pi$$
$$284$$ −8.00000 −0.474713
$$285$$ 0 0
$$286$$ −2.00000 −0.118262
$$287$$ 0 0
$$288$$ 3.00000 0.176777
$$289$$ −1.00000 −0.0588235
$$290$$ 8.00000 0.469776
$$291$$ 0 0
$$292$$ −4.00000 −0.234082
$$293$$ −18.0000 −1.05157 −0.525786 0.850617i $$-0.676229\pi$$
−0.525786 + 0.850617i $$0.676229\pi$$
$$294$$ 0 0
$$295$$ 48.0000 2.79467
$$296$$ −10.0000 −0.581238
$$297$$ 0 0
$$298$$ −2.00000 −0.115857
$$299$$ −8.00000 −0.462652
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 24.0000 1.38104
$$303$$ 0 0
$$304$$ 6.00000 0.344124
$$305$$ 56.0000 3.20655
$$306$$ 12.0000 0.685994
$$307$$ 10.0000 0.570730 0.285365 0.958419i $$-0.407885\pi$$
0.285365 + 0.958419i $$0.407885\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ −8.00000 −0.454369
$$311$$ −14.0000 −0.793867 −0.396934 0.917847i $$-0.629926\pi$$
−0.396934 + 0.917847i $$0.629926\pi$$
$$312$$ 0 0
$$313$$ 2.00000 0.113047 0.0565233 0.998401i $$-0.481998\pi$$
0.0565233 + 0.998401i $$0.481998\pi$$
$$314$$ −8.00000 −0.451466
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 6.00000 0.336994 0.168497 0.985702i $$-0.446109\pi$$
0.168497 + 0.985702i $$0.446109\pi$$
$$318$$ 0 0
$$319$$ 2.00000 0.111979
$$320$$ 4.00000 0.223607
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 24.0000 1.33540
$$324$$ 9.00000 0.500000
$$325$$ −22.0000 −1.22034
$$326$$ −4.00000 −0.221540
$$327$$ 0 0
$$328$$ 4.00000 0.220863
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −20.0000 −1.09930 −0.549650 0.835395i $$-0.685239\pi$$
−0.549650 + 0.835395i $$0.685239\pi$$
$$332$$ 6.00000 0.329293
$$333$$ −30.0000 −1.64399
$$334$$ 4.00000 0.218870
$$335$$ −48.0000 −2.62252
$$336$$ 0 0
$$337$$ −18.0000 −0.980522 −0.490261 0.871576i $$-0.663099\pi$$
−0.490261 + 0.871576i $$0.663099\pi$$
$$338$$ 9.00000 0.489535
$$339$$ 0 0
$$340$$ 16.0000 0.867722
$$341$$ −2.00000 −0.108306
$$342$$ 18.0000 0.973329
$$343$$ 0 0
$$344$$ 8.00000 0.431331
$$345$$ 0 0
$$346$$ −14.0000 −0.752645
$$347$$ 8.00000 0.429463 0.214731 0.976673i $$-0.431112\pi$$
0.214731 + 0.976673i $$0.431112\pi$$
$$348$$ 0 0
$$349$$ 10.0000 0.535288 0.267644 0.963518i $$-0.413755\pi$$
0.267644 + 0.963518i $$0.413755\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 1.00000 0.0533002
$$353$$ 6.00000 0.319348 0.159674 0.987170i $$-0.448956\pi$$
0.159674 + 0.987170i $$0.448956\pi$$
$$354$$ 0 0
$$355$$ −32.0000 −1.69838
$$356$$ 6.00000 0.317999
$$357$$ 0 0
$$358$$ −4.00000 −0.211407
$$359$$ −16.0000 −0.844448 −0.422224 0.906492i $$-0.638750\pi$$
−0.422224 + 0.906492i $$0.638750\pi$$
$$360$$ 12.0000 0.632456
$$361$$ 17.0000 0.894737
$$362$$ 20.0000 1.05118
$$363$$ 0 0
$$364$$ 0 0
$$365$$ −16.0000 −0.837478
$$366$$ 0 0
$$367$$ −22.0000 −1.14839 −0.574195 0.818718i $$-0.694685\pi$$
−0.574195 + 0.818718i $$0.694685\pi$$
$$368$$ 4.00000 0.208514
$$369$$ 12.0000 0.624695
$$370$$ −40.0000 −2.07950
$$371$$ 0 0
$$372$$ 0 0
$$373$$ −10.0000 −0.517780 −0.258890 0.965907i $$-0.583357\pi$$
−0.258890 + 0.965907i $$0.583357\pi$$
$$374$$ 4.00000 0.206835
$$375$$ 0 0
$$376$$ 2.00000 0.103142
$$377$$ 4.00000 0.206010
$$378$$ 0 0
$$379$$ 4.00000 0.205466 0.102733 0.994709i $$-0.467241\pi$$
0.102733 + 0.994709i $$0.467241\pi$$
$$380$$ 24.0000 1.23117
$$381$$ 0 0
$$382$$ 4.00000 0.204658
$$383$$ 10.0000 0.510976 0.255488 0.966812i $$-0.417764\pi$$
0.255488 + 0.966812i $$0.417764\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −2.00000 −0.101797
$$387$$ 24.0000 1.21999
$$388$$ 14.0000 0.710742
$$389$$ −30.0000 −1.52106 −0.760530 0.649303i $$-0.775061\pi$$
−0.760530 + 0.649303i $$0.775061\pi$$
$$390$$ 0 0
$$391$$ 16.0000 0.809155
$$392$$ 0 0
$$393$$ 0 0
$$394$$ −6.00000 −0.302276
$$395$$ 0 0
$$396$$ 3.00000 0.150756
$$397$$ −24.0000 −1.20453 −0.602263 0.798298i $$-0.705734\pi$$
−0.602263 + 0.798298i $$0.705734\pi$$
$$398$$ −14.0000 −0.701757
$$399$$ 0 0
$$400$$ 11.0000 0.550000
$$401$$ −18.0000 −0.898877 −0.449439 0.893311i $$-0.648376\pi$$
−0.449439 + 0.893311i $$0.648376\pi$$
$$402$$ 0 0
$$403$$ −4.00000 −0.199254
$$404$$ −6.00000 −0.298511
$$405$$ 36.0000 1.78885
$$406$$ 0 0
$$407$$ −10.0000 −0.495682
$$408$$ 0 0
$$409$$ −16.0000 −0.791149 −0.395575 0.918434i $$-0.629455\pi$$
−0.395575 + 0.918434i $$0.629455\pi$$
$$410$$ 16.0000 0.790184
$$411$$ 0 0
$$412$$ −18.0000 −0.886796
$$413$$ 0 0
$$414$$ 12.0000 0.589768
$$415$$ 24.0000 1.17811
$$416$$ 2.00000 0.0980581
$$417$$ 0 0
$$418$$ 6.00000 0.293470
$$419$$ 32.0000 1.56330 0.781651 0.623716i $$-0.214378\pi$$
0.781651 + 0.623716i $$0.214378\pi$$
$$420$$ 0 0
$$421$$ −2.00000 −0.0974740 −0.0487370 0.998812i $$-0.515520\pi$$
−0.0487370 + 0.998812i $$0.515520\pi$$
$$422$$ 8.00000 0.389434
$$423$$ 6.00000 0.291730
$$424$$ −6.00000 −0.291386
$$425$$ 44.0000 2.13431
$$426$$ 0 0
$$427$$ 0 0
$$428$$ −16.0000 −0.773389
$$429$$ 0 0
$$430$$ 32.0000 1.54318
$$431$$ 16.0000 0.770693 0.385346 0.922772i $$-0.374082\pi$$
0.385346 + 0.922772i $$0.374082\pi$$
$$432$$ 0 0
$$433$$ −2.00000 −0.0961139 −0.0480569 0.998845i $$-0.515303\pi$$
−0.0480569 + 0.998845i $$0.515303\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ −14.0000 −0.670478
$$437$$ 24.0000 1.14808
$$438$$ 0 0
$$439$$ −28.0000 −1.33637 −0.668184 0.743996i $$-0.732928\pi$$
−0.668184 + 0.743996i $$0.732928\pi$$
$$440$$ 4.00000 0.190693
$$441$$ 0 0
$$442$$ 8.00000 0.380521
$$443$$ −36.0000 −1.71041 −0.855206 0.518289i $$-0.826569\pi$$
−0.855206 + 0.518289i $$0.826569\pi$$
$$444$$ 0 0
$$445$$ 24.0000 1.13771
$$446$$ −2.00000 −0.0947027
$$447$$ 0 0
$$448$$ 0 0
$$449$$ −18.0000 −0.849473 −0.424736 0.905317i $$-0.639633\pi$$
−0.424736 + 0.905317i $$0.639633\pi$$
$$450$$ 33.0000 1.55563
$$451$$ 4.00000 0.188353
$$452$$ 14.0000 0.658505
$$453$$ 0 0
$$454$$ −2.00000 −0.0938647
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 2.00000 0.0935561 0.0467780 0.998905i $$-0.485105\pi$$
0.0467780 + 0.998905i $$0.485105\pi$$
$$458$$ 20.0000 0.934539
$$459$$ 0 0
$$460$$ 16.0000 0.746004
$$461$$ −30.0000 −1.39724 −0.698620 0.715493i $$-0.746202\pi$$
−0.698620 + 0.715493i $$0.746202\pi$$
$$462$$ 0 0
$$463$$ −32.0000 −1.48717 −0.743583 0.668644i $$-0.766875\pi$$
−0.743583 + 0.668644i $$0.766875\pi$$
$$464$$ −2.00000 −0.0928477
$$465$$ 0 0
$$466$$ −30.0000 −1.38972
$$467$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$468$$ 6.00000 0.277350
$$469$$ 0 0
$$470$$ 8.00000 0.369012
$$471$$ 0 0
$$472$$ −12.0000 −0.552345
$$473$$ 8.00000 0.367840
$$474$$ 0 0
$$475$$ 66.0000 3.02829
$$476$$ 0 0
$$477$$ −18.0000 −0.824163
$$478$$ −16.0000 −0.731823
$$479$$ 16.0000 0.731059 0.365529 0.930800i $$-0.380888\pi$$
0.365529 + 0.930800i $$0.380888\pi$$
$$480$$ 0 0
$$481$$ −20.0000 −0.911922
$$482$$ 12.0000 0.546585
$$483$$ 0 0
$$484$$ 1.00000 0.0454545
$$485$$ 56.0000 2.54283
$$486$$ 0 0
$$487$$ −28.0000 −1.26880 −0.634401 0.773004i $$-0.718753\pi$$
−0.634401 + 0.773004i $$0.718753\pi$$
$$488$$ −14.0000 −0.633750
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −36.0000 −1.62466 −0.812329 0.583200i $$-0.801800\pi$$
−0.812329 + 0.583200i $$0.801800\pi$$
$$492$$ 0 0
$$493$$ −8.00000 −0.360302
$$494$$ 12.0000 0.539906
$$495$$ 12.0000 0.539360
$$496$$ 2.00000 0.0898027
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 44.0000 1.96971 0.984855 0.173379i $$-0.0554684\pi$$
0.984855 + 0.173379i $$0.0554684\pi$$
$$500$$ 24.0000 1.07331
$$501$$ 0 0
$$502$$ 12.0000 0.535586
$$503$$ 36.0000 1.60516 0.802580 0.596544i $$-0.203460\pi$$
0.802580 + 0.596544i $$0.203460\pi$$
$$504$$ 0 0
$$505$$ −24.0000 −1.06799
$$506$$ 4.00000 0.177822
$$507$$ 0 0
$$508$$ 8.00000 0.354943
$$509$$ 28.0000 1.24108 0.620539 0.784176i $$-0.286914\pi$$
0.620539 + 0.784176i $$0.286914\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ −1.00000 −0.0441942
$$513$$ 0 0
$$514$$ −6.00000 −0.264649
$$515$$ −72.0000 −3.17270
$$516$$ 0 0
$$517$$ 2.00000 0.0879599
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 8.00000 0.350823
$$521$$ −10.0000 −0.438108 −0.219054 0.975713i $$-0.570297\pi$$
−0.219054 + 0.975713i $$0.570297\pi$$
$$522$$ −6.00000 −0.262613
$$523$$ 34.0000 1.48672 0.743358 0.668894i $$-0.233232\pi$$
0.743358 + 0.668894i $$0.233232\pi$$
$$524$$ −6.00000 −0.262111
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 8.00000 0.348485
$$528$$ 0 0
$$529$$ −7.00000 −0.304348
$$530$$ −24.0000 −1.04249
$$531$$ −36.0000 −1.56227
$$532$$ 0 0
$$533$$ 8.00000 0.346518
$$534$$ 0 0
$$535$$ −64.0000 −2.76696
$$536$$ 12.0000 0.518321
$$537$$ 0 0
$$538$$ 12.0000 0.517357
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 14.0000 0.601907 0.300954 0.953639i $$-0.402695\pi$$
0.300954 + 0.953639i $$0.402695\pi$$
$$542$$ −20.0000 −0.859074
$$543$$ 0 0
$$544$$ −4.00000 −0.171499
$$545$$ −56.0000 −2.39878
$$546$$ 0 0
$$547$$ −12.0000 −0.513083 −0.256541 0.966533i $$-0.582583\pi$$
−0.256541 + 0.966533i $$0.582583\pi$$
$$548$$ 6.00000 0.256307
$$549$$ −42.0000 −1.79252
$$550$$ 11.0000 0.469042
$$551$$ −12.0000 −0.511217
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 30.0000 1.27458
$$555$$ 0 0
$$556$$ −14.0000 −0.593732
$$557$$ 14.0000 0.593199 0.296600 0.955002i $$-0.404147\pi$$
0.296600 + 0.955002i $$0.404147\pi$$
$$558$$ 6.00000 0.254000
$$559$$ 16.0000 0.676728
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 10.0000 0.421825
$$563$$ 34.0000 1.43293 0.716465 0.697623i $$-0.245759\pi$$
0.716465 + 0.697623i $$0.245759\pi$$
$$564$$ 0 0
$$565$$ 56.0000 2.35594
$$566$$ 6.00000 0.252199
$$567$$ 0 0
$$568$$ 8.00000 0.335673
$$569$$ 6.00000 0.251533 0.125767 0.992060i $$-0.459861\pi$$
0.125767 + 0.992060i $$0.459861\pi$$
$$570$$ 0 0
$$571$$ 28.0000 1.17176 0.585882 0.810397i $$-0.300748\pi$$
0.585882 + 0.810397i $$0.300748\pi$$
$$572$$ 2.00000 0.0836242
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 44.0000 1.83493
$$576$$ −3.00000 −0.125000
$$577$$ −14.0000 −0.582828 −0.291414 0.956597i $$-0.594126\pi$$
−0.291414 + 0.956597i $$0.594126\pi$$
$$578$$ 1.00000 0.0415945
$$579$$ 0 0
$$580$$ −8.00000 −0.332182
$$581$$ 0 0
$$582$$ 0 0
$$583$$ −6.00000 −0.248495
$$584$$ 4.00000 0.165521
$$585$$ 24.0000 0.992278
$$586$$ 18.0000 0.743573
$$587$$ 12.0000 0.495293 0.247647 0.968850i $$-0.420343\pi$$
0.247647 + 0.968850i $$0.420343\pi$$
$$588$$ 0 0
$$589$$ 12.0000 0.494451
$$590$$ −48.0000 −1.97613
$$591$$ 0 0
$$592$$ 10.0000 0.410997
$$593$$ 12.0000 0.492781 0.246390 0.969171i $$-0.420755\pi$$
0.246390 + 0.969171i $$0.420755\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 2.00000 0.0819232
$$597$$ 0 0
$$598$$ 8.00000 0.327144
$$599$$ 24.0000 0.980613 0.490307 0.871550i $$-0.336885\pi$$
0.490307 + 0.871550i $$0.336885\pi$$
$$600$$ 0 0
$$601$$ −8.00000 −0.326327 −0.163163 0.986599i $$-0.552170\pi$$
−0.163163 + 0.986599i $$0.552170\pi$$
$$602$$ 0 0
$$603$$ 36.0000 1.46603
$$604$$ −24.0000 −0.976546
$$605$$ 4.00000 0.162623
$$606$$ 0 0
$$607$$ −8.00000 −0.324710 −0.162355 0.986732i $$-0.551909\pi$$
−0.162355 + 0.986732i $$0.551909\pi$$
$$608$$ −6.00000 −0.243332
$$609$$ 0 0
$$610$$ −56.0000 −2.26737
$$611$$ 4.00000 0.161823
$$612$$ −12.0000 −0.485071
$$613$$ 46.0000 1.85792 0.928961 0.370177i $$-0.120703\pi$$
0.928961 + 0.370177i $$0.120703\pi$$
$$614$$ −10.0000 −0.403567
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 18.0000 0.724653 0.362326 0.932051i $$-0.381983\pi$$
0.362326 + 0.932051i $$0.381983\pi$$
$$618$$ 0 0
$$619$$ −8.00000 −0.321547 −0.160774 0.986991i $$-0.551399\pi$$
−0.160774 + 0.986991i $$0.551399\pi$$
$$620$$ 8.00000 0.321288
$$621$$ 0 0
$$622$$ 14.0000 0.561349
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 41.0000 1.64000
$$626$$ −2.00000 −0.0799361
$$627$$ 0 0
$$628$$ 8.00000 0.319235
$$629$$ 40.0000 1.59490
$$630$$ 0 0
$$631$$ −12.0000 −0.477712 −0.238856 0.971055i $$-0.576772\pi$$
−0.238856 + 0.971055i $$0.576772\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ −6.00000 −0.238290
$$635$$ 32.0000 1.26988
$$636$$ 0 0
$$637$$ 0 0
$$638$$ −2.00000 −0.0791808
$$639$$ 24.0000 0.949425
$$640$$ −4.00000 −0.158114
$$641$$ −6.00000 −0.236986 −0.118493 0.992955i $$-0.537806\pi$$
−0.118493 + 0.992955i $$0.537806\pi$$
$$642$$ 0 0
$$643$$ −4.00000 −0.157745 −0.0788723 0.996885i $$-0.525132\pi$$
−0.0788723 + 0.996885i $$0.525132\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ −24.0000 −0.944267
$$647$$ −6.00000 −0.235884 −0.117942 0.993020i $$-0.537630\pi$$
−0.117942 + 0.993020i $$0.537630\pi$$
$$648$$ −9.00000 −0.353553
$$649$$ −12.0000 −0.471041
$$650$$ 22.0000 0.862911
$$651$$ 0 0
$$652$$ 4.00000 0.156652
$$653$$ 10.0000 0.391330 0.195665 0.980671i $$-0.437313\pi$$
0.195665 + 0.980671i $$0.437313\pi$$
$$654$$ 0 0
$$655$$ −24.0000 −0.937758
$$656$$ −4.00000 −0.156174
$$657$$ 12.0000 0.468165
$$658$$ 0 0
$$659$$ −8.00000 −0.311636 −0.155818 0.987786i $$-0.549801\pi$$
−0.155818 + 0.987786i $$0.549801\pi$$
$$660$$ 0 0
$$661$$ −20.0000 −0.777910 −0.388955 0.921257i $$-0.627164\pi$$
−0.388955 + 0.921257i $$0.627164\pi$$
$$662$$ 20.0000 0.777322
$$663$$ 0 0
$$664$$ −6.00000 −0.232845
$$665$$ 0 0
$$666$$ 30.0000 1.16248
$$667$$ −8.00000 −0.309761
$$668$$ −4.00000 −0.154765
$$669$$ 0 0
$$670$$ 48.0000 1.85440
$$671$$ −14.0000 −0.540464
$$672$$ 0 0
$$673$$ 22.0000 0.848038 0.424019 0.905653i $$-0.360619\pi$$
0.424019 + 0.905653i $$0.360619\pi$$
$$674$$ 18.0000 0.693334
$$675$$ 0 0
$$676$$ −9.00000 −0.346154
$$677$$ −26.0000 −0.999261 −0.499631 0.866239i $$-0.666531\pi$$
−0.499631 + 0.866239i $$0.666531\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ −16.0000 −0.613572
$$681$$ 0 0
$$682$$ 2.00000 0.0765840
$$683$$ 36.0000 1.37750 0.688751 0.724998i $$-0.258159\pi$$
0.688751 + 0.724998i $$0.258159\pi$$
$$684$$ −18.0000 −0.688247
$$685$$ 24.0000 0.916993
$$686$$ 0 0
$$687$$ 0 0
$$688$$ −8.00000 −0.304997
$$689$$ −12.0000 −0.457164
$$690$$ 0 0
$$691$$ −36.0000 −1.36950 −0.684752 0.728776i $$-0.740090\pi$$
−0.684752 + 0.728776i $$0.740090\pi$$
$$692$$ 14.0000 0.532200
$$693$$ 0 0
$$694$$ −8.00000 −0.303676
$$695$$ −56.0000 −2.12420
$$696$$ 0 0
$$697$$ −16.0000 −0.606043
$$698$$ −10.0000 −0.378506
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −30.0000 −1.13308 −0.566542 0.824033i $$-0.691719\pi$$
−0.566542 + 0.824033i $$0.691719\pi$$
$$702$$ 0 0
$$703$$ 60.0000 2.26294
$$704$$ −1.00000 −0.0376889
$$705$$ 0 0
$$706$$ −6.00000 −0.225813
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 18.0000 0.676004 0.338002 0.941145i $$-0.390249\pi$$
0.338002 + 0.941145i $$0.390249\pi$$
$$710$$ 32.0000 1.20094
$$711$$ 0 0
$$712$$ −6.00000 −0.224860
$$713$$ 8.00000 0.299602
$$714$$ 0 0
$$715$$ 8.00000 0.299183
$$716$$ 4.00000 0.149487
$$717$$ 0 0
$$718$$ 16.0000 0.597115
$$719$$ 26.0000 0.969636 0.484818 0.874615i $$-0.338886\pi$$
0.484818 + 0.874615i $$0.338886\pi$$
$$720$$ −12.0000 −0.447214
$$721$$ 0 0
$$722$$ −17.0000 −0.632674
$$723$$ 0 0
$$724$$ −20.0000 −0.743294
$$725$$ −22.0000 −0.817059
$$726$$ 0 0
$$727$$ 10.0000 0.370879 0.185440 0.982656i $$-0.440629\pi$$
0.185440 + 0.982656i $$0.440629\pi$$
$$728$$ 0 0
$$729$$ −27.0000 −1.00000
$$730$$ 16.0000 0.592187
$$731$$ −32.0000 −1.18356
$$732$$ 0 0
$$733$$ 22.0000 0.812589 0.406294 0.913742i $$-0.366821\pi$$
0.406294 + 0.913742i $$0.366821\pi$$
$$734$$ 22.0000 0.812035
$$735$$ 0 0
$$736$$ −4.00000 −0.147442
$$737$$ 12.0000 0.442026
$$738$$ −12.0000 −0.441726
$$739$$ −12.0000 −0.441427 −0.220714 0.975339i $$-0.570839\pi$$
−0.220714 + 0.975339i $$0.570839\pi$$
$$740$$ 40.0000 1.47043
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −24.0000 −0.880475 −0.440237 0.897881i $$-0.645106\pi$$
−0.440237 + 0.897881i $$0.645106\pi$$
$$744$$ 0 0
$$745$$ 8.00000 0.293097
$$746$$ 10.0000 0.366126
$$747$$ −18.0000 −0.658586
$$748$$ −4.00000 −0.146254
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 28.0000 1.02173 0.510867 0.859660i $$-0.329324\pi$$
0.510867 + 0.859660i $$0.329324\pi$$
$$752$$ −2.00000 −0.0729325
$$753$$ 0 0
$$754$$ −4.00000 −0.145671
$$755$$ −96.0000 −3.49380
$$756$$ 0 0
$$757$$ 26.0000 0.944986 0.472493 0.881334i $$-0.343354\pi$$
0.472493 + 0.881334i $$0.343354\pi$$
$$758$$ −4.00000 −0.145287
$$759$$ 0 0
$$760$$ −24.0000 −0.870572
$$761$$ 48.0000 1.74000 0.869999 0.493053i $$-0.164119\pi$$
0.869999 + 0.493053i $$0.164119\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ −4.00000 −0.144715
$$765$$ −48.0000 −1.73544
$$766$$ −10.0000 −0.361315
$$767$$ −24.0000 −0.866590
$$768$$ 0 0
$$769$$ −16.0000 −0.576975 −0.288487 0.957484i $$-0.593152\pi$$
−0.288487 + 0.957484i $$0.593152\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 2.00000 0.0719816
$$773$$ 48.0000 1.72644 0.863220 0.504828i $$-0.168444\pi$$
0.863220 + 0.504828i $$0.168444\pi$$
$$774$$ −24.0000 −0.862662
$$775$$ 22.0000 0.790263
$$776$$ −14.0000 −0.502571
$$777$$ 0 0
$$778$$ 30.0000 1.07555
$$779$$ −24.0000 −0.859889
$$780$$ 0 0
$$781$$ 8.00000 0.286263
$$782$$ −16.0000 −0.572159
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 32.0000 1.14213
$$786$$ 0 0
$$787$$ 22.0000 0.784215 0.392108 0.919919i $$-0.371746\pi$$
0.392108 + 0.919919i $$0.371746\pi$$
$$788$$ 6.00000 0.213741
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 0 0
$$792$$ −3.00000 −0.106600
$$793$$ −28.0000 −0.994309
$$794$$ 24.0000 0.851728
$$795$$ 0 0
$$796$$ 14.0000 0.496217
$$797$$ 16.0000 0.566749 0.283375 0.959009i $$-0.408546\pi$$
0.283375 + 0.959009i $$0.408546\pi$$
$$798$$ 0 0
$$799$$ −8.00000 −0.283020
$$800$$ −11.0000 −0.388909
$$801$$ −18.0000 −0.635999
$$802$$ 18.0000 0.635602
$$803$$ 4.00000 0.141157
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 4.00000 0.140894
$$807$$ 0 0
$$808$$ 6.00000 0.211079
$$809$$ −30.0000 −1.05474 −0.527372 0.849635i $$-0.676823\pi$$
−0.527372 + 0.849635i $$0.676823\pi$$
$$810$$ −36.0000 −1.26491
$$811$$ −38.0000 −1.33436 −0.667180 0.744896i $$-0.732499\pi$$
−0.667180 + 0.744896i $$0.732499\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 10.0000 0.350500
$$815$$ 16.0000 0.560456
$$816$$ 0 0
$$817$$ −48.0000 −1.67931
$$818$$ 16.0000 0.559427
$$819$$ 0 0
$$820$$ −16.0000 −0.558744
$$821$$ 18.0000 0.628204 0.314102 0.949389i $$-0.398297\pi$$
0.314102 + 0.949389i $$0.398297\pi$$
$$822$$ 0 0
$$823$$ −4.00000 −0.139431 −0.0697156 0.997567i $$-0.522209\pi$$
−0.0697156 + 0.997567i $$0.522209\pi$$
$$824$$ 18.0000 0.627060
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 20.0000 0.695468 0.347734 0.937593i $$-0.386951\pi$$
0.347734 + 0.937593i $$0.386951\pi$$
$$828$$ −12.0000 −0.417029
$$829$$ 20.0000 0.694629 0.347314 0.937749i $$-0.387094\pi$$
0.347314 + 0.937749i $$0.387094\pi$$
$$830$$ −24.0000 −0.833052
$$831$$ 0 0
$$832$$ −2.00000 −0.0693375
$$833$$ 0 0
$$834$$ 0 0
$$835$$ −16.0000 −0.553703
$$836$$ −6.00000 −0.207514
$$837$$ 0 0
$$838$$ −32.0000 −1.10542
$$839$$ −30.0000 −1.03572 −0.517858 0.855467i $$-0.673270\pi$$
−0.517858 + 0.855467i $$0.673270\pi$$
$$840$$ 0 0
$$841$$ −25.0000 −0.862069
$$842$$ 2.00000 0.0689246
$$843$$ 0 0
$$844$$ −8.00000 −0.275371
$$845$$ −36.0000 −1.23844
$$846$$ −6.00000 −0.206284
$$847$$ 0 0
$$848$$ 6.00000 0.206041
$$849$$ 0 0
$$850$$ −44.0000 −1.50919
$$851$$ 40.0000 1.37118
$$852$$ 0 0
$$853$$ −2.00000 −0.0684787 −0.0342393 0.999414i $$-0.510901\pi$$
−0.0342393 + 0.999414i $$0.510901\pi$$
$$854$$ 0 0
$$855$$ −72.0000 −2.46235
$$856$$ 16.0000 0.546869
$$857$$ −32.0000 −1.09310 −0.546550 0.837427i $$-0.684059\pi$$
−0.546550 + 0.837427i $$0.684059\pi$$
$$858$$ 0 0
$$859$$ −28.0000 −0.955348 −0.477674 0.878537i $$-0.658520\pi$$
−0.477674 + 0.878537i $$0.658520\pi$$
$$860$$ −32.0000 −1.09119
$$861$$ 0 0
$$862$$ −16.0000 −0.544962
$$863$$ −44.0000 −1.49778 −0.748889 0.662696i $$-0.769412\pi$$
−0.748889 + 0.662696i $$0.769412\pi$$
$$864$$ 0 0
$$865$$ 56.0000 1.90406
$$866$$ 2.00000 0.0679628
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 0 0
$$870$$ 0 0
$$871$$ 24.0000 0.813209
$$872$$ 14.0000 0.474100
$$873$$ −42.0000 −1.42148
$$874$$ −24.0000 −0.811812
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 34.0000 1.14810 0.574049 0.818821i $$-0.305372\pi$$
0.574049 + 0.818821i $$0.305372\pi$$
$$878$$ 28.0000 0.944954
$$879$$ 0 0
$$880$$ −4.00000 −0.134840
$$881$$ −30.0000 −1.01073 −0.505363 0.862907i $$-0.668641\pi$$
−0.505363 + 0.862907i $$0.668641\pi$$
$$882$$ 0 0
$$883$$ −44.0000 −1.48072 −0.740359 0.672212i $$-0.765344\pi$$
−0.740359 + 0.672212i $$0.765344\pi$$
$$884$$ −8.00000 −0.269069
$$885$$ 0 0
$$886$$ 36.0000 1.20944
$$887$$ −16.0000 −0.537227 −0.268614 0.963248i $$-0.586566\pi$$
−0.268614 + 0.963248i $$0.586566\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ −24.0000 −0.804482
$$891$$ −9.00000 −0.301511
$$892$$ 2.00000 0.0669650
$$893$$ −12.0000 −0.401565
$$894$$ 0 0
$$895$$ 16.0000 0.534821
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 18.0000 0.600668
$$899$$ −4.00000 −0.133407
$$900$$ −33.0000 −1.10000
$$901$$ 24.0000 0.799556
$$902$$ −4.00000 −0.133185
$$903$$ 0 0
$$904$$ −14.0000 −0.465633
$$905$$ −80.0000 −2.65929
$$906$$ 0 0
$$907$$ −52.0000 −1.72663 −0.863316 0.504664i $$-0.831616\pi$$
−0.863316 + 0.504664i $$0.831616\pi$$
$$908$$ 2.00000 0.0663723
$$909$$ 18.0000 0.597022
$$910$$ 0 0
$$911$$ −36.0000 −1.19273 −0.596367 0.802712i $$-0.703390\pi$$
−0.596367 + 0.802712i $$0.703390\pi$$
$$912$$ 0 0
$$913$$ −6.00000 −0.198571
$$914$$ −2.00000 −0.0661541
$$915$$ 0 0
$$916$$ −20.0000 −0.660819
$$917$$ 0 0
$$918$$ 0 0
$$919$$ 24.0000 0.791687 0.395843 0.918318i $$-0.370452\pi$$
0.395843 + 0.918318i $$0.370452\pi$$
$$920$$ −16.0000 −0.527504
$$921$$ 0 0
$$922$$ 30.0000 0.987997
$$923$$ 16.0000 0.526646
$$924$$ 0 0
$$925$$ 110.000 3.61678
$$926$$ 32.0000 1.05159
$$927$$ 54.0000 1.77359
$$928$$ 2.00000 0.0656532
$$929$$ 30.0000 0.984268 0.492134 0.870519i $$-0.336217\pi$$
0.492134 + 0.870519i $$0.336217\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 30.0000 0.982683
$$933$$ 0 0
$$934$$ 0 0
$$935$$ −16.0000 −0.523256
$$936$$ −6.00000 −0.196116
$$937$$ −12.0000 −0.392023 −0.196011 0.980602i $$-0.562799\pi$$
−0.196011 + 0.980602i $$0.562799\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ −8.00000 −0.260931
$$941$$ 14.0000 0.456387 0.228193 0.973616i $$-0.426718\pi$$
0.228193 + 0.973616i $$0.426718\pi$$
$$942$$ 0 0
$$943$$ −16.0000 −0.521032
$$944$$ 12.0000 0.390567
$$945$$ 0 0
$$946$$ −8.00000 −0.260102
$$947$$ −4.00000 −0.129983 −0.0649913 0.997886i $$-0.520702\pi$$
−0.0649913 + 0.997886i $$0.520702\pi$$
$$948$$ 0 0
$$949$$ 8.00000 0.259691
$$950$$ −66.0000 −2.14132
$$951$$ 0 0
$$952$$ 0 0
$$953$$ −22.0000 −0.712650 −0.356325 0.934362i $$-0.615970\pi$$
−0.356325 + 0.934362i $$0.615970\pi$$
$$954$$ 18.0000 0.582772
$$955$$ −16.0000 −0.517748
$$956$$ 16.0000 0.517477
$$957$$ 0 0
$$958$$ −16.0000 −0.516937
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −27.0000 −0.870968
$$962$$ 20.0000 0.644826
$$963$$ 48.0000 1.54678
$$964$$ −12.0000 −0.386494
$$965$$ 8.00000 0.257529
$$966$$ 0 0
$$967$$ 32.0000 1.02905 0.514525 0.857475i $$-0.327968\pi$$
0.514525 + 0.857475i $$0.327968\pi$$
$$968$$ −1.00000 −0.0321412
$$969$$ 0 0
$$970$$ −56.0000 −1.79805
$$971$$ −56.0000 −1.79713 −0.898563 0.438845i $$-0.855388\pi$$
−0.898563 + 0.438845i $$0.855388\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 28.0000 0.897178
$$975$$ 0 0
$$976$$ 14.0000 0.448129
$$977$$ 2.00000 0.0639857 0.0319928 0.999488i $$-0.489815\pi$$
0.0319928 + 0.999488i $$0.489815\pi$$
$$978$$ 0 0
$$979$$ −6.00000 −0.191761
$$980$$ 0 0
$$981$$ 42.0000 1.34096
$$982$$ 36.0000 1.14881
$$983$$ −18.0000 −0.574111 −0.287055 0.957914i $$-0.592676\pi$$
−0.287055 + 0.957914i $$0.592676\pi$$
$$984$$ 0 0
$$985$$ 24.0000 0.764704
$$986$$ 8.00000 0.254772
$$987$$ 0 0
$$988$$ −12.0000 −0.381771
$$989$$ −32.0000 −1.01754
$$990$$ −12.0000 −0.381385
$$991$$ 16.0000 0.508257 0.254128 0.967170i $$-0.418211\pi$$
0.254128 + 0.967170i $$0.418211\pi$$
$$992$$ −2.00000 −0.0635001
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 56.0000 1.77532
$$996$$ 0 0
$$997$$ 42.0000 1.33015 0.665077 0.746775i $$-0.268399\pi$$
0.665077 + 0.746775i $$0.268399\pi$$
$$998$$ −44.0000 −1.39280
$$999$$ 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 1078.2.a.d.1.1 1
3.2 odd 2 9702.2.a.ba.1.1 1
4.3 odd 2 8624.2.a.r.1.1 1
7.2 even 3 1078.2.e.i.67.1 2
7.3 odd 6 1078.2.e.j.177.1 2
7.4 even 3 1078.2.e.i.177.1 2
7.5 odd 6 1078.2.e.j.67.1 2
7.6 odd 2 154.2.a.a.1.1 1
21.20 even 2 1386.2.a.l.1.1 1
28.27 even 2 1232.2.a.e.1.1 1
35.13 even 4 3850.2.c.j.1849.2 2
35.27 even 4 3850.2.c.j.1849.1 2
35.34 odd 2 3850.2.a.u.1.1 1
56.13 odd 2 4928.2.a.v.1.1 1
56.27 even 2 4928.2.a.w.1.1 1
77.76 even 2 1694.2.a.g.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
154.2.a.a.1.1 1 7.6 odd 2
1078.2.a.d.1.1 1 1.1 even 1 trivial
1078.2.e.i.67.1 2 7.2 even 3
1078.2.e.i.177.1 2 7.4 even 3
1078.2.e.j.67.1 2 7.5 odd 6
1078.2.e.j.177.1 2 7.3 odd 6
1232.2.a.e.1.1 1 28.27 even 2
1386.2.a.l.1.1 1 21.20 even 2
1694.2.a.g.1.1 1 77.76 even 2
3850.2.a.u.1.1 1 35.34 odd 2
3850.2.c.j.1849.1 2 35.27 even 4
3850.2.c.j.1849.2 2 35.13 even 4
4928.2.a.v.1.1 1 56.13 odd 2
4928.2.a.w.1.1 1 56.27 even 2
8624.2.a.r.1.1 1 4.3 odd 2
9702.2.a.ba.1.1 1 3.2 odd 2