# Properties

 Label 525.2.d.a.274.2 Level $525$ Weight $2$ Character 525.274 Analytic conductor $4.192$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$525 = 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 525.d (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.19214610612$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 21) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 274.2 Root $$1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 525.274 Dual form 525.2.d.a.274.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000i q^{2} +1.00000i q^{3} +1.00000 q^{4} -1.00000 q^{6} +1.00000i q^{7} +3.00000i q^{8} -1.00000 q^{9} +O(q^{10})$$ $$q+1.00000i q^{2} +1.00000i q^{3} +1.00000 q^{4} -1.00000 q^{6} +1.00000i q^{7} +3.00000i q^{8} -1.00000 q^{9} +4.00000 q^{11} +1.00000i q^{12} -2.00000i q^{13} -1.00000 q^{14} -1.00000 q^{16} +6.00000i q^{17} -1.00000i q^{18} -4.00000 q^{19} -1.00000 q^{21} +4.00000i q^{22} -3.00000 q^{24} +2.00000 q^{26} -1.00000i q^{27} +1.00000i q^{28} +2.00000 q^{29} +5.00000i q^{32} +4.00000i q^{33} -6.00000 q^{34} -1.00000 q^{36} -6.00000i q^{37} -4.00000i q^{38} +2.00000 q^{39} +2.00000 q^{41} -1.00000i q^{42} -4.00000i q^{43} +4.00000 q^{44} -1.00000i q^{48} -1.00000 q^{49} -6.00000 q^{51} -2.00000i q^{52} +6.00000i q^{53} +1.00000 q^{54} -3.00000 q^{56} -4.00000i q^{57} +2.00000i q^{58} -12.0000 q^{59} -2.00000 q^{61} -1.00000i q^{63} -7.00000 q^{64} -4.00000 q^{66} -4.00000i q^{67} +6.00000i q^{68} -3.00000i q^{72} -6.00000i q^{73} +6.00000 q^{74} -4.00000 q^{76} +4.00000i q^{77} +2.00000i q^{78} +16.0000 q^{79} +1.00000 q^{81} +2.00000i q^{82} -12.0000i q^{83} -1.00000 q^{84} +4.00000 q^{86} +2.00000i q^{87} +12.0000i q^{88} +14.0000 q^{89} +2.00000 q^{91} -5.00000 q^{96} -18.0000i q^{97} -1.00000i q^{98} -4.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{4} - 2q^{6} - 2q^{9} + O(q^{10})$$ $$2q + 2q^{4} - 2q^{6} - 2q^{9} + 8q^{11} - 2q^{14} - 2q^{16} - 8q^{19} - 2q^{21} - 6q^{24} + 4q^{26} + 4q^{29} - 12q^{34} - 2q^{36} + 4q^{39} + 4q^{41} + 8q^{44} - 2q^{49} - 12q^{51} + 2q^{54} - 6q^{56} - 24q^{59} - 4q^{61} - 14q^{64} - 8q^{66} + 12q^{74} - 8q^{76} + 32q^{79} + 2q^{81} - 2q^{84} + 8q^{86} + 28q^{89} + 4q^{91} - 10q^{96} - 8q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/525\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$176$$ $$451$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.00000i 0.707107i 0.935414 + 0.353553i $$0.115027\pi$$
−0.935414 + 0.353553i $$0.884973\pi$$
$$3$$ 1.00000i 0.577350i
$$4$$ 1.00000 0.500000
$$5$$ 0 0
$$6$$ −1.00000 −0.408248
$$7$$ 1.00000i 0.377964i
$$8$$ 3.00000i 1.06066i
$$9$$ −1.00000 −0.333333
$$10$$ 0 0
$$11$$ 4.00000 1.20605 0.603023 0.797724i $$-0.293963\pi$$
0.603023 + 0.797724i $$0.293963\pi$$
$$12$$ 1.00000i 0.288675i
$$13$$ − 2.00000i − 0.554700i −0.960769 0.277350i $$-0.910544\pi$$
0.960769 0.277350i $$-0.0894562\pi$$
$$14$$ −1.00000 −0.267261
$$15$$ 0 0
$$16$$ −1.00000 −0.250000
$$17$$ 6.00000i 1.45521i 0.685994 + 0.727607i $$0.259367\pi$$
−0.685994 + 0.727607i $$0.740633\pi$$
$$18$$ − 1.00000i − 0.235702i
$$19$$ −4.00000 −0.917663 −0.458831 0.888523i $$-0.651732\pi$$
−0.458831 + 0.888523i $$0.651732\pi$$
$$20$$ 0 0
$$21$$ −1.00000 −0.218218
$$22$$ 4.00000i 0.852803i
$$23$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$24$$ −3.00000 −0.612372
$$25$$ 0 0
$$26$$ 2.00000 0.392232
$$27$$ − 1.00000i − 0.192450i
$$28$$ 1.00000i 0.188982i
$$29$$ 2.00000 0.371391 0.185695 0.982607i $$-0.440546\pi$$
0.185695 + 0.982607i $$0.440546\pi$$
$$30$$ 0 0
$$31$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$32$$ 5.00000i 0.883883i
$$33$$ 4.00000i 0.696311i
$$34$$ −6.00000 −1.02899
$$35$$ 0 0
$$36$$ −1.00000 −0.166667
$$37$$ − 6.00000i − 0.986394i −0.869918 0.493197i $$-0.835828\pi$$
0.869918 0.493197i $$-0.164172\pi$$
$$38$$ − 4.00000i − 0.648886i
$$39$$ 2.00000 0.320256
$$40$$ 0 0
$$41$$ 2.00000 0.312348 0.156174 0.987730i $$-0.450084\pi$$
0.156174 + 0.987730i $$0.450084\pi$$
$$42$$ − 1.00000i − 0.154303i
$$43$$ − 4.00000i − 0.609994i −0.952353 0.304997i $$-0.901344\pi$$
0.952353 0.304997i $$-0.0986555\pi$$
$$44$$ 4.00000 0.603023
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$48$$ − 1.00000i − 0.144338i
$$49$$ −1.00000 −0.142857
$$50$$ 0 0
$$51$$ −6.00000 −0.840168
$$52$$ − 2.00000i − 0.277350i
$$53$$ 6.00000i 0.824163i 0.911147 + 0.412082i $$0.135198\pi$$
−0.911147 + 0.412082i $$0.864802\pi$$
$$54$$ 1.00000 0.136083
$$55$$ 0 0
$$56$$ −3.00000 −0.400892
$$57$$ − 4.00000i − 0.529813i
$$58$$ 2.00000i 0.262613i
$$59$$ −12.0000 −1.56227 −0.781133 0.624364i $$-0.785358\pi$$
−0.781133 + 0.624364i $$0.785358\pi$$
$$60$$ 0 0
$$61$$ −2.00000 −0.256074 −0.128037 0.991769i $$-0.540868\pi$$
−0.128037 + 0.991769i $$0.540868\pi$$
$$62$$ 0 0
$$63$$ − 1.00000i − 0.125988i
$$64$$ −7.00000 −0.875000
$$65$$ 0 0
$$66$$ −4.00000 −0.492366
$$67$$ − 4.00000i − 0.488678i −0.969690 0.244339i $$-0.921429\pi$$
0.969690 0.244339i $$-0.0785709\pi$$
$$68$$ 6.00000i 0.727607i
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$72$$ − 3.00000i − 0.353553i
$$73$$ − 6.00000i − 0.702247i −0.936329 0.351123i $$-0.885800\pi$$
0.936329 0.351123i $$-0.114200\pi$$
$$74$$ 6.00000 0.697486
$$75$$ 0 0
$$76$$ −4.00000 −0.458831
$$77$$ 4.00000i 0.455842i
$$78$$ 2.00000i 0.226455i
$$79$$ 16.0000 1.80014 0.900070 0.435745i $$-0.143515\pi$$
0.900070 + 0.435745i $$0.143515\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 2.00000i 0.220863i
$$83$$ − 12.0000i − 1.31717i −0.752506 0.658586i $$-0.771155\pi$$
0.752506 0.658586i $$-0.228845\pi$$
$$84$$ −1.00000 −0.109109
$$85$$ 0 0
$$86$$ 4.00000 0.431331
$$87$$ 2.00000i 0.214423i
$$88$$ 12.0000i 1.27920i
$$89$$ 14.0000 1.48400 0.741999 0.670402i $$-0.233878\pi$$
0.741999 + 0.670402i $$0.233878\pi$$
$$90$$ 0 0
$$91$$ 2.00000 0.209657
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 0 0
$$96$$ −5.00000 −0.510310
$$97$$ − 18.0000i − 1.82762i −0.406138 0.913812i $$-0.633125\pi$$
0.406138 0.913812i $$-0.366875\pi$$
$$98$$ − 1.00000i − 0.101015i
$$99$$ −4.00000 −0.402015
$$100$$ 0 0
$$101$$ 14.0000 1.39305 0.696526 0.717532i $$-0.254728\pi$$
0.696526 + 0.717532i $$0.254728\pi$$
$$102$$ − 6.00000i − 0.594089i
$$103$$ 8.00000i 0.788263i 0.919054 + 0.394132i $$0.128955\pi$$
−0.919054 + 0.394132i $$0.871045\pi$$
$$104$$ 6.00000 0.588348
$$105$$ 0 0
$$106$$ −6.00000 −0.582772
$$107$$ − 4.00000i − 0.386695i −0.981130 0.193347i $$-0.938066\pi$$
0.981130 0.193347i $$-0.0619344\pi$$
$$108$$ − 1.00000i − 0.0962250i
$$109$$ 18.0000 1.72409 0.862044 0.506834i $$-0.169184\pi$$
0.862044 + 0.506834i $$0.169184\pi$$
$$110$$ 0 0
$$111$$ 6.00000 0.569495
$$112$$ − 1.00000i − 0.0944911i
$$113$$ − 14.0000i − 1.31701i −0.752577 0.658505i $$-0.771189\pi$$
0.752577 0.658505i $$-0.228811\pi$$
$$114$$ 4.00000 0.374634
$$115$$ 0 0
$$116$$ 2.00000 0.185695
$$117$$ 2.00000i 0.184900i
$$118$$ − 12.0000i − 1.10469i
$$119$$ −6.00000 −0.550019
$$120$$ 0 0
$$121$$ 5.00000 0.454545
$$122$$ − 2.00000i − 0.181071i
$$123$$ 2.00000i 0.180334i
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 1.00000 0.0890871
$$127$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$128$$ 3.00000i 0.265165i
$$129$$ 4.00000 0.352180
$$130$$ 0 0
$$131$$ 4.00000 0.349482 0.174741 0.984614i $$-0.444091\pi$$
0.174741 + 0.984614i $$0.444091\pi$$
$$132$$ 4.00000i 0.348155i
$$133$$ − 4.00000i − 0.346844i
$$134$$ 4.00000 0.345547
$$135$$ 0 0
$$136$$ −18.0000 −1.54349
$$137$$ 6.00000i 0.512615i 0.966595 + 0.256307i $$0.0825059\pi$$
−0.966595 + 0.256307i $$0.917494\pi$$
$$138$$ 0 0
$$139$$ −12.0000 −1.01783 −0.508913 0.860818i $$-0.669953\pi$$
−0.508913 + 0.860818i $$0.669953\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ − 8.00000i − 0.668994i
$$144$$ 1.00000 0.0833333
$$145$$ 0 0
$$146$$ 6.00000 0.496564
$$147$$ − 1.00000i − 0.0824786i
$$148$$ − 6.00000i − 0.493197i
$$149$$ −6.00000 −0.491539 −0.245770 0.969328i $$-0.579041\pi$$
−0.245770 + 0.969328i $$0.579041\pi$$
$$150$$ 0 0
$$151$$ 8.00000 0.651031 0.325515 0.945537i $$-0.394462\pi$$
0.325515 + 0.945537i $$0.394462\pi$$
$$152$$ − 12.0000i − 0.973329i
$$153$$ − 6.00000i − 0.485071i
$$154$$ −4.00000 −0.322329
$$155$$ 0 0
$$156$$ 2.00000 0.160128
$$157$$ 2.00000i 0.159617i 0.996810 + 0.0798087i $$0.0254309\pi$$
−0.996810 + 0.0798087i $$0.974569\pi$$
$$158$$ 16.0000i 1.27289i
$$159$$ −6.00000 −0.475831
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 1.00000i 0.0785674i
$$163$$ 4.00000i 0.313304i 0.987654 + 0.156652i $$0.0500701\pi$$
−0.987654 + 0.156652i $$0.949930\pi$$
$$164$$ 2.00000 0.156174
$$165$$ 0 0
$$166$$ 12.0000 0.931381
$$167$$ 8.00000i 0.619059i 0.950890 + 0.309529i $$0.100171\pi$$
−0.950890 + 0.309529i $$0.899829\pi$$
$$168$$ − 3.00000i − 0.231455i
$$169$$ 9.00000 0.692308
$$170$$ 0 0
$$171$$ 4.00000 0.305888
$$172$$ − 4.00000i − 0.304997i
$$173$$ − 10.0000i − 0.760286i −0.924928 0.380143i $$-0.875875\pi$$
0.924928 0.380143i $$-0.124125\pi$$
$$174$$ −2.00000 −0.151620
$$175$$ 0 0
$$176$$ −4.00000 −0.301511
$$177$$ − 12.0000i − 0.901975i
$$178$$ 14.0000i 1.04934i
$$179$$ 4.00000 0.298974 0.149487 0.988764i $$-0.452238\pi$$
0.149487 + 0.988764i $$0.452238\pi$$
$$180$$ 0 0
$$181$$ −26.0000 −1.93256 −0.966282 0.257485i $$-0.917106\pi$$
−0.966282 + 0.257485i $$0.917106\pi$$
$$182$$ 2.00000i 0.148250i
$$183$$ − 2.00000i − 0.147844i
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 24.0000i 1.75505i
$$188$$ 0 0
$$189$$ 1.00000 0.0727393
$$190$$ 0 0
$$191$$ −8.00000 −0.578860 −0.289430 0.957199i $$-0.593466\pi$$
−0.289430 + 0.957199i $$0.593466\pi$$
$$192$$ − 7.00000i − 0.505181i
$$193$$ 2.00000i 0.143963i 0.997406 + 0.0719816i $$0.0229323\pi$$
−0.997406 + 0.0719816i $$0.977068\pi$$
$$194$$ 18.0000 1.29232
$$195$$ 0 0
$$196$$ −1.00000 −0.0714286
$$197$$ − 22.0000i − 1.56744i −0.621117 0.783718i $$-0.713321\pi$$
0.621117 0.783718i $$-0.286679\pi$$
$$198$$ − 4.00000i − 0.284268i
$$199$$ −24.0000 −1.70131 −0.850657 0.525720i $$-0.823796\pi$$
−0.850657 + 0.525720i $$0.823796\pi$$
$$200$$ 0 0
$$201$$ 4.00000 0.282138
$$202$$ 14.0000i 0.985037i
$$203$$ 2.00000i 0.140372i
$$204$$ −6.00000 −0.420084
$$205$$ 0 0
$$206$$ −8.00000 −0.557386
$$207$$ 0 0
$$208$$ 2.00000i 0.138675i
$$209$$ −16.0000 −1.10674
$$210$$ 0 0
$$211$$ 4.00000 0.275371 0.137686 0.990476i $$-0.456034\pi$$
0.137686 + 0.990476i $$0.456034\pi$$
$$212$$ 6.00000i 0.412082i
$$213$$ 0 0
$$214$$ 4.00000 0.273434
$$215$$ 0 0
$$216$$ 3.00000 0.204124
$$217$$ 0 0
$$218$$ 18.0000i 1.21911i
$$219$$ 6.00000 0.405442
$$220$$ 0 0
$$221$$ 12.0000 0.807207
$$222$$ 6.00000i 0.402694i
$$223$$ 16.0000i 1.07144i 0.844396 + 0.535720i $$0.179960\pi$$
−0.844396 + 0.535720i $$0.820040\pi$$
$$224$$ −5.00000 −0.334077
$$225$$ 0 0
$$226$$ 14.0000 0.931266
$$227$$ 12.0000i 0.796468i 0.917284 + 0.398234i $$0.130377\pi$$
−0.917284 + 0.398234i $$0.869623\pi$$
$$228$$ − 4.00000i − 0.264906i
$$229$$ 10.0000 0.660819 0.330409 0.943838i $$-0.392813\pi$$
0.330409 + 0.943838i $$0.392813\pi$$
$$230$$ 0 0
$$231$$ −4.00000 −0.263181
$$232$$ 6.00000i 0.393919i
$$233$$ − 6.00000i − 0.393073i −0.980497 0.196537i $$-0.937031\pi$$
0.980497 0.196537i $$-0.0629694\pi$$
$$234$$ −2.00000 −0.130744
$$235$$ 0 0
$$236$$ −12.0000 −0.781133
$$237$$ 16.0000i 1.03931i
$$238$$ − 6.00000i − 0.388922i
$$239$$ −24.0000 −1.55243 −0.776215 0.630468i $$-0.782863\pi$$
−0.776215 + 0.630468i $$0.782863\pi$$
$$240$$ 0 0
$$241$$ 2.00000 0.128831 0.0644157 0.997923i $$-0.479482\pi$$
0.0644157 + 0.997923i $$0.479482\pi$$
$$242$$ 5.00000i 0.321412i
$$243$$ 1.00000i 0.0641500i
$$244$$ −2.00000 −0.128037
$$245$$ 0 0
$$246$$ −2.00000 −0.127515
$$247$$ 8.00000i 0.509028i
$$248$$ 0 0
$$249$$ 12.0000 0.760469
$$250$$ 0 0
$$251$$ −20.0000 −1.26239 −0.631194 0.775625i $$-0.717435\pi$$
−0.631194 + 0.775625i $$0.717435\pi$$
$$252$$ − 1.00000i − 0.0629941i
$$253$$ 0 0
$$254$$ 0 0
$$255$$ 0 0
$$256$$ −17.0000 −1.06250
$$257$$ − 26.0000i − 1.62184i −0.585160 0.810918i $$-0.698968\pi$$
0.585160 0.810918i $$-0.301032\pi$$
$$258$$ 4.00000i 0.249029i
$$259$$ 6.00000 0.372822
$$260$$ 0 0
$$261$$ −2.00000 −0.123797
$$262$$ 4.00000i 0.247121i
$$263$$ 16.0000i 0.986602i 0.869859 + 0.493301i $$0.164210\pi$$
−0.869859 + 0.493301i $$0.835790\pi$$
$$264$$ −12.0000 −0.738549
$$265$$ 0 0
$$266$$ 4.00000 0.245256
$$267$$ 14.0000i 0.856786i
$$268$$ − 4.00000i − 0.244339i
$$269$$ −6.00000 −0.365826 −0.182913 0.983129i $$-0.558553\pi$$
−0.182913 + 0.983129i $$0.558553\pi$$
$$270$$ 0 0
$$271$$ 16.0000 0.971931 0.485965 0.873978i $$-0.338468\pi$$
0.485965 + 0.873978i $$0.338468\pi$$
$$272$$ − 6.00000i − 0.363803i
$$273$$ 2.00000i 0.121046i
$$274$$ −6.00000 −0.362473
$$275$$ 0 0
$$276$$ 0 0
$$277$$ − 22.0000i − 1.32185i −0.750451 0.660926i $$-0.770164\pi$$
0.750451 0.660926i $$-0.229836\pi$$
$$278$$ − 12.0000i − 0.719712i
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −22.0000 −1.31241 −0.656205 0.754583i $$-0.727839\pi$$
−0.656205 + 0.754583i $$0.727839\pi$$
$$282$$ 0 0
$$283$$ − 20.0000i − 1.18888i −0.804141 0.594438i $$-0.797374\pi$$
0.804141 0.594438i $$-0.202626\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 8.00000 0.473050
$$287$$ 2.00000i 0.118056i
$$288$$ − 5.00000i − 0.294628i
$$289$$ −19.0000 −1.11765
$$290$$ 0 0
$$291$$ 18.0000 1.05518
$$292$$ − 6.00000i − 0.351123i
$$293$$ 14.0000i 0.817889i 0.912559 + 0.408944i $$0.134103\pi$$
−0.912559 + 0.408944i $$0.865897\pi$$
$$294$$ 1.00000 0.0583212
$$295$$ 0 0
$$296$$ 18.0000 1.04623
$$297$$ − 4.00000i − 0.232104i
$$298$$ − 6.00000i − 0.347571i
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 4.00000 0.230556
$$302$$ 8.00000i 0.460348i
$$303$$ 14.0000i 0.804279i
$$304$$ 4.00000 0.229416
$$305$$ 0 0
$$306$$ 6.00000 0.342997
$$307$$ − 4.00000i − 0.228292i −0.993464 0.114146i $$-0.963587\pi$$
0.993464 0.114146i $$-0.0364132\pi$$
$$308$$ 4.00000i 0.227921i
$$309$$ −8.00000 −0.455104
$$310$$ 0 0
$$311$$ −24.0000 −1.36092 −0.680458 0.732787i $$-0.738219\pi$$
−0.680458 + 0.732787i $$0.738219\pi$$
$$312$$ 6.00000i 0.339683i
$$313$$ 26.0000i 1.46961i 0.678280 + 0.734803i $$0.262726\pi$$
−0.678280 + 0.734803i $$0.737274\pi$$
$$314$$ −2.00000 −0.112867
$$315$$ 0 0
$$316$$ 16.0000 0.900070
$$317$$ 18.0000i 1.01098i 0.862832 + 0.505490i $$0.168688\pi$$
−0.862832 + 0.505490i $$0.831312\pi$$
$$318$$ − 6.00000i − 0.336463i
$$319$$ 8.00000 0.447914
$$320$$ 0 0
$$321$$ 4.00000 0.223258
$$322$$ 0 0
$$323$$ − 24.0000i − 1.33540i
$$324$$ 1.00000 0.0555556
$$325$$ 0 0
$$326$$ −4.00000 −0.221540
$$327$$ 18.0000i 0.995402i
$$328$$ 6.00000i 0.331295i
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −4.00000 −0.219860 −0.109930 0.993939i $$-0.535063\pi$$
−0.109930 + 0.993939i $$0.535063\pi$$
$$332$$ − 12.0000i − 0.658586i
$$333$$ 6.00000i 0.328798i
$$334$$ −8.00000 −0.437741
$$335$$ 0 0
$$336$$ 1.00000 0.0545545
$$337$$ 14.0000i 0.762629i 0.924445 + 0.381314i $$0.124528\pi$$
−0.924445 + 0.381314i $$0.875472\pi$$
$$338$$ 9.00000i 0.489535i
$$339$$ 14.0000 0.760376
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 4.00000i 0.216295i
$$343$$ − 1.00000i − 0.0539949i
$$344$$ 12.0000 0.646997
$$345$$ 0 0
$$346$$ 10.0000 0.537603
$$347$$ 28.0000i 1.50312i 0.659665 + 0.751559i $$0.270698\pi$$
−0.659665 + 0.751559i $$0.729302\pi$$
$$348$$ 2.00000i 0.107211i
$$349$$ 2.00000 0.107058 0.0535288 0.998566i $$-0.482953\pi$$
0.0535288 + 0.998566i $$0.482953\pi$$
$$350$$ 0 0
$$351$$ −2.00000 −0.106752
$$352$$ 20.0000i 1.06600i
$$353$$ 10.0000i 0.532246i 0.963939 + 0.266123i $$0.0857428\pi$$
−0.963939 + 0.266123i $$0.914257\pi$$
$$354$$ 12.0000 0.637793
$$355$$ 0 0
$$356$$ 14.0000 0.741999
$$357$$ − 6.00000i − 0.317554i
$$358$$ 4.00000i 0.211407i
$$359$$ −32.0000 −1.68890 −0.844448 0.535638i $$-0.820071\pi$$
−0.844448 + 0.535638i $$0.820071\pi$$
$$360$$ 0 0
$$361$$ −3.00000 −0.157895
$$362$$ − 26.0000i − 1.36653i
$$363$$ 5.00000i 0.262432i
$$364$$ 2.00000 0.104828
$$365$$ 0 0
$$366$$ 2.00000 0.104542
$$367$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$368$$ 0 0
$$369$$ −2.00000 −0.104116
$$370$$ 0 0
$$371$$ −6.00000 −0.311504
$$372$$ 0 0
$$373$$ − 10.0000i − 0.517780i −0.965907 0.258890i $$-0.916643\pi$$
0.965907 0.258890i $$-0.0833568\pi$$
$$374$$ −24.0000 −1.24101
$$375$$ 0 0
$$376$$ 0 0
$$377$$ − 4.00000i − 0.206010i
$$378$$ 1.00000i 0.0514344i
$$379$$ −12.0000 −0.616399 −0.308199 0.951322i $$-0.599726\pi$$
−0.308199 + 0.951322i $$0.599726\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ − 8.00000i − 0.409316i
$$383$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$384$$ −3.00000 −0.153093
$$385$$ 0 0
$$386$$ −2.00000 −0.101797
$$387$$ 4.00000i 0.203331i
$$388$$ − 18.0000i − 0.913812i
$$389$$ −6.00000 −0.304212 −0.152106 0.988364i $$-0.548606\pi$$
−0.152106 + 0.988364i $$0.548606\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ − 3.00000i − 0.151523i
$$393$$ 4.00000i 0.201773i
$$394$$ 22.0000 1.10834
$$395$$ 0 0
$$396$$ −4.00000 −0.201008
$$397$$ 18.0000i 0.903394i 0.892171 + 0.451697i $$0.149181\pi$$
−0.892171 + 0.451697i $$0.850819\pi$$
$$398$$ − 24.0000i − 1.20301i
$$399$$ 4.00000 0.200250
$$400$$ 0 0
$$401$$ −30.0000 −1.49813 −0.749064 0.662497i $$-0.769497\pi$$
−0.749064 + 0.662497i $$0.769497\pi$$
$$402$$ 4.00000i 0.199502i
$$403$$ 0 0
$$404$$ 14.0000 0.696526
$$405$$ 0 0
$$406$$ −2.00000 −0.0992583
$$407$$ − 24.0000i − 1.18964i
$$408$$ − 18.0000i − 0.891133i
$$409$$ 22.0000 1.08783 0.543915 0.839140i $$-0.316941\pi$$
0.543915 + 0.839140i $$0.316941\pi$$
$$410$$ 0 0
$$411$$ −6.00000 −0.295958
$$412$$ 8.00000i 0.394132i
$$413$$ − 12.0000i − 0.590481i
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 10.0000 0.490290
$$417$$ − 12.0000i − 0.587643i
$$418$$ − 16.0000i − 0.782586i
$$419$$ 12.0000 0.586238 0.293119 0.956076i $$-0.405307\pi$$
0.293119 + 0.956076i $$0.405307\pi$$
$$420$$ 0 0
$$421$$ 38.0000 1.85201 0.926003 0.377515i $$-0.123221\pi$$
0.926003 + 0.377515i $$0.123221\pi$$
$$422$$ 4.00000i 0.194717i
$$423$$ 0 0
$$424$$ −18.0000 −0.874157
$$425$$ 0 0
$$426$$ 0 0
$$427$$ − 2.00000i − 0.0967868i
$$428$$ − 4.00000i − 0.193347i
$$429$$ 8.00000 0.386244
$$430$$ 0 0
$$431$$ −24.0000 −1.15604 −0.578020 0.816023i $$-0.696174\pi$$
−0.578020 + 0.816023i $$0.696174\pi$$
$$432$$ 1.00000i 0.0481125i
$$433$$ − 14.0000i − 0.672797i −0.941720 0.336399i $$-0.890791\pi$$
0.941720 0.336399i $$-0.109209\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 18.0000 0.862044
$$437$$ 0 0
$$438$$ 6.00000i 0.286691i
$$439$$ 24.0000 1.14546 0.572729 0.819745i $$-0.305885\pi$$
0.572729 + 0.819745i $$0.305885\pi$$
$$440$$ 0 0
$$441$$ 1.00000 0.0476190
$$442$$ 12.0000i 0.570782i
$$443$$ 36.0000i 1.71041i 0.518289 + 0.855206i $$0.326569\pi$$
−0.518289 + 0.855206i $$0.673431\pi$$
$$444$$ 6.00000 0.284747
$$445$$ 0 0
$$446$$ −16.0000 −0.757622
$$447$$ − 6.00000i − 0.283790i
$$448$$ − 7.00000i − 0.330719i
$$449$$ 30.0000 1.41579 0.707894 0.706319i $$-0.249646\pi$$
0.707894 + 0.706319i $$0.249646\pi$$
$$450$$ 0 0
$$451$$ 8.00000 0.376705
$$452$$ − 14.0000i − 0.658505i
$$453$$ 8.00000i 0.375873i
$$454$$ −12.0000 −0.563188
$$455$$ 0 0
$$456$$ 12.0000 0.561951
$$457$$ − 10.0000i − 0.467780i −0.972263 0.233890i $$-0.924854\pi$$
0.972263 0.233890i $$-0.0751456\pi$$
$$458$$ 10.0000i 0.467269i
$$459$$ 6.00000 0.280056
$$460$$ 0 0
$$461$$ −10.0000 −0.465746 −0.232873 0.972507i $$-0.574813\pi$$
−0.232873 + 0.972507i $$0.574813\pi$$
$$462$$ − 4.00000i − 0.186097i
$$463$$ 16.0000i 0.743583i 0.928316 + 0.371792i $$0.121256\pi$$
−0.928316 + 0.371792i $$0.878744\pi$$
$$464$$ −2.00000 −0.0928477
$$465$$ 0 0
$$466$$ 6.00000 0.277945
$$467$$ − 36.0000i − 1.66588i −0.553362 0.832941i $$-0.686655\pi$$
0.553362 0.832941i $$-0.313345\pi$$
$$468$$ 2.00000i 0.0924500i
$$469$$ 4.00000 0.184703
$$470$$ 0 0
$$471$$ −2.00000 −0.0921551
$$472$$ − 36.0000i − 1.65703i
$$473$$ − 16.0000i − 0.735681i
$$474$$ −16.0000 −0.734904
$$475$$ 0 0
$$476$$ −6.00000 −0.275010
$$477$$ − 6.00000i − 0.274721i
$$478$$ − 24.0000i − 1.09773i
$$479$$ 16.0000 0.731059 0.365529 0.930800i $$-0.380888\pi$$
0.365529 + 0.930800i $$0.380888\pi$$
$$480$$ 0 0
$$481$$ −12.0000 −0.547153
$$482$$ 2.00000i 0.0910975i
$$483$$ 0 0
$$484$$ 5.00000 0.227273
$$485$$ 0 0
$$486$$ −1.00000 −0.0453609
$$487$$ 8.00000i 0.362515i 0.983436 + 0.181257i $$0.0580167\pi$$
−0.983436 + 0.181257i $$0.941983\pi$$
$$488$$ − 6.00000i − 0.271607i
$$489$$ −4.00000 −0.180886
$$490$$ 0 0
$$491$$ 20.0000 0.902587 0.451294 0.892375i $$-0.350963\pi$$
0.451294 + 0.892375i $$0.350963\pi$$
$$492$$ 2.00000i 0.0901670i
$$493$$ 12.0000i 0.540453i
$$494$$ −8.00000 −0.359937
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 12.0000i 0.537733i
$$499$$ −4.00000 −0.179065 −0.0895323 0.995984i $$-0.528537\pi$$
−0.0895323 + 0.995984i $$0.528537\pi$$
$$500$$ 0 0
$$501$$ −8.00000 −0.357414
$$502$$ − 20.0000i − 0.892644i
$$503$$ 24.0000i 1.07011i 0.844818 + 0.535054i $$0.179709\pi$$
−0.844818 + 0.535054i $$0.820291\pi$$
$$504$$ 3.00000 0.133631
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 9.00000i 0.399704i
$$508$$ 0 0
$$509$$ 10.0000 0.443242 0.221621 0.975133i $$-0.428865\pi$$
0.221621 + 0.975133i $$0.428865\pi$$
$$510$$ 0 0
$$511$$ 6.00000 0.265424
$$512$$ − 11.0000i − 0.486136i
$$513$$ 4.00000i 0.176604i
$$514$$ 26.0000 1.14681
$$515$$ 0 0
$$516$$ 4.00000 0.176090
$$517$$ 0 0
$$518$$ 6.00000i 0.263625i
$$519$$ 10.0000 0.438951
$$520$$ 0 0
$$521$$ 18.0000 0.788594 0.394297 0.918983i $$-0.370988\pi$$
0.394297 + 0.918983i $$0.370988\pi$$
$$522$$ − 2.00000i − 0.0875376i
$$523$$ − 20.0000i − 0.874539i −0.899331 0.437269i $$-0.855946\pi$$
0.899331 0.437269i $$-0.144054\pi$$
$$524$$ 4.00000 0.174741
$$525$$ 0 0
$$526$$ −16.0000 −0.697633
$$527$$ 0 0
$$528$$ − 4.00000i − 0.174078i
$$529$$ 23.0000 1.00000
$$530$$ 0 0
$$531$$ 12.0000 0.520756
$$532$$ − 4.00000i − 0.173422i
$$533$$ − 4.00000i − 0.173259i
$$534$$ −14.0000 −0.605839
$$535$$ 0 0
$$536$$ 12.0000 0.518321
$$537$$ 4.00000i 0.172613i
$$538$$ − 6.00000i − 0.258678i
$$539$$ −4.00000 −0.172292
$$540$$ 0 0
$$541$$ −34.0000 −1.46177 −0.730887 0.682498i $$-0.760893\pi$$
−0.730887 + 0.682498i $$0.760893\pi$$
$$542$$ 16.0000i 0.687259i
$$543$$ − 26.0000i − 1.11577i
$$544$$ −30.0000 −1.28624
$$545$$ 0 0
$$546$$ −2.00000 −0.0855921
$$547$$ − 4.00000i − 0.171028i −0.996337 0.0855138i $$-0.972747\pi$$
0.996337 0.0855138i $$-0.0272532\pi$$
$$548$$ 6.00000i 0.256307i
$$549$$ 2.00000 0.0853579
$$550$$ 0 0
$$551$$ −8.00000 −0.340811
$$552$$ 0 0
$$553$$ 16.0000i 0.680389i
$$554$$ 22.0000 0.934690
$$555$$ 0 0
$$556$$ −12.0000 −0.508913
$$557$$ 2.00000i 0.0847427i 0.999102 + 0.0423714i $$0.0134913\pi$$
−0.999102 + 0.0423714i $$0.986509\pi$$
$$558$$ 0 0
$$559$$ −8.00000 −0.338364
$$560$$ 0 0
$$561$$ −24.0000 −1.01328
$$562$$ − 22.0000i − 0.928014i
$$563$$ 4.00000i 0.168580i 0.996441 + 0.0842900i $$0.0268622\pi$$
−0.996441 + 0.0842900i $$0.973138\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 20.0000 0.840663
$$567$$ 1.00000i 0.0419961i
$$568$$ 0 0
$$569$$ −10.0000 −0.419222 −0.209611 0.977785i $$-0.567220\pi$$
−0.209611 + 0.977785i $$0.567220\pi$$
$$570$$ 0 0
$$571$$ −4.00000 −0.167395 −0.0836974 0.996491i $$-0.526673\pi$$
−0.0836974 + 0.996491i $$0.526673\pi$$
$$572$$ − 8.00000i − 0.334497i
$$573$$ − 8.00000i − 0.334205i
$$574$$ −2.00000 −0.0834784
$$575$$ 0 0
$$576$$ 7.00000 0.291667
$$577$$ − 34.0000i − 1.41544i −0.706494 0.707719i $$-0.749724\pi$$
0.706494 0.707719i $$-0.250276\pi$$
$$578$$ − 19.0000i − 0.790296i
$$579$$ −2.00000 −0.0831172
$$580$$ 0 0
$$581$$ 12.0000 0.497844
$$582$$ 18.0000i 0.746124i
$$583$$ 24.0000i 0.993978i
$$584$$ 18.0000 0.744845
$$585$$ 0 0
$$586$$ −14.0000 −0.578335
$$587$$ − 28.0000i − 1.15568i −0.816149 0.577842i $$-0.803895\pi$$
0.816149 0.577842i $$-0.196105\pi$$
$$588$$ − 1.00000i − 0.0412393i
$$589$$ 0 0
$$590$$ 0 0
$$591$$ 22.0000 0.904959
$$592$$ 6.00000i 0.246598i
$$593$$ − 6.00000i − 0.246390i −0.992382 0.123195i $$-0.960686\pi$$
0.992382 0.123195i $$-0.0393141\pi$$
$$594$$ 4.00000 0.164122
$$595$$ 0 0
$$596$$ −6.00000 −0.245770
$$597$$ − 24.0000i − 0.982255i
$$598$$ 0 0
$$599$$ −48.0000 −1.96123 −0.980613 0.195952i $$-0.937220\pi$$
−0.980613 + 0.195952i $$0.937220\pi$$
$$600$$ 0 0
$$601$$ −6.00000 −0.244745 −0.122373 0.992484i $$-0.539050\pi$$
−0.122373 + 0.992484i $$0.539050\pi$$
$$602$$ 4.00000i 0.163028i
$$603$$ 4.00000i 0.162893i
$$604$$ 8.00000 0.325515
$$605$$ 0 0
$$606$$ −14.0000 −0.568711
$$607$$ 16.0000i 0.649420i 0.945814 + 0.324710i $$0.105267\pi$$
−0.945814 + 0.324710i $$0.894733\pi$$
$$608$$ − 20.0000i − 0.811107i
$$609$$ −2.00000 −0.0810441
$$610$$ 0 0
$$611$$ 0 0
$$612$$ − 6.00000i − 0.242536i
$$613$$ − 26.0000i − 1.05013i −0.851062 0.525065i $$-0.824041\pi$$
0.851062 0.525065i $$-0.175959\pi$$
$$614$$ 4.00000 0.161427
$$615$$ 0 0
$$616$$ −12.0000 −0.483494
$$617$$ 6.00000i 0.241551i 0.992680 + 0.120775i $$0.0385381\pi$$
−0.992680 + 0.120775i $$0.961462\pi$$
$$618$$ − 8.00000i − 0.321807i
$$619$$ 20.0000 0.803868 0.401934 0.915669i $$-0.368338\pi$$
0.401934 + 0.915669i $$0.368338\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ − 24.0000i − 0.962312i
$$623$$ 14.0000i 0.560898i
$$624$$ −2.00000 −0.0800641
$$625$$ 0 0
$$626$$ −26.0000 −1.03917
$$627$$ − 16.0000i − 0.638978i
$$628$$ 2.00000i 0.0798087i
$$629$$ 36.0000 1.43541
$$630$$ 0 0
$$631$$ −40.0000 −1.59237 −0.796187 0.605050i $$-0.793153\pi$$
−0.796187 + 0.605050i $$0.793153\pi$$
$$632$$ 48.0000i 1.90934i
$$633$$ 4.00000i 0.158986i
$$634$$ −18.0000 −0.714871
$$635$$ 0 0
$$636$$ −6.00000 −0.237915
$$637$$ 2.00000i 0.0792429i
$$638$$ 8.00000i 0.316723i
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 18.0000 0.710957 0.355479 0.934684i $$-0.384318\pi$$
0.355479 + 0.934684i $$0.384318\pi$$
$$642$$ 4.00000i 0.157867i
$$643$$ 20.0000i 0.788723i 0.918955 + 0.394362i $$0.129034\pi$$
−0.918955 + 0.394362i $$0.870966\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 24.0000 0.944267
$$647$$ 40.0000i 1.57256i 0.617869 + 0.786281i $$0.287996\pi$$
−0.617869 + 0.786281i $$0.712004\pi$$
$$648$$ 3.00000i 0.117851i
$$649$$ −48.0000 −1.88416
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 4.00000i 0.156652i
$$653$$ − 18.0000i − 0.704394i −0.935926 0.352197i $$-0.885435\pi$$
0.935926 0.352197i $$-0.114565\pi$$
$$654$$ −18.0000 −0.703856
$$655$$ 0 0
$$656$$ −2.00000 −0.0780869
$$657$$ 6.00000i 0.234082i
$$658$$ 0 0
$$659$$ −12.0000 −0.467454 −0.233727 0.972302i $$-0.575092\pi$$
−0.233727 + 0.972302i $$0.575092\pi$$
$$660$$ 0 0
$$661$$ 22.0000 0.855701 0.427850 0.903850i $$-0.359271\pi$$
0.427850 + 0.903850i $$0.359271\pi$$
$$662$$ − 4.00000i − 0.155464i
$$663$$ 12.0000i 0.466041i
$$664$$ 36.0000 1.39707
$$665$$ 0 0
$$666$$ −6.00000 −0.232495
$$667$$ 0 0
$$668$$ 8.00000i 0.309529i
$$669$$ −16.0000 −0.618596
$$670$$ 0 0
$$671$$ −8.00000 −0.308837
$$672$$ − 5.00000i − 0.192879i
$$673$$ 34.0000i 1.31060i 0.755367 + 0.655302i $$0.227459\pi$$
−0.755367 + 0.655302i $$0.772541\pi$$
$$674$$ −14.0000 −0.539260
$$675$$ 0 0
$$676$$ 9.00000 0.346154
$$677$$ 18.0000i 0.691796i 0.938272 + 0.345898i $$0.112426\pi$$
−0.938272 + 0.345898i $$0.887574\pi$$
$$678$$ 14.0000i 0.537667i
$$679$$ 18.0000 0.690777
$$680$$ 0 0
$$681$$ −12.0000 −0.459841
$$682$$ 0 0
$$683$$ − 12.0000i − 0.459167i −0.973289 0.229584i $$-0.926264\pi$$
0.973289 0.229584i $$-0.0737364\pi$$
$$684$$ 4.00000 0.152944
$$685$$ 0 0
$$686$$ 1.00000 0.0381802
$$687$$ 10.0000i 0.381524i
$$688$$ 4.00000i 0.152499i
$$689$$ 12.0000 0.457164
$$690$$ 0 0
$$691$$ 20.0000 0.760836 0.380418 0.924815i $$-0.375780\pi$$
0.380418 + 0.924815i $$0.375780\pi$$
$$692$$ − 10.0000i − 0.380143i
$$693$$ − 4.00000i − 0.151947i
$$694$$ −28.0000 −1.06287
$$695$$ 0 0
$$696$$ −6.00000 −0.227429
$$697$$ 12.0000i 0.454532i
$$698$$ 2.00000i 0.0757011i
$$699$$ 6.00000 0.226941
$$700$$ 0 0
$$701$$ 30.0000 1.13308 0.566542 0.824033i $$-0.308281\pi$$
0.566542 + 0.824033i $$0.308281\pi$$
$$702$$ − 2.00000i − 0.0754851i
$$703$$ 24.0000i 0.905177i
$$704$$ −28.0000 −1.05529
$$705$$ 0 0
$$706$$ −10.0000 −0.376355
$$707$$ 14.0000i 0.526524i
$$708$$ − 12.0000i − 0.450988i
$$709$$ −6.00000 −0.225335 −0.112667 0.993633i $$-0.535939\pi$$
−0.112667 + 0.993633i $$0.535939\pi$$
$$710$$ 0 0
$$711$$ −16.0000 −0.600047
$$712$$ 42.0000i 1.57402i
$$713$$ 0 0
$$714$$ 6.00000 0.224544
$$715$$ 0 0
$$716$$ 4.00000 0.149487
$$717$$ − 24.0000i − 0.896296i
$$718$$ − 32.0000i − 1.19423i
$$719$$ 48.0000 1.79010 0.895049 0.445968i $$-0.147140\pi$$
0.895049 + 0.445968i $$0.147140\pi$$
$$720$$ 0 0
$$721$$ −8.00000 −0.297936
$$722$$ − 3.00000i − 0.111648i
$$723$$ 2.00000i 0.0743808i
$$724$$ −26.0000 −0.966282
$$725$$ 0 0
$$726$$ −5.00000 −0.185567
$$727$$ 40.0000i 1.48352i 0.670667 + 0.741759i $$0.266008\pi$$
−0.670667 + 0.741759i $$0.733992\pi$$
$$728$$ 6.00000i 0.222375i
$$729$$ −1.00000 −0.0370370
$$730$$ 0 0
$$731$$ 24.0000 0.887672
$$732$$ − 2.00000i − 0.0739221i
$$733$$ − 18.0000i − 0.664845i −0.943131 0.332423i $$-0.892134\pi$$
0.943131 0.332423i $$-0.107866\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ − 16.0000i − 0.589368i
$$738$$ − 2.00000i − 0.0736210i
$$739$$ −36.0000 −1.32428 −0.662141 0.749380i $$-0.730352\pi$$
−0.662141 + 0.749380i $$0.730352\pi$$
$$740$$ 0 0
$$741$$ −8.00000 −0.293887
$$742$$ − 6.00000i − 0.220267i
$$743$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 10.0000 0.366126
$$747$$ 12.0000i 0.439057i
$$748$$ 24.0000i 0.877527i
$$749$$ 4.00000 0.146157
$$750$$ 0 0
$$751$$ −32.0000 −1.16770 −0.583848 0.811863i $$-0.698454\pi$$
−0.583848 + 0.811863i $$0.698454\pi$$
$$752$$ 0 0
$$753$$ − 20.0000i − 0.728841i
$$754$$ 4.00000 0.145671
$$755$$ 0 0
$$756$$ 1.00000 0.0363696
$$757$$ 10.0000i 0.363456i 0.983349 + 0.181728i $$0.0581691\pi$$
−0.983349 + 0.181728i $$0.941831\pi$$
$$758$$ − 12.0000i − 0.435860i
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 18.0000 0.652499 0.326250 0.945284i $$-0.394215\pi$$
0.326250 + 0.945284i $$0.394215\pi$$
$$762$$ 0 0
$$763$$ 18.0000i 0.651644i
$$764$$ −8.00000 −0.289430
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 24.0000i 0.866590i
$$768$$ − 17.0000i − 0.613435i
$$769$$ −2.00000 −0.0721218 −0.0360609 0.999350i $$-0.511481\pi$$
−0.0360609 + 0.999350i $$0.511481\pi$$
$$770$$ 0 0
$$771$$ 26.0000 0.936367
$$772$$ 2.00000i 0.0719816i
$$773$$ 14.0000i 0.503545i 0.967786 + 0.251773i $$0.0810135\pi$$
−0.967786 + 0.251773i $$0.918987\pi$$
$$774$$ −4.00000 −0.143777
$$775$$ 0 0
$$776$$ 54.0000 1.93849
$$777$$ 6.00000i 0.215249i
$$778$$ − 6.00000i − 0.215110i
$$779$$ −8.00000 −0.286630
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ − 2.00000i − 0.0714742i
$$784$$ 1.00000 0.0357143
$$785$$ 0 0
$$786$$ −4.00000 −0.142675
$$787$$ 44.0000i 1.56843i 0.620489 + 0.784215i $$0.286934\pi$$
−0.620489 + 0.784215i $$0.713066\pi$$
$$788$$ − 22.0000i − 0.783718i
$$789$$ −16.0000 −0.569615
$$790$$ 0 0
$$791$$ 14.0000 0.497783
$$792$$ − 12.0000i − 0.426401i
$$793$$ 4.00000i 0.142044i
$$794$$ −18.0000 −0.638796
$$795$$ 0 0
$$796$$ −24.0000 −0.850657
$$797$$ 26.0000i 0.920967i 0.887668 + 0.460484i $$0.152324\pi$$
−0.887668 + 0.460484i $$0.847676\pi$$
$$798$$ 4.00000i 0.141598i
$$799$$ 0 0
$$800$$ 0 0
$$801$$ −14.0000 −0.494666
$$802$$ − 30.0000i − 1.05934i
$$803$$ − 24.0000i − 0.846942i
$$804$$ 4.00000 0.141069
$$805$$ 0 0
$$806$$ 0 0
$$807$$ − 6.00000i − 0.211210i
$$808$$ 42.0000i 1.47755i
$$809$$ −42.0000 −1.47664 −0.738321 0.674450i $$-0.764381\pi$$
−0.738321 + 0.674450i $$0.764381\pi$$
$$810$$ 0 0
$$811$$ 44.0000 1.54505 0.772524 0.634985i $$-0.218994\pi$$
0.772524 + 0.634985i $$0.218994\pi$$
$$812$$ 2.00000i 0.0701862i
$$813$$ 16.0000i 0.561144i
$$814$$ 24.0000 0.841200
$$815$$ 0 0
$$816$$ 6.00000 0.210042
$$817$$ 16.0000i 0.559769i
$$818$$ 22.0000i 0.769212i
$$819$$ −2.00000 −0.0698857
$$820$$ 0 0
$$821$$ 38.0000 1.32621 0.663105 0.748527i $$-0.269238\pi$$
0.663105 + 0.748527i $$0.269238\pi$$
$$822$$ − 6.00000i − 0.209274i
$$823$$ 24.0000i 0.836587i 0.908312 + 0.418294i $$0.137372\pi$$
−0.908312 + 0.418294i $$0.862628\pi$$
$$824$$ −24.0000 −0.836080
$$825$$ 0 0
$$826$$ 12.0000 0.417533
$$827$$ 12.0000i 0.417281i 0.977992 + 0.208640i $$0.0669038\pi$$
−0.977992 + 0.208640i $$0.933096\pi$$
$$828$$ 0 0
$$829$$ −14.0000 −0.486240 −0.243120 0.969996i $$-0.578171\pi$$
−0.243120 + 0.969996i $$0.578171\pi$$
$$830$$ 0 0
$$831$$ 22.0000 0.763172
$$832$$ 14.0000i 0.485363i
$$833$$ − 6.00000i − 0.207888i
$$834$$ 12.0000 0.415526
$$835$$ 0 0
$$836$$ −16.0000 −0.553372
$$837$$ 0 0
$$838$$ 12.0000i 0.414533i
$$839$$ 8.00000 0.276191 0.138095 0.990419i $$-0.455902\pi$$
0.138095 + 0.990419i $$0.455902\pi$$
$$840$$ 0 0
$$841$$ −25.0000 −0.862069
$$842$$ 38.0000i 1.30957i
$$843$$ − 22.0000i − 0.757720i
$$844$$ 4.00000 0.137686
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 5.00000i 0.171802i
$$848$$ − 6.00000i − 0.206041i
$$849$$ 20.0000 0.686398
$$850$$ 0 0
$$851$$ 0 0
$$852$$ 0 0
$$853$$ − 10.0000i − 0.342393i −0.985237 0.171197i $$-0.945237\pi$$
0.985237 0.171197i $$-0.0547634\pi$$
$$854$$ 2.00000 0.0684386
$$855$$ 0 0
$$856$$ 12.0000 0.410152
$$857$$ 14.0000i 0.478231i 0.970991 + 0.239115i $$0.0768574\pi$$
−0.970991 + 0.239115i $$0.923143\pi$$
$$858$$ 8.00000i 0.273115i
$$859$$ −44.0000 −1.50126 −0.750630 0.660722i $$-0.770250\pi$$
−0.750630 + 0.660722i $$0.770250\pi$$
$$860$$ 0 0
$$861$$ −2.00000 −0.0681598
$$862$$ − 24.0000i − 0.817443i
$$863$$ − 24.0000i − 0.816970i −0.912765 0.408485i $$-0.866057\pi$$
0.912765 0.408485i $$-0.133943\pi$$
$$864$$ 5.00000 0.170103
$$865$$ 0 0
$$866$$ 14.0000 0.475739
$$867$$ − 19.0000i − 0.645274i
$$868$$ 0 0
$$869$$ 64.0000 2.17105
$$870$$ 0 0
$$871$$ −8.00000 −0.271070
$$872$$ 54.0000i 1.82867i
$$873$$ 18.0000i 0.609208i
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 6.00000 0.202721
$$877$$ − 46.0000i − 1.55331i −0.629926 0.776655i $$-0.716915\pi$$
0.629926 0.776655i $$-0.283085\pi$$
$$878$$ 24.0000i 0.809961i
$$879$$ −14.0000 −0.472208
$$880$$ 0 0
$$881$$ −6.00000 −0.202145 −0.101073 0.994879i $$-0.532227\pi$$
−0.101073 + 0.994879i $$0.532227\pi$$
$$882$$ 1.00000i 0.0336718i
$$883$$ − 28.0000i − 0.942275i −0.882060 0.471138i $$-0.843844\pi$$
0.882060 0.471138i $$-0.156156\pi$$
$$884$$ 12.0000 0.403604
$$885$$ 0 0
$$886$$ −36.0000 −1.20944
$$887$$ − 8.00000i − 0.268614i −0.990940 0.134307i $$-0.957119\pi$$
0.990940 0.134307i $$-0.0428808\pi$$
$$888$$ 18.0000i 0.604040i
$$889$$ 0 0
$$890$$ 0 0
$$891$$ 4.00000 0.134005
$$892$$ 16.0000i 0.535720i
$$893$$ 0 0
$$894$$ 6.00000 0.200670
$$895$$ 0 0
$$896$$ −3.00000 −0.100223
$$897$$ 0 0
$$898$$ 30.0000i 1.00111i
$$899$$ 0 0
$$900$$ 0 0
$$901$$ −36.0000 −1.19933
$$902$$ 8.00000i 0.266371i
$$903$$ 4.00000i 0.133112i
$$904$$ 42.0000 1.39690
$$905$$ 0 0
$$906$$ −8.00000 −0.265782
$$907$$ 4.00000i 0.132818i 0.997792 + 0.0664089i $$0.0211542\pi$$
−0.997792 + 0.0664089i $$0.978846\pi$$
$$908$$ 12.0000i 0.398234i
$$909$$ −14.0000 −0.464351
$$910$$ 0 0
$$911$$ −24.0000 −0.795155 −0.397578 0.917568i $$-0.630149\pi$$
−0.397578 + 0.917568i $$0.630149\pi$$
$$912$$ 4.00000i 0.132453i
$$913$$ − 48.0000i − 1.58857i
$$914$$ 10.0000 0.330771
$$915$$ 0 0
$$916$$ 10.0000 0.330409
$$917$$ 4.00000i 0.132092i
$$918$$ 6.00000i 0.198030i
$$919$$ −8.00000 −0.263896 −0.131948 0.991257i $$-0.542123\pi$$
−0.131948 + 0.991257i $$0.542123\pi$$
$$920$$ 0 0
$$921$$ 4.00000 0.131804
$$922$$ − 10.0000i − 0.329332i
$$923$$ 0 0
$$924$$ −4.00000 −0.131590
$$925$$ 0 0
$$926$$ −16.0000 −0.525793
$$927$$ − 8.00000i − 0.262754i
$$928$$ 10.0000i 0.328266i
$$929$$ −26.0000 −0.853032 −0.426516 0.904480i $$-0.640259\pi$$
−0.426516 + 0.904480i $$0.640259\pi$$
$$930$$ 0 0
$$931$$ 4.00000 0.131095
$$932$$ − 6.00000i − 0.196537i
$$933$$ − 24.0000i − 0.785725i
$$934$$ 36.0000 1.17796
$$935$$ 0 0
$$936$$ −6.00000 −0.196116
$$937$$ − 42.0000i − 1.37208i −0.727564 0.686040i $$-0.759347\pi$$
0.727564 0.686040i $$-0.240653\pi$$
$$938$$ 4.00000i 0.130605i
$$939$$ −26.0000 −0.848478
$$940$$ 0 0
$$941$$ 38.0000 1.23876 0.619382 0.785090i $$-0.287383\pi$$
0.619382 + 0.785090i $$0.287383\pi$$
$$942$$ − 2.00000i − 0.0651635i
$$943$$ 0 0
$$944$$ 12.0000 0.390567
$$945$$ 0 0
$$946$$ 16.0000 0.520205
$$947$$ − 44.0000i − 1.42981i −0.699223 0.714904i $$-0.746470\pi$$
0.699223 0.714904i $$-0.253530\pi$$
$$948$$ 16.0000i 0.519656i
$$949$$ −12.0000 −0.389536
$$950$$ 0 0
$$951$$ −18.0000 −0.583690
$$952$$ − 18.0000i − 0.583383i
$$953$$ 26.0000i 0.842223i 0.907009 + 0.421111i $$0.138360\pi$$
−0.907009 + 0.421111i $$0.861640\pi$$
$$954$$ 6.00000 0.194257
$$955$$ 0 0
$$956$$ −24.0000 −0.776215
$$957$$ 8.00000i 0.258603i
$$958$$ 16.0000i 0.516937i
$$959$$ −6.00000 −0.193750
$$960$$ 0 0
$$961$$ −31.0000 −1.00000
$$962$$ − 12.0000i − 0.386896i
$$963$$ 4.00000i 0.128898i
$$964$$ 2.00000 0.0644157
$$965$$ 0 0
$$966$$ 0 0
$$967$$ − 40.0000i − 1.28631i −0.765735 0.643157i $$-0.777624\pi$$
0.765735 0.643157i $$-0.222376\pi$$
$$968$$ 15.0000i 0.482118i
$$969$$ 24.0000 0.770991
$$970$$ 0 0
$$971$$ 12.0000 0.385098 0.192549 0.981287i $$-0.438325\pi$$
0.192549 + 0.981287i $$0.438325\pi$$
$$972$$ 1.00000i 0.0320750i
$$973$$ − 12.0000i − 0.384702i
$$974$$ −8.00000 −0.256337
$$975$$ 0 0
$$976$$ 2.00000 0.0640184
$$977$$ 30.0000i 0.959785i 0.877327 + 0.479893i $$0.159324\pi$$
−0.877327 + 0.479893i $$0.840676\pi$$
$$978$$ − 4.00000i − 0.127906i
$$979$$ 56.0000 1.78977
$$980$$ 0 0
$$981$$ −18.0000 −0.574696
$$982$$ 20.0000i 0.638226i
$$983$$ 24.0000i 0.765481i 0.923856 + 0.382741i $$0.125020\pi$$
−0.923856 + 0.382741i $$0.874980\pi$$
$$984$$ −6.00000 −0.191273
$$985$$ 0 0
$$986$$ −12.0000 −0.382158
$$987$$ 0 0
$$988$$ 8.00000i 0.254514i
$$989$$ 0 0
$$990$$ 0 0
$$991$$ −16.0000 −0.508257 −0.254128 0.967170i $$-0.581789\pi$$
−0.254128 + 0.967170i $$0.581789\pi$$
$$992$$ 0 0
$$993$$ − 4.00000i − 0.126936i
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 12.0000 0.380235
$$997$$ 26.0000i 0.823428i 0.911313 + 0.411714i $$0.135070\pi$$
−0.911313 + 0.411714i $$0.864930\pi$$
$$998$$ − 4.00000i − 0.126618i
$$999$$ −6.00000 −0.189832
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 525.2.d.a.274.2 2
3.2 odd 2 1575.2.d.a.1324.1 2
5.2 odd 4 21.2.a.a.1.1 1
5.3 odd 4 525.2.a.d.1.1 1
5.4 even 2 inner 525.2.d.a.274.1 2
15.2 even 4 63.2.a.a.1.1 1
15.8 even 4 1575.2.a.c.1.1 1
15.14 odd 2 1575.2.d.a.1324.2 2
20.3 even 4 8400.2.a.bn.1.1 1
20.7 even 4 336.2.a.a.1.1 1
35.2 odd 12 147.2.e.b.67.1 2
35.12 even 12 147.2.e.c.67.1 2
35.13 even 4 3675.2.a.n.1.1 1
35.17 even 12 147.2.e.c.79.1 2
35.27 even 4 147.2.a.a.1.1 1
35.32 odd 12 147.2.e.b.79.1 2
40.27 even 4 1344.2.a.s.1.1 1
40.37 odd 4 1344.2.a.g.1.1 1
45.2 even 12 567.2.f.b.190.1 2
45.7 odd 12 567.2.f.g.190.1 2
45.22 odd 12 567.2.f.g.379.1 2
45.32 even 12 567.2.f.b.379.1 2
55.32 even 4 2541.2.a.j.1.1 1
60.47 odd 4 1008.2.a.l.1.1 1
65.12 odd 4 3549.2.a.c.1.1 1
80.27 even 4 5376.2.c.l.2689.2 2
80.37 odd 4 5376.2.c.r.2689.1 2
80.67 even 4 5376.2.c.l.2689.1 2
80.77 odd 4 5376.2.c.r.2689.2 2
85.67 odd 4 6069.2.a.b.1.1 1
95.37 even 4 7581.2.a.d.1.1 1
105.2 even 12 441.2.e.a.361.1 2
105.17 odd 12 441.2.e.b.226.1 2
105.32 even 12 441.2.e.a.226.1 2
105.47 odd 12 441.2.e.b.361.1 2
105.62 odd 4 441.2.a.f.1.1 1
120.77 even 4 4032.2.a.h.1.1 1
120.107 odd 4 4032.2.a.k.1.1 1
140.27 odd 4 2352.2.a.v.1.1 1
140.47 odd 12 2352.2.q.e.1537.1 2
140.67 even 12 2352.2.q.x.961.1 2
140.87 odd 12 2352.2.q.e.961.1 2
140.107 even 12 2352.2.q.x.1537.1 2
165.32 odd 4 7623.2.a.g.1.1 1
280.27 odd 4 9408.2.a.m.1.1 1
280.237 even 4 9408.2.a.bv.1.1 1
420.167 even 4 7056.2.a.p.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
21.2.a.a.1.1 1 5.2 odd 4
63.2.a.a.1.1 1 15.2 even 4
147.2.a.a.1.1 1 35.27 even 4
147.2.e.b.67.1 2 35.2 odd 12
147.2.e.b.79.1 2 35.32 odd 12
147.2.e.c.67.1 2 35.12 even 12
147.2.e.c.79.1 2 35.17 even 12
336.2.a.a.1.1 1 20.7 even 4
441.2.a.f.1.1 1 105.62 odd 4
441.2.e.a.226.1 2 105.32 even 12
441.2.e.a.361.1 2 105.2 even 12
441.2.e.b.226.1 2 105.17 odd 12
441.2.e.b.361.1 2 105.47 odd 12
525.2.a.d.1.1 1 5.3 odd 4
525.2.d.a.274.1 2 5.4 even 2 inner
525.2.d.a.274.2 2 1.1 even 1 trivial
567.2.f.b.190.1 2 45.2 even 12
567.2.f.b.379.1 2 45.32 even 12
567.2.f.g.190.1 2 45.7 odd 12
567.2.f.g.379.1 2 45.22 odd 12
1008.2.a.l.1.1 1 60.47 odd 4
1344.2.a.g.1.1 1 40.37 odd 4
1344.2.a.s.1.1 1 40.27 even 4
1575.2.a.c.1.1 1 15.8 even 4
1575.2.d.a.1324.1 2 3.2 odd 2
1575.2.d.a.1324.2 2 15.14 odd 2
2352.2.a.v.1.1 1 140.27 odd 4
2352.2.q.e.961.1 2 140.87 odd 12
2352.2.q.e.1537.1 2 140.47 odd 12
2352.2.q.x.961.1 2 140.67 even 12
2352.2.q.x.1537.1 2 140.107 even 12
2541.2.a.j.1.1 1 55.32 even 4
3549.2.a.c.1.1 1 65.12 odd 4
3675.2.a.n.1.1 1 35.13 even 4
4032.2.a.h.1.1 1 120.77 even 4
4032.2.a.k.1.1 1 120.107 odd 4
5376.2.c.l.2689.1 2 80.67 even 4
5376.2.c.l.2689.2 2 80.27 even 4
5376.2.c.r.2689.1 2 80.37 odd 4
5376.2.c.r.2689.2 2 80.77 odd 4
6069.2.a.b.1.1 1 85.67 odd 4
7056.2.a.p.1.1 1 420.167 even 4
7581.2.a.d.1.1 1 95.37 even 4
7623.2.a.g.1.1 1 165.32 odd 4
8400.2.a.bn.1.1 1 20.3 even 4
9408.2.a.m.1.1 1 280.27 odd 4
9408.2.a.bv.1.1 1 280.237 even 4