# Properties

 Label 1694.2.a.g.1.1 Level $1694$ Weight $2$ Character 1694.1 Self dual yes Analytic conductor $13.527$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1694 = 2 \cdot 7 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1694.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$13.5266581024$$ 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 $$\chi$$ $$=$$ 1694.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} +1.00000 q^{4} -4.00000 q^{5} +1.00000 q^{7} +1.00000 q^{8} -3.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} +1.00000 q^{4} -4.00000 q^{5} +1.00000 q^{7} +1.00000 q^{8} -3.00000 q^{9} -4.00000 q^{10} -2.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} +4.00000 q^{17} -3.00000 q^{18} +6.00000 q^{19} -4.00000 q^{20} +4.00000 q^{23} +11.0000 q^{25} -2.00000 q^{26} +1.00000 q^{28} +2.00000 q^{29} -2.00000 q^{31} +1.00000 q^{32} +4.00000 q^{34} -4.00000 q^{35} -3.00000 q^{36} +10.0000 q^{37} +6.00000 q^{38} -4.00000 q^{40} -4.00000 q^{41} +8.00000 q^{43} +12.0000 q^{45} +4.00000 q^{46} +2.00000 q^{47} +1.00000 q^{49} +11.0000 q^{50} -2.00000 q^{52} +6.00000 q^{53} +1.00000 q^{56} +2.00000 q^{58} -12.0000 q^{59} +14.0000 q^{61} -2.00000 q^{62} -3.00000 q^{63} +1.00000 q^{64} +8.00000 q^{65} -12.0000 q^{67} +4.00000 q^{68} -4.00000 q^{70} -8.00000 q^{71} -3.00000 q^{72} -4.00000 q^{73} +10.0000 q^{74} +6.00000 q^{76} -4.00000 q^{80} +9.00000 q^{81} -4.00000 q^{82} +6.00000 q^{83} -16.0000 q^{85} +8.00000 q^{86} -6.00000 q^{89} +12.0000 q^{90} -2.00000 q^{91} +4.00000 q^{92} +2.00000 q^{94} -24.0000 q^{95} -14.0000 q^{97} +1.00000 q^{98} +O(q^{100})$$

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.00000 0.707107
$$3$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$4$$ 1.00000 0.500000
$$5$$ −4.00000 −1.78885 −0.894427 0.447214i $$-0.852416\pi$$
−0.894427 + 0.447214i $$0.852416\pi$$
$$6$$ 0 0
$$7$$ 1.00000 0.377964
$$8$$ 1.00000 0.353553
$$9$$ −3.00000 −1.00000
$$10$$ −4.00000 −1.26491
$$11$$ 0 0
$$12$$ 0 0
$$13$$ −2.00000 −0.554700 −0.277350 0.960769i $$-0.589456\pi$$
−0.277350 + 0.960769i $$0.589456\pi$$
$$14$$ 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 $$0.338794\pi$$
$$18$$ −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
$$22$$ 0 0
$$23$$ 4.00000 0.834058 0.417029 0.908893i $$-0.363071\pi$$
0.417029 + 0.908893i $$0.363071\pi$$
$$24$$ 0 0
$$25$$ 11.0000 2.20000
$$26$$ −2.00000 −0.392232
$$27$$ 0 0
$$28$$ 1.00000 0.188982
$$29$$ 2.00000 0.371391 0.185695 0.982607i $$-0.440546\pi$$
0.185695 + 0.982607i $$0.440546\pi$$
$$30$$ 0 0
$$31$$ −2.00000 −0.359211 −0.179605 0.983739i $$-0.557482\pi$$
−0.179605 + 0.983739i $$0.557482\pi$$
$$32$$ 1.00000 0.176777
$$33$$ 0 0
$$34$$ 4.00000 0.685994
$$35$$ −4.00000 −0.676123
$$36$$ −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 $$-0.291172\pi$$
0.609994 + 0.792406i $$0.291172\pi$$
$$44$$ 0 0
$$45$$ 12.0000 1.78885
$$46$$ 4.00000 0.589768
$$47$$ 2.00000 0.291730 0.145865 0.989305i $$-0.453403\pi$$
0.145865 + 0.989305i $$0.453403\pi$$
$$48$$ 0 0
$$49$$ 1.00000 0.142857
$$50$$ 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
$$55$$ 0 0
$$56$$ 1.00000 0.133631
$$57$$ 0 0
$$58$$ 2.00000 0.262613
$$59$$ −12.0000 −1.56227 −0.781133 0.624364i $$-0.785358\pi$$
−0.781133 + 0.624364i $$0.785358\pi$$
$$60$$ 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
$$63$$ −3.00000 −0.377964
$$64$$ 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
$$70$$ −4.00000 −0.478091
$$71$$ −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
$$88$$ 0 0
$$89$$ −6.00000 −0.635999 −0.317999 0.948091i $$-0.603011\pi$$
−0.317999 + 0.948091i $$0.603011\pi$$
$$90$$ 12.0000 1.26491
$$91$$ −2.00000 −0.209657
$$92$$ 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 $$-0.751641\pi$$
−0.710742 + 0.703452i $$0.751641\pi$$
$$98$$ 1.00000 0.101015
$$99$$ 0 0
$$100$$ 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 $$-0.152926\pi$$
0.886796 + 0.462160i $$0.152926\pi$$
$$104$$ −2.00000 −0.196116
$$105$$ 0 0
$$106$$ 6.00000 0.582772
$$107$$ 16.0000 1.54678 0.773389 0.633932i $$-0.218560\pi$$
0.773389 + 0.633932i $$0.218560\pi$$
$$108$$ 0 0
$$109$$ 14.0000 1.34096 0.670478 0.741929i $$-0.266089\pi$$
0.670478 + 0.741929i $$0.266089\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 1.00000 0.0944911
$$113$$ 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
$$119$$ 4.00000 0.366679
$$120$$ 0 0
$$121$$ 0 0
$$122$$ 14.0000 1.26750
$$123$$ 0 0
$$124$$ −2.00000 −0.179605
$$125$$ −24.0000 −2.14663
$$126$$ −3.00000 −0.267261
$$127$$ −8.00000 −0.709885 −0.354943 0.934888i $$-0.615500\pi$$
−0.354943 + 0.934888i $$0.615500\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ 0 0
$$130$$ 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
$$133$$ 6.00000 0.520266
$$134$$ −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 $$0.702346\pi$$
$$140$$ −4.00000 −0.338062
$$141$$ 0 0
$$142$$ −8.00000 −0.671345
$$143$$ 0 0
$$144$$ −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 $$-0.526106\pi$$
−0.0819232 + 0.996639i $$0.526106\pi$$
$$150$$ 0 0
$$151$$ 24.0000 1.95309 0.976546 0.215308i $$-0.0690756\pi$$
0.976546 + 0.215308i $$0.0690756\pi$$
$$152$$ 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 $$-0.603426\pi$$
−0.319235 + 0.947676i $$0.603426\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ −4.00000 −0.316228
$$161$$ 4.00000 0.315244
$$162$$ 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
$$175$$ 11.0000 0.831522
$$176$$ 0 0
$$177$$ 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 $$-0.233262\pi$$
0.743294 + 0.668965i $$0.233262\pi$$
$$182$$ −2.00000 −0.148250
$$183$$ 0 0
$$184$$ 4.00000 0.294884
$$185$$ −40.0000 −2.94086
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 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 $$-0.522932\pi$$
−0.0719816 + 0.997406i $$0.522932\pi$$
$$194$$ −14.0000 −1.00514
$$195$$ 0 0
$$196$$ 1.00000 0.0714286
$$197$$ −6.00000 −0.427482 −0.213741 0.976890i $$-0.568565\pi$$
−0.213741 + 0.976890i $$0.568565\pi$$
$$198$$ 0 0
$$199$$ −14.0000 −0.992434 −0.496217 0.868199i $$-0.665278\pi$$
−0.496217 + 0.868199i $$0.665278\pi$$
$$200$$ 11.0000 0.777817
$$201$$ 0 0
$$202$$ −6.00000 −0.422159
$$203$$ 2.00000 0.140372
$$204$$ 0 0
$$205$$ 16.0000 1.11749
$$206$$ 18.0000 1.25412
$$207$$ −12.0000 −0.834058
$$208$$ −2.00000 −0.138675
$$209$$ 0 0
$$210$$ 0 0
$$211$$ 8.00000 0.550743 0.275371 0.961338i $$-0.411199\pi$$
0.275371 + 0.961338i $$0.411199\pi$$
$$212$$ 6.00000 0.412082
$$213$$ 0 0
$$214$$ 16.0000 1.09374
$$215$$ −32.0000 −2.18238
$$216$$ 0 0
$$217$$ −2.00000 −0.135769
$$218$$ 14.0000 0.948200
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −8.00000 −0.538138
$$222$$ 0 0
$$223$$ −2.00000 −0.133930 −0.0669650 0.997755i $$-0.521332\pi$$
−0.0669650 + 0.997755i $$0.521332\pi$$
$$224$$ 1.00000 0.0668153
$$225$$ −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 $$-0.270209\pi$$
0.660819 + 0.750546i $$0.270209\pi$$
$$230$$ −16.0000 −1.05501
$$231$$ 0 0
$$232$$ 2.00000 0.131306
$$233$$ −30.0000 −1.96537 −0.982683 0.185296i $$-0.940675\pi$$
−0.982683 + 0.185296i $$0.940675\pi$$
$$234$$ 6.00000 0.392232
$$235$$ −8.00000 −0.521862
$$236$$ −12.0000 −0.781133
$$237$$ 0 0
$$238$$ 4.00000 0.259281
$$239$$ −16.0000 −1.03495 −0.517477 0.855697i $$-0.673129\pi$$
−0.517477 + 0.855697i $$0.673129\pi$$
$$240$$ 0 0
$$241$$ −12.0000 −0.772988 −0.386494 0.922292i $$-0.626314\pi$$
−0.386494 + 0.922292i $$0.626314\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 14.0000 0.896258
$$245$$ −4.00000 −0.255551
$$246$$ 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 $$-0.376365\pi$$
0.378717 + 0.925513i $$0.376365\pi$$
$$252$$ −3.00000 −0.188982
$$253$$ 0 0
$$254$$ −8.00000 −0.501965
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ −6.00000 −0.374270 −0.187135 0.982334i $$-0.559920\pi$$
−0.187135 + 0.982334i $$0.559920\pi$$
$$258$$ 0 0
$$259$$ 10.0000 0.621370
$$260$$ 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
$$266$$ 6.00000 0.367884
$$267$$ 0 0
$$268$$ −12.0000 −0.733017
$$269$$ 12.0000 0.731653 0.365826 0.930683i $$-0.380786\pi$$
0.365826 + 0.930683i $$0.380786\pi$$
$$270$$ 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
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 30.0000 1.80253 0.901263 0.433273i $$-0.142641\pi$$
0.901263 + 0.433273i $$0.142641\pi$$
$$278$$ −14.0000 −0.839664
$$279$$ 6.00000 0.359211
$$280$$ −4.00000 −0.239046
$$281$$ 10.0000 0.596550 0.298275 0.954480i $$-0.403589\pi$$
0.298275 + 0.954480i $$0.403589\pi$$
$$282$$ 0 0
$$283$$ −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
$$286$$ 0 0
$$287$$ −4.00000 −0.236113
$$288$$ −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
$$301$$ 8.00000 0.461112
$$302$$ 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 $$-0.370074\pi$$
0.396934 + 0.917847i $$0.370074\pi$$
$$312$$ 0 0
$$313$$ −2.00000 −0.113047 −0.0565233 0.998401i $$-0.518002\pi$$
−0.0565233 + 0.998401i $$0.518002\pi$$
$$314$$ −8.00000 −0.451466
$$315$$ 12.0000 0.676123
$$316$$ 0 0
$$317$$ 6.00000 0.336994 0.168497 0.985702i $$-0.446109\pi$$
0.168497 + 0.985702i $$0.446109\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ −4.00000 −0.223607
$$321$$ 0 0
$$322$$ 4.00000 0.222911
$$323$$ 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
$$329$$ 2.00000 0.110264
$$330$$ 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 $$-0.336901\pi$$
0.490261 + 0.871576i $$0.336901\pi$$
$$338$$ −9.00000 −0.489535
$$339$$ 0 0
$$340$$ −16.0000 −0.867722
$$341$$ 0 0
$$342$$ −18.0000 −0.973329
$$343$$ 1.00000 0.0539949
$$344$$ 8.00000 0.431331
$$345$$ 0 0
$$346$$ 14.0000 0.752645
$$347$$ −8.00000 −0.429463 −0.214731 0.976673i $$-0.568888\pi$$
−0.214731 + 0.976673i $$0.568888\pi$$
$$348$$ 0 0
$$349$$ 10.0000 0.535288 0.267644 0.963518i $$-0.413755\pi$$
0.267644 + 0.963518i $$0.413755\pi$$
$$350$$ 11.0000 0.587975
$$351$$ 0 0
$$352$$ 0 0
$$353$$ −6.00000 −0.319348 −0.159674 0.987170i $$-0.551044\pi$$
−0.159674 + 0.987170i $$0.551044\pi$$
$$354$$ 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 $$-0.361250\pi$$
0.422224 + 0.906492i $$0.361250\pi$$
$$360$$ 12.0000 0.632456
$$361$$ 17.0000 0.894737
$$362$$ 20.0000 1.05118
$$363$$ 0 0
$$364$$ −2.00000 −0.104828
$$365$$ 16.0000 0.837478
$$366$$ 0 0
$$367$$ 22.0000 1.14839 0.574195 0.818718i $$-0.305315\pi$$
0.574195 + 0.818718i $$0.305315\pi$$
$$368$$ 4.00000 0.208514
$$369$$ 12.0000 0.624695
$$370$$ −40.0000 −2.07950
$$371$$ 6.00000 0.311504
$$372$$ 0 0
$$373$$ 10.0000 0.517780 0.258890 0.965907i $$-0.416643\pi$$
0.258890 + 0.965907i $$0.416643\pi$$
$$374$$ 0 0
$$375$$ 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 $$-0.582236\pi$$
−0.255488 + 0.966812i $$0.582236\pi$$
$$384$$ 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
$$392$$ 1.00000 0.0505076
$$393$$ 0 0
$$394$$ −6.00000 −0.302276
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 24.0000 1.20453 0.602263 0.798298i $$-0.294266\pi$$
0.602263 + 0.798298i $$0.294266\pi$$
$$398$$ −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
$$406$$ 2.00000 0.0992583
$$407$$ 0 0
$$408$$ 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
$$413$$ −12.0000 −0.590481
$$414$$ −12.0000 −0.589768
$$415$$ −24.0000 −1.17811
$$416$$ −2.00000 −0.0980581
$$417$$ 0 0
$$418$$ 0 0
$$419$$ −32.0000 −1.56330 −0.781651 0.623716i $$-0.785622\pi$$
−0.781651 + 0.623716i $$0.785622\pi$$
$$420$$ 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
$$427$$ 14.0000 0.677507
$$428$$ 16.0000 0.773389
$$429$$ 0 0
$$430$$ −32.0000 −1.54318
$$431$$ −16.0000 −0.770693 −0.385346 0.922772i $$-0.625918\pi$$
−0.385346 + 0.922772i $$0.625918\pi$$
$$432$$ 0 0
$$433$$ 2.00000 0.0961139 0.0480569 0.998845i $$-0.484697\pi$$
0.0480569 + 0.998845i $$0.484697\pi$$
$$434$$ −2.00000 −0.0960031
$$435$$ 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 $$0.732928\pi$$
$$440$$ 0 0
$$441$$ −3.00000 −0.142857
$$442$$ −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
$$448$$ 1.00000 0.0472456
$$449$$ −18.0000 −0.849473 −0.424736 0.905317i $$-0.639633\pi$$
−0.424736 + 0.905317i $$0.639633\pi$$
$$450$$ −33.0000 −1.55563
$$451$$ 0 0
$$452$$ 14.0000 0.658505
$$453$$ 0 0
$$454$$ 2.00000 0.0938647
$$455$$ 8.00000 0.375046
$$456$$ 0 0
$$457$$ −2.00000 −0.0935561 −0.0467780 0.998905i $$-0.514895\pi$$
−0.0467780 + 0.998905i $$0.514895\pi$$
$$458$$ 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
$$469$$ −12.0000 −0.554109
$$470$$ −8.00000 −0.369012
$$471$$ 0 0
$$472$$ −12.0000 −0.552345
$$473$$ 0 0
$$474$$ 0 0
$$475$$ 66.0000 3.02829
$$476$$ 4.00000 0.183340
$$477$$ −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
$$484$$ 0 0
$$485$$ 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
$$490$$ −4.00000 −0.180702
$$491$$ 36.0000 1.62466 0.812329 0.583200i $$-0.198200\pi$$
0.812329 + 0.583200i $$0.198200\pi$$
$$492$$ 0 0
$$493$$ 8.00000 0.360302
$$494$$ −12.0000 −0.539906
$$495$$ 0 0
$$496$$ −2.00000 −0.0898027
$$497$$ −8.00000 −0.358849
$$498$$ 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 $$0.203460\pi$$
$$504$$ −3.00000 −0.133631
$$505$$ 24.0000 1.06799
$$506$$ 0 0
$$507$$ 0 0
$$508$$ −8.00000 −0.354943
$$509$$ −28.0000 −1.24108 −0.620539 0.784176i $$-0.713086\pi$$
−0.620539 + 0.784176i $$0.713086\pi$$
$$510$$ 0 0
$$511$$ −4.00000 −0.176950
$$512$$ 1.00000 0.0441942
$$513$$ 0 0
$$514$$ −6.00000 −0.264649
$$515$$ −72.0000 −3.17270
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 10.0000 0.439375
$$519$$ 0 0
$$520$$ 8.00000 0.350823
$$521$$ 10.0000 0.438108 0.219054 0.975713i $$-0.429703\pi$$
0.219054 + 0.975713i $$0.429703\pi$$
$$522$$ −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
$$532$$ 6.00000 0.260133
$$533$$ 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 $$-0.597305\pi$$
−0.300954 + 0.953639i $$0.597305\pi$$
$$542$$ 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 $$-0.417417\pi$$
0.256541 + 0.966533i $$0.417417\pi$$
$$548$$ 6.00000 0.256307
$$549$$ −42.0000 −1.79252
$$550$$ 0 0
$$551$$ 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 $$-0.595853\pi$$
−0.296600 + 0.955002i $$0.595853\pi$$
$$558$$ 6.00000 0.254000
$$559$$ −16.0000 −0.676728
$$560$$ −4.00000 −0.169031
$$561$$ 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
$$567$$ 9.00000 0.377964
$$568$$ −8.00000 −0.335673
$$569$$ −6.00000 −0.251533 −0.125767 0.992060i $$-0.540139\pi$$
−0.125767 + 0.992060i $$0.540139\pi$$
$$570$$ 0 0
$$571$$ −28.0000 −1.17176 −0.585882 0.810397i $$-0.699252\pi$$
−0.585882 + 0.810397i $$0.699252\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ −4.00000 −0.166957
$$575$$ 44.0000 1.83493
$$576$$ −3.00000 −0.125000
$$577$$ 14.0000 0.582828 0.291414 0.956597i $$-0.405874\pi$$
0.291414 + 0.956597i $$0.405874\pi$$
$$578$$ −1.00000 −0.0415945
$$579$$ 0 0
$$580$$ −8.00000 −0.332182
$$581$$ 6.00000 0.248922
$$582$$ 0 0
$$583$$ 0 0
$$584$$ −4.00000 −0.165521
$$585$$ −24.0000 −0.992278
$$586$$ −18.0000 −0.743573
$$587$$ −12.0000 −0.495293 −0.247647 0.968850i $$-0.579657\pi$$
−0.247647 + 0.968850i $$0.579657\pi$$
$$588$$ 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
$$595$$ −16.0000 −0.655936
$$596$$ −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 $$0.552170\pi$$
$$602$$ 8.00000 0.326056
$$603$$ 36.0000 1.46603
$$604$$ 24.0000 0.976546
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −8.00000 −0.324710 −0.162355 0.986732i $$-0.551909\pi$$
−0.162355 + 0.986732i $$0.551909\pi$$
$$608$$ 6.00000 0.243332
$$609$$ 0 0
$$610$$ −56.0000 −2.26737
$$611$$ −4.00000 −0.161823
$$612$$ −12.0000 −0.485071
$$613$$ −46.0000 −1.85792 −0.928961 0.370177i $$-0.879297\pi$$
−0.928961 + 0.370177i $$0.879297\pi$$
$$614$$ 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 $$-0.448601\pi$$
0.160774 + 0.986991i $$0.448601\pi$$
$$620$$ 8.00000 0.321288
$$621$$ 0 0
$$622$$ 14.0000 0.561349
$$623$$ −6.00000 −0.240385
$$624$$ 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
$$630$$ 12.0000 0.478091
$$631$$ −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
$$637$$ −2.00000 −0.0792429
$$638$$ 0 0
$$639$$ 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 $$-0.474868\pi$$
0.0788723 + 0.996885i $$0.474868\pi$$
$$644$$ 4.00000 0.157622
$$645$$ 0 0
$$646$$ 24.0000 0.944267
$$647$$ 6.00000 0.235884 0.117942 0.993020i $$-0.462370\pi$$
0.117942 + 0.993020i $$0.462370\pi$$
$$648$$ 9.00000 0.353553
$$649$$ 0 0
$$650$$ −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
$$658$$ 2.00000 0.0779681
$$659$$ 8.00000 0.311636 0.155818 0.987786i $$-0.450199\pi$$
0.155818 + 0.987786i $$0.450199\pi$$
$$660$$ 0 0
$$661$$ 20.0000 0.777910 0.388955 0.921257i $$-0.372836\pi$$
0.388955 + 0.921257i $$0.372836\pi$$
$$662$$ −20.0000 −0.777322
$$663$$ 0 0
$$664$$ 6.00000 0.232845
$$665$$ −24.0000 −0.930680
$$666$$ −30.0000 −1.16248
$$667$$ 8.00000 0.309761
$$668$$ −4.00000 −0.154765
$$669$$ 0 0
$$670$$ 48.0000 1.85440
$$671$$ 0 0
$$672$$ 0 0
$$673$$ −22.0000 −0.848038 −0.424019 0.905653i $$-0.639381\pi$$
−0.424019 + 0.905653i $$0.639381\pi$$
$$674$$ 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
$$679$$ −14.0000 −0.537271
$$680$$ −16.0000 −0.613572
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 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
$$686$$ 1.00000 0.0381802
$$687$$ 0 0
$$688$$ 8.00000 0.304997
$$689$$ −12.0000 −0.457164
$$690$$ 0 0
$$691$$ 36.0000 1.36950 0.684752 0.728776i $$-0.259910\pi$$
0.684752 + 0.728776i $$0.259910\pi$$
$$692$$ 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
$$700$$ 11.0000 0.415761
$$701$$ 30.0000 1.13308 0.566542 0.824033i $$-0.308281\pi$$
0.566542 + 0.824033i $$0.308281\pi$$
$$702$$ 0 0
$$703$$ 60.0000 2.26294
$$704$$ 0 0
$$705$$ 0 0
$$706$$ −6.00000 −0.225813
$$707$$ −6.00000 −0.225653
$$708$$ 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
$$715$$ 0 0
$$716$$ 4.00000 0.149487
$$717$$ 0 0
$$718$$ 16.0000 0.597115
$$719$$ −26.0000 −0.969636 −0.484818 0.874615i $$-0.661114\pi$$
−0.484818 + 0.874615i $$0.661114\pi$$
$$720$$ 12.0000 0.447214
$$721$$ 18.0000 0.670355
$$722$$ 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 $$-0.559371\pi$$
−0.185440 + 0.982656i $$0.559371\pi$$
$$728$$ −2.00000 −0.0741249
$$729$$ −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
$$737$$ 0 0
$$738$$ 12.0000 0.441726
$$739$$ 12.0000 0.441427 0.220714 0.975339i $$-0.429161\pi$$
0.220714 + 0.975339i $$0.429161\pi$$
$$740$$ −40.0000 −1.47043
$$741$$ 0 0
$$742$$ 6.00000 0.220267
$$743$$ 24.0000 0.880475 0.440237 0.897881i $$-0.354894\pi$$
0.440237 + 0.897881i $$0.354894\pi$$
$$744$$ 0 0
$$745$$ 8.00000 0.293097
$$746$$ 10.0000 0.366126
$$747$$ −18.0000 −0.658586
$$748$$ 0 0
$$749$$ 16.0000 0.584627
$$750$$ 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
$$763$$ 14.0000 0.506834
$$764$$ −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 $$-0.831556\pi$$
−0.863220 + 0.504828i $$0.831556\pi$$
$$774$$ −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
$$781$$ 0 0
$$782$$ 16.0000 0.572159
$$783$$ 0 0
$$784$$ 1.00000 0.0357143
$$785$$ 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
$$791$$ 14.0000 0.497783
$$792$$ 0 0
$$793$$ −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 $$-0.591454\pi$$
−0.283375 + 0.959009i $$0.591454\pi$$
$$798$$ 0 0
$$799$$ 8.00000 0.283020
$$800$$ 11.0000 0.388909
$$801$$ 18.0000 0.635999
$$802$$ −18.0000 −0.635602
$$803$$ 0 0
$$804$$ 0 0
$$805$$ −16.0000 −0.563926
$$806$$ 4.00000 0.140894
$$807$$ 0 0
$$808$$ −6.00000 −0.211079
$$809$$ 30.0000 1.05474 0.527372 0.849635i $$-0.323177\pi$$
0.527372 + 0.849635i $$0.323177\pi$$
$$810$$ −36.0000 −1.26491
$$811$$ −38.0000 −1.33436 −0.667180 0.744896i $$-0.732499\pi$$
−0.667180 + 0.744896i $$0.732499\pi$$
$$812$$ 2.00000 0.0701862
$$813$$ 0 0
$$814$$ 0 0
$$815$$ −16.0000 −0.560456
$$816$$ 0 0
$$817$$ 48.0000 1.67931
$$818$$ −16.0000 −0.559427
$$819$$ 6.00000 0.209657
$$820$$ 16.0000 0.558744
$$821$$ −18.0000 −0.628204 −0.314102 0.949389i $$-0.601703\pi$$
−0.314102 + 0.949389i $$0.601703\pi$$
$$822$$ 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
$$826$$ −12.0000 −0.417533
$$827$$ −20.0000 −0.695468 −0.347734 0.937593i $$-0.613049\pi$$
−0.347734 + 0.937593i $$0.613049\pi$$
$$828$$ −12.0000 −0.417029
$$829$$ −20.0000 −0.694629 −0.347314 0.937749i $$-0.612906\pi$$
−0.347314 + 0.937749i $$0.612906\pi$$
$$830$$ −24.0000 −0.833052
$$831$$ 0 0
$$832$$ −2.00000 −0.0693375
$$833$$ 4.00000 0.138592
$$834$$ 0 0
$$835$$ 16.0000 0.553703
$$836$$ 0 0
$$837$$ 0 0
$$838$$ −32.0000 −1.10542
$$839$$ 30.0000 1.03572 0.517858 0.855467i $$-0.326730\pi$$
0.517858 + 0.855467i $$0.326730\pi$$
$$840$$ 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 $$0.510901\pi$$
$$854$$ 14.0000 0.479070
$$855$$ 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 $$-0.341480\pi$$
0.477674 + 0.878537i $$0.341480\pi$$
$$860$$ −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
$$868$$ −2.00000 −0.0678844
$$869$$ 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
$$875$$ −24.0000 −0.811348
$$876$$ 0 0
$$877$$ −34.0000 −1.14810 −0.574049 0.818821i $$-0.694628\pi$$
−0.574049 + 0.818821i $$0.694628\pi$$
$$878$$ −28.0000 −0.944954
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 30.0000 1.01073 0.505363 0.862907i $$-0.331359\pi$$
0.505363 + 0.862907i $$0.331359\pi$$
$$882$$ −3.00000 −0.101015
$$883$$ −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
$$889$$ −8.00000 −0.268311
$$890$$ 24.0000 0.804482
$$891$$ 0 0
$$892$$ −2.00000 −0.0669650
$$893$$ 12.0000 0.401565
$$894$$ 0 0
$$895$$ −16.0000 −0.534821
$$896$$ 1.00000 0.0334077
$$897$$ 0 0
$$898$$ −18.0000 −0.600668
$$899$$ −4.00000 −0.133407
$$900$$ −33.0000 −1.10000
$$901$$ 24.0000 0.799556
$$902$$ 0 0
$$903$$ 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
$$910$$ 8.00000 0.265197
$$911$$ −36.0000 −1.19273 −0.596367 0.802712i $$-0.703390\pi$$
−0.596367 + 0.802712i $$0.703390\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ −2.00000 −0.0661541
$$915$$ 0 0
$$916$$ 20.0000 0.660819
$$917$$ −6.00000 −0.198137
$$918$$ 0 0
$$919$$ −24.0000 −0.791687 −0.395843 0.918318i $$-0.629548\pi$$
−0.395843 + 0.918318i $$0.629548\pi$$
$$920$$ −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 $$-0.663783\pi$$
−0.492134 + 0.870519i $$0.663783\pi$$
$$930$$ 0 0
$$931$$ 6.00000 0.196642
$$932$$ −30.0000 −0.982683
$$933$$ 0 0
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 6.00000 0.196116
$$937$$ −12.0000 −0.392023 −0.196011 0.980602i $$-0.562799\pi$$
−0.196011 + 0.980602i $$0.562799\pi$$
$$938$$ −12.0000 −0.391814
$$939$$ 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
$$946$$ 0 0
$$947$$ −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
$$952$$ 4.00000 0.129641
$$953$$ 22.0000 0.712650 0.356325 0.934362i $$-0.384030\pi$$
0.356325 + 0.934362i $$0.384030\pi$$
$$954$$ −18.0000 −0.582772
$$955$$ 16.0000 0.517748
$$956$$ −16.0000 −0.517477
$$957$$ 0 0
$$958$$ 16.0000 0.516937
$$959$$ 6.00000 0.193750
$$960$$ 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 $$-0.672032\pi$$
−0.514525 + 0.857475i $$0.672032\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 56.0000 1.79805
$$971$$ 56.0000 1.79713 0.898563 0.438845i $$-0.144612\pi$$
0.898563 + 0.438845i $$0.144612\pi$$
$$972$$ 0 0
$$973$$ −14.0000 −0.448819
$$974$$ −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
$$979$$ 0 0
$$980$$ −4.00000 −0.127775
$$981$$ −42.0000 −1.34096
$$982$$ 36.0000 1.14881
$$983$$ 18.0000 0.574111 0.287055 0.957914i $$-0.407324\pi$$
0.287055 + 0.957914i $$0.407324\pi$$
$$984$$ 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
$$990$$ 0 0
$$991$$ 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
$$994$$ −8.00000 −0.253745
$$995$$ 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 1694.2.a.g.1.1 1
11.10 odd 2 154.2.a.a.1.1 1
33.32 even 2 1386.2.a.l.1.1 1
44.43 even 2 1232.2.a.e.1.1 1
55.32 even 4 3850.2.c.j.1849.1 2
55.43 even 4 3850.2.c.j.1849.2 2
55.54 odd 2 3850.2.a.u.1.1 1
77.10 even 6 1078.2.e.i.177.1 2
77.32 odd 6 1078.2.e.j.177.1 2
77.54 even 6 1078.2.e.i.67.1 2
77.65 odd 6 1078.2.e.j.67.1 2
77.76 even 2 1078.2.a.d.1.1 1
88.21 odd 2 4928.2.a.v.1.1 1
88.43 even 2 4928.2.a.w.1.1 1
231.230 odd 2 9702.2.a.ba.1.1 1
308.307 odd 2 8624.2.a.r.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
154.2.a.a.1.1 1 11.10 odd 2
1078.2.a.d.1.1 1 77.76 even 2
1078.2.e.i.67.1 2 77.54 even 6
1078.2.e.i.177.1 2 77.10 even 6
1078.2.e.j.67.1 2 77.65 odd 6
1078.2.e.j.177.1 2 77.32 odd 6
1232.2.a.e.1.1 1 44.43 even 2
1386.2.a.l.1.1 1 33.32 even 2
1694.2.a.g.1.1 1 1.1 even 1 trivial
3850.2.a.u.1.1 1 55.54 odd 2
3850.2.c.j.1849.1 2 55.32 even 4
3850.2.c.j.1849.2 2 55.43 even 4
4928.2.a.v.1.1 1 88.21 odd 2
4928.2.a.w.1.1 1 88.43 even 2
8624.2.a.r.1.1 1 308.307 odd 2
9702.2.a.ba.1.1 1 231.230 odd 2