# Properties

 Label 2925.2.c.g.2224.1 Level $2925$ Weight $2$ Character 2925.2224 Analytic conductor $23.356$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2925 = 3^{2} \cdot 5^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2925.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$23.3562425912$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 585) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 2224.1 Root $$-1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 2925.2224 Dual form 2925.2.c.g.2224.2

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000i q^{2} +1.00000 q^{4} +2.00000i q^{7} -3.00000i q^{8} +O(q^{10})$$ $$q-1.00000i q^{2} +1.00000 q^{4} +2.00000i q^{7} -3.00000i q^{8} -4.00000 q^{11} +1.00000i q^{13} +2.00000 q^{14} -1.00000 q^{16} -4.00000i q^{17} -6.00000 q^{19} +4.00000i q^{22} +1.00000 q^{26} +2.00000i q^{28} +4.00000 q^{29} -10.0000 q^{31} -5.00000i q^{32} -4.00000 q^{34} -2.00000i q^{37} +6.00000i q^{38} -6.00000 q^{41} +8.00000i q^{43} -4.00000 q^{44} -8.00000i q^{47} +3.00000 q^{49} +1.00000i q^{52} +4.00000i q^{53} +6.00000 q^{56} -4.00000i q^{58} -12.0000 q^{59} +2.00000 q^{61} +10.0000i q^{62} -7.00000 q^{64} -10.0000i q^{67} -4.00000i q^{68} +6.00000i q^{73} -2.00000 q^{74} -6.00000 q^{76} -8.00000i q^{77} -12.0000 q^{79} +6.00000i q^{82} +4.00000i q^{83} +8.00000 q^{86} +12.0000i q^{88} -14.0000 q^{89} -2.00000 q^{91} -8.00000 q^{94} -14.0000i q^{97} -3.00000i q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{4}+O(q^{10})$$ 2 * q + 2 * q^4 $$2 q + 2 q^{4} - 8 q^{11} + 4 q^{14} - 2 q^{16} - 12 q^{19} + 2 q^{26} + 8 q^{29} - 20 q^{31} - 8 q^{34} - 12 q^{41} - 8 q^{44} + 6 q^{49} + 12 q^{56} - 24 q^{59} + 4 q^{61} - 14 q^{64} - 4 q^{74} - 12 q^{76} - 24 q^{79} + 16 q^{86} - 28 q^{89} - 4 q^{91} - 16 q^{94}+O(q^{100})$$ 2 * q + 2 * q^4 - 8 * q^11 + 4 * q^14 - 2 * q^16 - 12 * q^19 + 2 * q^26 + 8 * q^29 - 20 * q^31 - 8 * q^34 - 12 * q^41 - 8 * q^44 + 6 * q^49 + 12 * q^56 - 24 * q^59 + 4 * q^61 - 14 * q^64 - 4 * q^74 - 12 * q^76 - 24 * q^79 + 16 * q^86 - 28 * q^89 - 4 * q^91 - 16 * q^94

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2925\mathbb{Z}\right)^\times$$.

 $$n$$ $$326$$ $$352$$ $$2251$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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 $$-0.884973\pi$$
0.935414 0.353553i $$-0.115027\pi$$
$$3$$ 0 0
$$4$$ 1.00000 0.500000
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 2.00000i 0.755929i 0.925820 + 0.377964i $$0.123376\pi$$
−0.925820 + 0.377964i $$0.876624\pi$$
$$8$$ − 3.00000i − 1.06066i
$$9$$ 0 0
$$10$$ 0 0
$$11$$ −4.00000 −1.20605 −0.603023 0.797724i $$-0.706037\pi$$
−0.603023 + 0.797724i $$0.706037\pi$$
$$12$$ 0 0
$$13$$ 1.00000i 0.277350i
$$14$$ 2.00000 0.534522
$$15$$ 0 0
$$16$$ −1.00000 −0.250000
$$17$$ − 4.00000i − 0.970143i −0.874475 0.485071i $$-0.838794\pi$$
0.874475 0.485071i $$-0.161206\pi$$
$$18$$ 0 0
$$19$$ −6.00000 −1.37649 −0.688247 0.725476i $$-0.741620\pi$$
−0.688247 + 0.725476i $$0.741620\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 4.00000i 0.852803i
$$23$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 1.00000 0.196116
$$27$$ 0 0
$$28$$ 2.00000i 0.377964i
$$29$$ 4.00000 0.742781 0.371391 0.928477i $$-0.378881\pi$$
0.371391 + 0.928477i $$0.378881\pi$$
$$30$$ 0 0
$$31$$ −10.0000 −1.79605 −0.898027 0.439941i $$-0.854999\pi$$
−0.898027 + 0.439941i $$0.854999\pi$$
$$32$$ − 5.00000i − 0.883883i
$$33$$ 0 0
$$34$$ −4.00000 −0.685994
$$35$$ 0 0
$$36$$ 0 0
$$37$$ − 2.00000i − 0.328798i −0.986394 0.164399i $$-0.947432\pi$$
0.986394 0.164399i $$-0.0525685\pi$$
$$38$$ 6.00000i 0.973329i
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −6.00000 −0.937043 −0.468521 0.883452i $$-0.655213\pi$$
−0.468521 + 0.883452i $$0.655213\pi$$
$$42$$ 0 0
$$43$$ 8.00000i 1.21999i 0.792406 + 0.609994i $$0.208828\pi$$
−0.792406 + 0.609994i $$0.791172\pi$$
$$44$$ −4.00000 −0.603023
$$45$$ 0 0
$$46$$ 0 0
$$47$$ − 8.00000i − 1.16692i −0.812142 0.583460i $$-0.801699\pi$$
0.812142 0.583460i $$-0.198301\pi$$
$$48$$ 0 0
$$49$$ 3.00000 0.428571
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 1.00000i 0.138675i
$$53$$ 4.00000i 0.549442i 0.961524 + 0.274721i $$0.0885855\pi$$
−0.961524 + 0.274721i $$0.911414\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 6.00000 0.801784
$$57$$ 0 0
$$58$$ − 4.00000i − 0.525226i
$$59$$ −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 $$-0.459132\pi$$
0.128037 + 0.991769i $$0.459132\pi$$
$$62$$ 10.0000i 1.27000i
$$63$$ 0 0
$$64$$ −7.00000 −0.875000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ − 10.0000i − 1.22169i −0.791748 0.610847i $$-0.790829\pi$$
0.791748 0.610847i $$-0.209171\pi$$
$$68$$ − 4.00000i − 0.485071i
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$72$$ 0 0
$$73$$ 6.00000i 0.702247i 0.936329 + 0.351123i $$0.114200\pi$$
−0.936329 + 0.351123i $$0.885800\pi$$
$$74$$ −2.00000 −0.232495
$$75$$ 0 0
$$76$$ −6.00000 −0.688247
$$77$$ − 8.00000i − 0.911685i
$$78$$ 0 0
$$79$$ −12.0000 −1.35011 −0.675053 0.737769i $$-0.735879\pi$$
−0.675053 + 0.737769i $$0.735879\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 6.00000i 0.662589i
$$83$$ 4.00000i 0.439057i 0.975606 + 0.219529i $$0.0704519\pi$$
−0.975606 + 0.219529i $$0.929548\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 8.00000 0.862662
$$87$$ 0 0
$$88$$ 12.0000i 1.27920i
$$89$$ −14.0000 −1.48400 −0.741999 0.670402i $$-0.766122\pi$$
−0.741999 + 0.670402i $$0.766122\pi$$
$$90$$ 0 0
$$91$$ −2.00000 −0.209657
$$92$$ 0 0
$$93$$ 0 0
$$94$$ −8.00000 −0.825137
$$95$$ 0 0
$$96$$ 0 0
$$97$$ − 14.0000i − 1.42148i −0.703452 0.710742i $$-0.748359\pi$$
0.703452 0.710742i $$-0.251641\pi$$
$$98$$ − 3.00000i − 0.303046i
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 12.0000 1.19404 0.597022 0.802225i $$-0.296350\pi$$
0.597022 + 0.802225i $$0.296350\pi$$
$$102$$ 0 0
$$103$$ 4.00000i 0.394132i 0.980390 + 0.197066i $$0.0631413\pi$$
−0.980390 + 0.197066i $$0.936859\pi$$
$$104$$ 3.00000 0.294174
$$105$$ 0 0
$$106$$ 4.00000 0.388514
$$107$$ 4.00000i 0.386695i 0.981130 + 0.193347i $$0.0619344\pi$$
−0.981130 + 0.193347i $$0.938066\pi$$
$$108$$ 0 0
$$109$$ 2.00000 0.191565 0.0957826 0.995402i $$-0.469465\pi$$
0.0957826 + 0.995402i $$0.469465\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ − 2.00000i − 0.188982i
$$113$$ − 16.0000i − 1.50515i −0.658505 0.752577i $$-0.728811\pi$$
0.658505 0.752577i $$-0.271189\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 4.00000 0.371391
$$117$$ 0 0
$$118$$ 12.0000i 1.10469i
$$119$$ 8.00000 0.733359
$$120$$ 0 0
$$121$$ 5.00000 0.454545
$$122$$ − 2.00000i − 0.181071i
$$123$$ 0 0
$$124$$ −10.0000 −0.898027
$$125$$ 0 0
$$126$$ 0 0
$$127$$ − 8.00000i − 0.709885i −0.934888 0.354943i $$-0.884500\pi$$
0.934888 0.354943i $$-0.115500\pi$$
$$128$$ − 3.00000i − 0.265165i
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −16.0000 −1.39793 −0.698963 0.715158i $$-0.746355\pi$$
−0.698963 + 0.715158i $$0.746355\pi$$
$$132$$ 0 0
$$133$$ − 12.0000i − 1.04053i
$$134$$ −10.0000 −0.863868
$$135$$ 0 0
$$136$$ −12.0000 −1.02899
$$137$$ 10.0000i 0.854358i 0.904167 + 0.427179i $$0.140493\pi$$
−0.904167 + 0.427179i $$0.859507\pi$$
$$138$$ 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
$$143$$ − 4.00000i − 0.334497i
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 6.00000 0.496564
$$147$$ 0 0
$$148$$ − 2.00000i − 0.164399i
$$149$$ 18.0000 1.47462 0.737309 0.675556i $$-0.236096\pi$$
0.737309 + 0.675556i $$0.236096\pi$$
$$150$$ 0 0
$$151$$ 10.0000 0.813788 0.406894 0.913475i $$-0.366612\pi$$
0.406894 + 0.913475i $$0.366612\pi$$
$$152$$ 18.0000i 1.45999i
$$153$$ 0 0
$$154$$ −8.00000 −0.644658
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 18.0000i 1.43656i 0.695756 + 0.718278i $$0.255069\pi$$
−0.695756 + 0.718278i $$0.744931\pi$$
$$158$$ 12.0000i 0.954669i
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ − 6.00000i − 0.469956i −0.972001 0.234978i $$-0.924498\pi$$
0.972001 0.234978i $$-0.0755019\pi$$
$$164$$ −6.00000 −0.468521
$$165$$ 0 0
$$166$$ 4.00000 0.310460
$$167$$ − 24.0000i − 1.85718i −0.371113 0.928588i $$-0.621024\pi$$
0.371113 0.928588i $$-0.378976\pi$$
$$168$$ 0 0
$$169$$ −1.00000 −0.0769231
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 8.00000i 0.609994i
$$173$$ − 24.0000i − 1.82469i −0.409426 0.912343i $$-0.634271\pi$$
0.409426 0.912343i $$-0.365729\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 4.00000 0.301511
$$177$$ 0 0
$$178$$ 14.0000i 1.04934i
$$179$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$180$$ 0 0
$$181$$ −10.0000 −0.743294 −0.371647 0.928374i $$-0.621207\pi$$
−0.371647 + 0.928374i $$0.621207\pi$$
$$182$$ 2.00000i 0.148250i
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 16.0000i 1.17004i
$$188$$ − 8.00000i − 0.583460i
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −12.0000 −0.868290 −0.434145 0.900843i $$-0.642949\pi$$
−0.434145 + 0.900843i $$0.642949\pi$$
$$192$$ 0 0
$$193$$ 26.0000i 1.87152i 0.352636 + 0.935760i $$0.385285\pi$$
−0.352636 + 0.935760i $$0.614715\pi$$
$$194$$ −14.0000 −1.00514
$$195$$ 0 0
$$196$$ 3.00000 0.214286
$$197$$ 18.0000i 1.28245i 0.767354 + 0.641223i $$0.221573\pi$$
−0.767354 + 0.641223i $$0.778427\pi$$
$$198$$ 0 0
$$199$$ −8.00000 −0.567105 −0.283552 0.958957i $$-0.591513\pi$$
−0.283552 + 0.958957i $$0.591513\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ − 12.0000i − 0.844317i
$$203$$ 8.00000i 0.561490i
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 4.00000 0.278693
$$207$$ 0 0
$$208$$ − 1.00000i − 0.0693375i
$$209$$ 24.0000 1.66011
$$210$$ 0 0
$$211$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$212$$ 4.00000i 0.274721i
$$213$$ 0 0
$$214$$ 4.00000 0.273434
$$215$$ 0 0
$$216$$ 0 0
$$217$$ − 20.0000i − 1.35769i
$$218$$ − 2.00000i − 0.135457i
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 4.00000 0.269069
$$222$$ 0 0
$$223$$ 14.0000i 0.937509i 0.883328 + 0.468755i $$0.155297\pi$$
−0.883328 + 0.468755i $$0.844703\pi$$
$$224$$ 10.0000 0.668153
$$225$$ 0 0
$$226$$ −16.0000 −1.06430
$$227$$ − 20.0000i − 1.32745i −0.747978 0.663723i $$-0.768975\pi$$
0.747978 0.663723i $$-0.231025\pi$$
$$228$$ 0 0
$$229$$ 10.0000 0.660819 0.330409 0.943838i $$-0.392813\pi$$
0.330409 + 0.943838i $$0.392813\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ − 12.0000i − 0.787839i
$$233$$ 8.00000i 0.524097i 0.965055 + 0.262049i $$0.0843981\pi$$
−0.965055 + 0.262049i $$0.915602\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ −12.0000 −0.781133
$$237$$ 0 0
$$238$$ − 8.00000i − 0.518563i
$$239$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$240$$ 0 0
$$241$$ 22.0000 1.41714 0.708572 0.705638i $$-0.249340\pi$$
0.708572 + 0.705638i $$0.249340\pi$$
$$242$$ − 5.00000i − 0.321412i
$$243$$ 0 0
$$244$$ 2.00000 0.128037
$$245$$ 0 0
$$246$$ 0 0
$$247$$ − 6.00000i − 0.381771i
$$248$$ 30.0000i 1.90500i
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 12.0000 0.757433 0.378717 0.925513i $$-0.376365\pi$$
0.378717 + 0.925513i $$0.376365\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ −8.00000 −0.501965
$$255$$ 0 0
$$256$$ −17.0000 −1.06250
$$257$$ 8.00000i 0.499026i 0.968371 + 0.249513i $$0.0802706\pi$$
−0.968371 + 0.249513i $$0.919729\pi$$
$$258$$ 0 0
$$259$$ 4.00000 0.248548
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 16.0000i 0.988483i
$$263$$ 4.00000i 0.246651i 0.992366 + 0.123325i $$0.0393559\pi$$
−0.992366 + 0.123325i $$0.960644\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ −12.0000 −0.735767
$$267$$ 0 0
$$268$$ − 10.0000i − 0.610847i
$$269$$ −12.0000 −0.731653 −0.365826 0.930683i $$-0.619214\pi$$
−0.365826 + 0.930683i $$0.619214\pi$$
$$270$$ 0 0
$$271$$ 14.0000 0.850439 0.425220 0.905090i $$-0.360197\pi$$
0.425220 + 0.905090i $$0.360197\pi$$
$$272$$ 4.00000i 0.242536i
$$273$$ 0 0
$$274$$ 10.0000 0.604122
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 10.0000i 0.600842i 0.953807 + 0.300421i $$0.0971271\pi$$
−0.953807 + 0.300421i $$0.902873\pi$$
$$278$$ 12.0000i 0.719712i
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −30.0000 −1.78965 −0.894825 0.446417i $$-0.852700\pi$$
−0.894825 + 0.446417i $$0.852700\pi$$
$$282$$ 0 0
$$283$$ 4.00000i 0.237775i 0.992908 + 0.118888i $$0.0379328\pi$$
−0.992908 + 0.118888i $$0.962067\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ −4.00000 −0.236525
$$287$$ − 12.0000i − 0.708338i
$$288$$ 0 0
$$289$$ 1.00000 0.0588235
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 6.00000i 0.351123i
$$293$$ 2.00000i 0.116841i 0.998292 + 0.0584206i $$0.0186065\pi$$
−0.998292 + 0.0584206i $$0.981394\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ −6.00000 −0.348743
$$297$$ 0 0
$$298$$ − 18.0000i − 1.04271i
$$299$$ 0 0
$$300$$ 0 0
$$301$$ −16.0000 −0.922225
$$302$$ − 10.0000i − 0.575435i
$$303$$ 0 0
$$304$$ 6.00000 0.344124
$$305$$ 0 0
$$306$$ 0 0
$$307$$ − 22.0000i − 1.25561i −0.778372 0.627803i $$-0.783954\pi$$
0.778372 0.627803i $$-0.216046\pi$$
$$308$$ − 8.00000i − 0.455842i
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −4.00000 −0.226819 −0.113410 0.993548i $$-0.536177\pi$$
−0.113410 + 0.993548i $$0.536177\pi$$
$$312$$ 0 0
$$313$$ − 2.00000i − 0.113047i −0.998401 0.0565233i $$-0.981998\pi$$
0.998401 0.0565233i $$-0.0180015\pi$$
$$314$$ 18.0000 1.01580
$$315$$ 0 0
$$316$$ −12.0000 −0.675053
$$317$$ 18.0000i 1.01098i 0.862832 + 0.505490i $$0.168688\pi$$
−0.862832 + 0.505490i $$0.831312\pi$$
$$318$$ 0 0
$$319$$ −16.0000 −0.895828
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 24.0000i 1.33540i
$$324$$ 0 0
$$325$$ 0 0
$$326$$ −6.00000 −0.332309
$$327$$ 0 0
$$328$$ 18.0000i 0.993884i
$$329$$ 16.0000 0.882109
$$330$$ 0 0
$$331$$ 18.0000 0.989369 0.494685 0.869072i $$-0.335284\pi$$
0.494685 + 0.869072i $$0.335284\pi$$
$$332$$ 4.00000i 0.219529i
$$333$$ 0 0
$$334$$ −24.0000 −1.31322
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 14.0000i 0.762629i 0.924445 + 0.381314i $$0.124528\pi$$
−0.924445 + 0.381314i $$0.875472\pi$$
$$338$$ 1.00000i 0.0543928i
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 40.0000 2.16612
$$342$$ 0 0
$$343$$ 20.0000i 1.07990i
$$344$$ 24.0000 1.29399
$$345$$ 0 0
$$346$$ −24.0000 −1.29025
$$347$$ 32.0000i 1.71785i 0.512101 + 0.858925i $$0.328867\pi$$
−0.512101 + 0.858925i $$0.671133\pi$$
$$348$$ 0 0
$$349$$ 6.00000 0.321173 0.160586 0.987022i $$-0.448662\pi$$
0.160586 + 0.987022i $$0.448662\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 20.0000i 1.06600i
$$353$$ 6.00000i 0.319348i 0.987170 + 0.159674i $$0.0510443\pi$$
−0.987170 + 0.159674i $$0.948956\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ −14.0000 −0.741999
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 16.0000 0.844448 0.422224 0.906492i $$-0.361250\pi$$
0.422224 + 0.906492i $$0.361250\pi$$
$$360$$ 0 0
$$361$$ 17.0000 0.894737
$$362$$ 10.0000i 0.525588i
$$363$$ 0 0
$$364$$ −2.00000 −0.104828
$$365$$ 0 0
$$366$$ 0 0
$$367$$ − 8.00000i − 0.417597i −0.977959 0.208798i $$-0.933045\pi$$
0.977959 0.208798i $$-0.0669552\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −8.00000 −0.415339
$$372$$ 0 0
$$373$$ − 22.0000i − 1.13912i −0.821951 0.569558i $$-0.807114\pi$$
0.821951 0.569558i $$-0.192886\pi$$
$$374$$ 16.0000 0.827340
$$375$$ 0 0
$$376$$ −24.0000 −1.23771
$$377$$ 4.00000i 0.206010i
$$378$$ 0 0
$$379$$ 2.00000 0.102733 0.0513665 0.998680i $$-0.483642\pi$$
0.0513665 + 0.998680i $$0.483642\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 12.0000i 0.613973i
$$383$$ 24.0000i 1.22634i 0.789950 + 0.613171i $$0.210106\pi$$
−0.789950 + 0.613171i $$0.789894\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 26.0000 1.32337
$$387$$ 0 0
$$388$$ − 14.0000i − 0.710742i
$$389$$ −8.00000 −0.405616 −0.202808 0.979219i $$-0.565007\pi$$
−0.202808 + 0.979219i $$0.565007\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ − 9.00000i − 0.454569i
$$393$$ 0 0
$$394$$ 18.0000 0.906827
$$395$$ 0 0
$$396$$ 0 0
$$397$$ − 18.0000i − 0.903394i −0.892171 0.451697i $$-0.850819\pi$$
0.892171 0.451697i $$-0.149181\pi$$
$$398$$ 8.00000i 0.401004i
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 22.0000 1.09863 0.549314 0.835616i $$-0.314889\pi$$
0.549314 + 0.835616i $$0.314889\pi$$
$$402$$ 0 0
$$403$$ − 10.0000i − 0.498135i
$$404$$ 12.0000 0.597022
$$405$$ 0 0
$$406$$ 8.00000 0.397033
$$407$$ 8.00000i 0.396545i
$$408$$ 0 0
$$409$$ −30.0000 −1.48340 −0.741702 0.670729i $$-0.765981\pi$$
−0.741702 + 0.670729i $$0.765981\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 4.00000i 0.197066i
$$413$$ − 24.0000i − 1.18096i
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 5.00000 0.245145
$$417$$ 0 0
$$418$$ − 24.0000i − 1.17388i
$$419$$ 16.0000 0.781651 0.390826 0.920465i $$-0.372190\pi$$
0.390826 + 0.920465i $$0.372190\pi$$
$$420$$ 0 0
$$421$$ 22.0000 1.07221 0.536107 0.844150i $$-0.319894\pi$$
0.536107 + 0.844150i $$0.319894\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 12.0000 0.582772
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 4.00000i 0.193574i
$$428$$ 4.00000i 0.193347i
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −40.0000 −1.92673 −0.963366 0.268190i $$-0.913575\pi$$
−0.963366 + 0.268190i $$0.913575\pi$$
$$432$$ 0 0
$$433$$ 2.00000i 0.0961139i 0.998845 + 0.0480569i $$0.0153029\pi$$
−0.998845 + 0.0480569i $$0.984697\pi$$
$$434$$ −20.0000 −0.960031
$$435$$ 0 0
$$436$$ 2.00000 0.0957826
$$437$$ 0 0
$$438$$ 0 0
$$439$$ 36.0000 1.71819 0.859093 0.511819i $$-0.171028\pi$$
0.859093 + 0.511819i $$0.171028\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ − 4.00000i − 0.190261i
$$443$$ − 20.0000i − 0.950229i −0.879924 0.475114i $$-0.842407\pi$$
0.879924 0.475114i $$-0.157593\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 14.0000 0.662919
$$447$$ 0 0
$$448$$ − 14.0000i − 0.661438i
$$449$$ −30.0000 −1.41579 −0.707894 0.706319i $$-0.750354\pi$$
−0.707894 + 0.706319i $$0.750354\pi$$
$$450$$ 0 0
$$451$$ 24.0000 1.13012
$$452$$ − 16.0000i − 0.752577i
$$453$$ 0 0
$$454$$ −20.0000 −0.938647
$$455$$ 0 0
$$456$$ 0 0
$$457$$ − 22.0000i − 1.02912i −0.857455 0.514558i $$-0.827956\pi$$
0.857455 0.514558i $$-0.172044\pi$$
$$458$$ − 10.0000i − 0.467269i
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −14.0000 −0.652045 −0.326023 0.945362i $$-0.605709\pi$$
−0.326023 + 0.945362i $$0.605709\pi$$
$$462$$ 0 0
$$463$$ − 18.0000i − 0.836531i −0.908325 0.418265i $$-0.862638\pi$$
0.908325 0.418265i $$-0.137362\pi$$
$$464$$ −4.00000 −0.185695
$$465$$ 0 0
$$466$$ 8.00000 0.370593
$$467$$ 8.00000i 0.370196i 0.982720 + 0.185098i $$0.0592602\pi$$
−0.982720 + 0.185098i $$0.940740\pi$$
$$468$$ 0 0
$$469$$ 20.0000 0.923514
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 36.0000i 1.65703i
$$473$$ − 32.0000i − 1.47136i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 8.00000 0.366679
$$477$$ 0 0
$$478$$ 0 0
$$479$$ 24.0000 1.09659 0.548294 0.836286i $$-0.315277\pi$$
0.548294 + 0.836286i $$0.315277\pi$$
$$480$$ 0 0
$$481$$ 2.00000 0.0911922
$$482$$ − 22.0000i − 1.00207i
$$483$$ 0 0
$$484$$ 5.00000 0.227273
$$485$$ 0 0
$$486$$ 0 0
$$487$$ − 22.0000i − 0.996915i −0.866914 0.498458i $$-0.833900\pi$$
0.866914 0.498458i $$-0.166100\pi$$
$$488$$ − 6.00000i − 0.271607i
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 36.0000 1.62466 0.812329 0.583200i $$-0.198200\pi$$
0.812329 + 0.583200i $$0.198200\pi$$
$$492$$ 0 0
$$493$$ − 16.0000i − 0.720604i
$$494$$ −6.00000 −0.269953
$$495$$ 0 0
$$496$$ 10.0000 0.449013
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 38.0000 1.70111 0.850557 0.525883i $$-0.176265\pi$$
0.850557 + 0.525883i $$0.176265\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ − 12.0000i − 0.535586i
$$503$$ − 12.0000i − 0.535054i −0.963550 0.267527i $$-0.913794\pi$$
0.963550 0.267527i $$-0.0862064\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 0 0
$$508$$ − 8.00000i − 0.354943i
$$509$$ −42.0000 −1.86162 −0.930809 0.365507i $$-0.880896\pi$$
−0.930809 + 0.365507i $$0.880896\pi$$
$$510$$ 0 0
$$511$$ −12.0000 −0.530849
$$512$$ 11.0000i 0.486136i
$$513$$ 0 0
$$514$$ 8.00000 0.352865
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 32.0000i 1.40736i
$$518$$ − 4.00000i − 0.175750i
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 12.0000 0.525730 0.262865 0.964833i $$-0.415333\pi$$
0.262865 + 0.964833i $$0.415333\pi$$
$$522$$ 0 0
$$523$$ 12.0000i 0.524723i 0.964970 + 0.262362i $$0.0845013\pi$$
−0.964970 + 0.262362i $$0.915499\pi$$
$$524$$ −16.0000 −0.698963
$$525$$ 0 0
$$526$$ 4.00000 0.174408
$$527$$ 40.0000i 1.74243i
$$528$$ 0 0
$$529$$ 23.0000 1.00000
$$530$$ 0 0
$$531$$ 0 0
$$532$$ − 12.0000i − 0.520266i
$$533$$ − 6.00000i − 0.259889i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ −30.0000 −1.29580
$$537$$ 0 0
$$538$$ 12.0000i 0.517357i
$$539$$ −12.0000 −0.516877
$$540$$ 0 0
$$541$$ 14.0000 0.601907 0.300954 0.953639i $$-0.402695\pi$$
0.300954 + 0.953639i $$0.402695\pi$$
$$542$$ − 14.0000i − 0.601351i
$$543$$ 0 0
$$544$$ −20.0000 −0.857493
$$545$$ 0 0
$$546$$ 0 0
$$547$$ − 36.0000i − 1.53925i −0.638497 0.769624i $$-0.720443\pi$$
0.638497 0.769624i $$-0.279557\pi$$
$$548$$ 10.0000i 0.427179i
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −24.0000 −1.02243
$$552$$ 0 0
$$553$$ − 24.0000i − 1.02058i
$$554$$ 10.0000 0.424859
$$555$$ 0 0
$$556$$ −12.0000 −0.508913
$$557$$ − 26.0000i − 1.10166i −0.834619 0.550828i $$-0.814312\pi$$
0.834619 0.550828i $$-0.185688\pi$$
$$558$$ 0 0
$$559$$ −8.00000 −0.338364
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 30.0000i 1.26547i
$$563$$ 4.00000i 0.168580i 0.996441 + 0.0842900i $$0.0268622\pi$$
−0.996441 + 0.0842900i $$0.973138\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 4.00000 0.168133
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −28.0000 −1.17382 −0.586911 0.809652i $$-0.699656\pi$$
−0.586911 + 0.809652i $$0.699656\pi$$
$$570$$ 0 0
$$571$$ −12.0000 −0.502184 −0.251092 0.967963i $$-0.580790\pi$$
−0.251092 + 0.967963i $$0.580790\pi$$
$$572$$ − 4.00000i − 0.167248i
$$573$$ 0 0
$$574$$ −12.0000 −0.500870
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 38.0000i 1.58196i 0.611842 + 0.790980i $$0.290429\pi$$
−0.611842 + 0.790980i $$0.709571\pi$$
$$578$$ − 1.00000i − 0.0415945i
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −8.00000 −0.331896
$$582$$ 0 0
$$583$$ − 16.0000i − 0.662652i
$$584$$ 18.0000 0.744845
$$585$$ 0 0
$$586$$ 2.00000 0.0826192
$$587$$ − 20.0000i − 0.825488i −0.910847 0.412744i $$-0.864570\pi$$
0.910847 0.412744i $$-0.135430\pi$$
$$588$$ 0 0
$$589$$ 60.0000 2.47226
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 2.00000i 0.0821995i
$$593$$ − 14.0000i − 0.574911i −0.957794 0.287456i $$-0.907191\pi$$
0.957794 0.287456i $$-0.0928094\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 18.0000 0.737309
$$597$$ 0 0
$$598$$ 0 0
$$599$$ −8.00000 −0.326871 −0.163436 0.986554i $$-0.552258\pi$$
−0.163436 + 0.986554i $$0.552258\pi$$
$$600$$ 0 0
$$601$$ 10.0000 0.407909 0.203954 0.978980i $$-0.434621\pi$$
0.203954 + 0.978980i $$0.434621\pi$$
$$602$$ 16.0000i 0.652111i
$$603$$ 0 0
$$604$$ 10.0000 0.406894
$$605$$ 0 0
$$606$$ 0 0
$$607$$ − 8.00000i − 0.324710i −0.986732 0.162355i $$-0.948091\pi$$
0.986732 0.162355i $$-0.0519090\pi$$
$$608$$ 30.0000i 1.21666i
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 8.00000 0.323645
$$612$$ 0 0
$$613$$ − 6.00000i − 0.242338i −0.992632 0.121169i $$-0.961336\pi$$
0.992632 0.121169i $$-0.0386643\pi$$
$$614$$ −22.0000 −0.887848
$$615$$ 0 0
$$616$$ −24.0000 −0.966988
$$617$$ 18.0000i 0.724653i 0.932051 + 0.362326i $$0.118017\pi$$
−0.932051 + 0.362326i $$0.881983\pi$$
$$618$$ 0 0
$$619$$ −22.0000 −0.884255 −0.442127 0.896952i $$-0.645776\pi$$
−0.442127 + 0.896952i $$0.645776\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 4.00000i 0.160385i
$$623$$ − 28.0000i − 1.12180i
$$624$$ 0 0
$$625$$ 0 0
$$626$$ −2.00000 −0.0799361
$$627$$ 0 0
$$628$$ 18.0000i 0.718278i
$$629$$ −8.00000 −0.318981
$$630$$ 0 0
$$631$$ −14.0000 −0.557331 −0.278666 0.960388i $$-0.589892\pi$$
−0.278666 + 0.960388i $$0.589892\pi$$
$$632$$ 36.0000i 1.43200i
$$633$$ 0 0
$$634$$ 18.0000 0.714871
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 3.00000i 0.118864i
$$638$$ 16.0000i 0.633446i
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −40.0000 −1.57991 −0.789953 0.613168i $$-0.789895\pi$$
−0.789953 + 0.613168i $$0.789895\pi$$
$$642$$ 0 0
$$643$$ − 14.0000i − 0.552106i −0.961142 0.276053i $$-0.910973\pi$$
0.961142 0.276053i $$-0.0890266\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 24.0000 0.944267
$$647$$ − 28.0000i − 1.10079i −0.834903 0.550397i $$-0.814476\pi$$
0.834903 0.550397i $$-0.185524\pi$$
$$648$$ 0 0
$$649$$ 48.0000 1.88416
$$650$$ 0 0
$$651$$ 0 0
$$652$$ − 6.00000i − 0.234978i
$$653$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 6.00000 0.234261
$$657$$ 0 0
$$658$$ − 16.0000i − 0.623745i
$$659$$ −36.0000 −1.40236 −0.701180 0.712984i $$-0.747343\pi$$
−0.701180 + 0.712984i $$0.747343\pi$$
$$660$$ 0 0
$$661$$ 26.0000 1.01128 0.505641 0.862744i $$-0.331256\pi$$
0.505641 + 0.862744i $$0.331256\pi$$
$$662$$ − 18.0000i − 0.699590i
$$663$$ 0 0
$$664$$ 12.0000 0.465690
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 0 0
$$668$$ − 24.0000i − 0.928588i
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −8.00000 −0.308837
$$672$$ 0 0
$$673$$ 18.0000i 0.693849i 0.937893 + 0.346925i $$0.112774\pi$$
−0.937893 + 0.346925i $$0.887226\pi$$
$$674$$ 14.0000 0.539260
$$675$$ 0 0
$$676$$ −1.00000 −0.0384615
$$677$$ − 36.0000i − 1.38359i −0.722093 0.691796i $$-0.756820\pi$$
0.722093 0.691796i $$-0.243180\pi$$
$$678$$ 0 0
$$679$$ 28.0000 1.07454
$$680$$ 0 0
$$681$$ 0 0
$$682$$ − 40.0000i − 1.53168i
$$683$$ − 4.00000i − 0.153056i −0.997067 0.0765279i $$-0.975617\pi$$
0.997067 0.0765279i $$-0.0243834\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 20.0000 0.763604
$$687$$ 0 0
$$688$$ − 8.00000i − 0.304997i
$$689$$ −4.00000 −0.152388
$$690$$ 0 0
$$691$$ 34.0000 1.29342 0.646710 0.762736i $$-0.276144\pi$$
0.646710 + 0.762736i $$0.276144\pi$$
$$692$$ − 24.0000i − 0.912343i
$$693$$ 0 0
$$694$$ 32.0000 1.21470
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 24.0000i 0.909065i
$$698$$ − 6.00000i − 0.227103i
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 40.0000 1.51078 0.755390 0.655276i $$-0.227448\pi$$
0.755390 + 0.655276i $$0.227448\pi$$
$$702$$ 0 0
$$703$$ 12.0000i 0.452589i
$$704$$ 28.0000 1.05529
$$705$$ 0 0
$$706$$ 6.00000 0.225813
$$707$$ 24.0000i 0.902613i
$$708$$ 0 0
$$709$$ 50.0000 1.87779 0.938895 0.344204i $$-0.111851\pi$$
0.938895 + 0.344204i $$0.111851\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 42.0000i 1.57402i
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 0 0
$$718$$ − 16.0000i − 0.597115i
$$719$$ 20.0000 0.745874 0.372937 0.927857i $$-0.378351\pi$$
0.372937 + 0.927857i $$0.378351\pi$$
$$720$$ 0 0
$$721$$ −8.00000 −0.297936
$$722$$ − 17.0000i − 0.632674i
$$723$$ 0 0
$$724$$ −10.0000 −0.371647
$$725$$ 0 0
$$726$$ 0 0
$$727$$ − 24.0000i − 0.890111i −0.895503 0.445055i $$-0.853184\pi$$
0.895503 0.445055i $$-0.146816\pi$$
$$728$$ 6.00000i 0.222375i
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 32.0000 1.18356
$$732$$ 0 0
$$733$$ − 34.0000i − 1.25582i −0.778287 0.627909i $$-0.783911\pi$$
0.778287 0.627909i $$-0.216089\pi$$
$$734$$ −8.00000 −0.295285
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 40.0000i 1.47342i
$$738$$ 0 0
$$739$$ −30.0000 −1.10357 −0.551784 0.833987i $$-0.686053\pi$$
−0.551784 + 0.833987i $$0.686053\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 8.00000i 0.293689i
$$743$$ − 24.0000i − 0.880475i −0.897881 0.440237i $$-0.854894\pi$$
0.897881 0.440237i $$-0.145106\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ −22.0000 −0.805477
$$747$$ 0 0
$$748$$ 16.0000i 0.585018i
$$749$$ −8.00000 −0.292314
$$750$$ 0 0
$$751$$ 20.0000 0.729810 0.364905 0.931045i $$-0.381101\pi$$
0.364905 + 0.931045i $$0.381101\pi$$
$$752$$ 8.00000i 0.291730i
$$753$$ 0 0
$$754$$ 4.00000 0.145671
$$755$$ 0 0
$$756$$ 0 0
$$757$$ − 34.0000i − 1.23575i −0.786276 0.617876i $$-0.787994\pi$$
0.786276 0.617876i $$-0.212006\pi$$
$$758$$ − 2.00000i − 0.0726433i
$$759$$ 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
$$763$$ 4.00000i 0.144810i
$$764$$ −12.0000 −0.434145
$$765$$ 0 0
$$766$$ 24.0000 0.867155
$$767$$ − 12.0000i − 0.433295i
$$768$$ 0 0
$$769$$ −2.00000 −0.0721218 −0.0360609 0.999350i $$-0.511481\pi$$
−0.0360609 + 0.999350i $$0.511481\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 26.0000i 0.935760i
$$773$$ 50.0000i 1.79838i 0.437564 + 0.899188i $$0.355842\pi$$
−0.437564 + 0.899188i $$0.644158\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ −42.0000 −1.50771
$$777$$ 0 0
$$778$$ 8.00000i 0.286814i
$$779$$ 36.0000 1.28983
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ 0 0
$$784$$ −3.00000 −0.107143
$$785$$ 0 0
$$786$$ 0 0
$$787$$ − 22.0000i − 0.784215i −0.919919 0.392108i $$-0.871746\pi$$
0.919919 0.392108i $$-0.128254\pi$$
$$788$$ 18.0000i 0.641223i
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 32.0000 1.13779
$$792$$ 0 0
$$793$$ 2.00000i 0.0710221i
$$794$$ −18.0000 −0.638796
$$795$$ 0 0
$$796$$ −8.00000 −0.283552
$$797$$ − 12.0000i − 0.425062i −0.977154 0.212531i $$-0.931829\pi$$
0.977154 0.212531i $$-0.0681706\pi$$
$$798$$ 0 0
$$799$$ −32.0000 −1.13208
$$800$$ 0 0
$$801$$ 0 0
$$802$$ − 22.0000i − 0.776847i
$$803$$ − 24.0000i − 0.846942i
$$804$$ 0 0
$$805$$ 0 0
$$806$$ −10.0000 −0.352235
$$807$$ 0 0
$$808$$ − 36.0000i − 1.26648i
$$809$$ −28.0000 −0.984428 −0.492214 0.870474i $$-0.663812\pi$$
−0.492214 + 0.870474i $$0.663812\pi$$
$$810$$ 0 0
$$811$$ 30.0000 1.05344 0.526721 0.850038i $$-0.323421\pi$$
0.526721 + 0.850038i $$0.323421\pi$$
$$812$$ 8.00000i 0.280745i
$$813$$ 0 0
$$814$$ 8.00000 0.280400
$$815$$ 0 0
$$816$$ 0 0
$$817$$ − 48.0000i − 1.67931i
$$818$$ 30.0000i 1.04893i
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −10.0000 −0.349002 −0.174501 0.984657i $$-0.555831\pi$$
−0.174501 + 0.984657i $$0.555831\pi$$
$$822$$ 0 0
$$823$$ 4.00000i 0.139431i 0.997567 + 0.0697156i $$0.0222092\pi$$
−0.997567 + 0.0697156i $$0.977791\pi$$
$$824$$ 12.0000 0.418040
$$825$$ 0 0
$$826$$ −24.0000 −0.835067
$$827$$ − 20.0000i − 0.695468i −0.937593 0.347734i $$-0.886951\pi$$
0.937593 0.347734i $$-0.113049\pi$$
$$828$$ 0 0
$$829$$ 2.00000 0.0694629 0.0347314 0.999397i $$-0.488942\pi$$
0.0347314 + 0.999397i $$0.488942\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ − 7.00000i − 0.242681i
$$833$$ − 12.0000i − 0.415775i
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 24.0000 0.830057
$$837$$ 0 0
$$838$$ − 16.0000i − 0.552711i
$$839$$ 40.0000 1.38095 0.690477 0.723355i $$-0.257401\pi$$
0.690477 + 0.723355i $$0.257401\pi$$
$$840$$ 0 0
$$841$$ −13.0000 −0.448276
$$842$$ − 22.0000i − 0.758170i
$$843$$ 0 0
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 10.0000i 0.343604i
$$848$$ − 4.00000i − 0.137361i
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 0 0
$$852$$ 0 0
$$853$$ − 30.0000i − 1.02718i −0.858036 0.513590i $$-0.828315\pi$$
0.858036 0.513590i $$-0.171685\pi$$
$$854$$ 4.00000 0.136877
$$855$$ 0 0
$$856$$ 12.0000 0.410152
$$857$$ − 48.0000i − 1.63965i −0.572615 0.819824i $$-0.694071\pi$$
0.572615 0.819824i $$-0.305929\pi$$
$$858$$ 0 0
$$859$$ −16.0000 −0.545913 −0.272956 0.962026i $$-0.588002\pi$$
−0.272956 + 0.962026i $$0.588002\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 40.0000i 1.36241i
$$863$$ 48.0000i 1.63394i 0.576681 + 0.816970i $$0.304348\pi$$
−0.576681 + 0.816970i $$0.695652\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 2.00000 0.0679628
$$867$$ 0 0
$$868$$ − 20.0000i − 0.678844i
$$869$$ 48.0000 1.62829
$$870$$ 0 0
$$871$$ 10.0000 0.338837
$$872$$ − 6.00000i − 0.203186i
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ − 42.0000i − 1.41824i −0.705088 0.709120i $$-0.749093\pi$$
0.705088 0.709120i $$-0.250907\pi$$
$$878$$ − 36.0000i − 1.21494i
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −28.0000 −0.943344 −0.471672 0.881774i $$-0.656349\pi$$
−0.471672 + 0.881774i $$0.656349\pi$$
$$882$$ 0 0
$$883$$ 4.00000i 0.134611i 0.997732 + 0.0673054i $$0.0214402\pi$$
−0.997732 + 0.0673054i $$0.978560\pi$$
$$884$$ 4.00000 0.134535
$$885$$ 0 0
$$886$$ −20.0000 −0.671913
$$887$$ − 56.0000i − 1.88030i −0.340766 0.940148i $$-0.610687\pi$$
0.340766 0.940148i $$-0.389313\pi$$
$$888$$ 0 0
$$889$$ 16.0000 0.536623
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 14.0000i 0.468755i
$$893$$ 48.0000i 1.60626i
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 6.00000 0.200446
$$897$$ 0 0
$$898$$ 30.0000i 1.00111i
$$899$$ −40.0000 −1.33407
$$900$$ 0 0
$$901$$ 16.0000 0.533037
$$902$$ − 24.0000i − 0.799113i
$$903$$ 0 0
$$904$$ −48.0000 −1.59646
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 44.0000i 1.46100i 0.682915 + 0.730498i $$0.260712\pi$$
−0.682915 + 0.730498i $$0.739288\pi$$
$$908$$ − 20.0000i − 0.663723i
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 12.0000 0.397578 0.198789 0.980042i $$-0.436299\pi$$
0.198789 + 0.980042i $$0.436299\pi$$
$$912$$ 0 0
$$913$$ − 16.0000i − 0.529523i
$$914$$ −22.0000 −0.727695
$$915$$ 0 0
$$916$$ 10.0000 0.330409
$$917$$ − 32.0000i − 1.05673i
$$918$$ 0 0
$$919$$ −44.0000 −1.45143 −0.725713 0.687998i $$-0.758490\pi$$
−0.725713 + 0.687998i $$0.758490\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 14.0000i 0.461065i
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 0 0
$$926$$ −18.0000 −0.591517
$$927$$ 0 0
$$928$$ − 20.0000i − 0.656532i
$$929$$ −10.0000 −0.328089 −0.164045 0.986453i $$-0.552454\pi$$
−0.164045 + 0.986453i $$0.552454\pi$$
$$930$$ 0 0
$$931$$ −18.0000 −0.589926
$$932$$ 8.00000i 0.262049i
$$933$$ 0 0
$$934$$ 8.00000 0.261768
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 54.0000i 1.76410i 0.471153 + 0.882052i $$0.343838\pi$$
−0.471153 + 0.882052i $$0.656162\pi$$
$$938$$ − 20.0000i − 0.653023i
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 6.00000 0.195594 0.0977972 0.995206i $$-0.468820\pi$$
0.0977972 + 0.995206i $$0.468820\pi$$
$$942$$ 0 0
$$943$$ 0 0
$$944$$ 12.0000 0.390567
$$945$$ 0 0
$$946$$ −32.0000 −1.04041
$$947$$ 12.0000i 0.389948i 0.980808 + 0.194974i $$0.0624622\pi$$
−0.980808 + 0.194974i $$0.937538\pi$$
$$948$$ 0 0
$$949$$ −6.00000 −0.194768
$$950$$ 0 0
$$951$$ 0 0
$$952$$ − 24.0000i − 0.777844i
$$953$$ 48.0000i 1.55487i 0.628962 + 0.777436i $$0.283480\pi$$
−0.628962 + 0.777436i $$0.716520\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 0 0
$$958$$ − 24.0000i − 0.775405i
$$959$$ −20.0000 −0.645834
$$960$$ 0 0
$$961$$ 69.0000 2.22581
$$962$$ − 2.00000i − 0.0644826i
$$963$$ 0 0
$$964$$ 22.0000 0.708572
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 58.0000i 1.86515i 0.360971 + 0.932577i $$0.382445\pi$$
−0.360971 + 0.932577i $$0.617555\pi$$
$$968$$ − 15.0000i − 0.482118i
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −52.0000 −1.66876 −0.834380 0.551190i $$-0.814174\pi$$
−0.834380 + 0.551190i $$0.814174\pi$$
$$972$$ 0 0
$$973$$ − 24.0000i − 0.769405i
$$974$$ −22.0000 −0.704925
$$975$$ 0 0
$$976$$ −2.00000 −0.0640184
$$977$$ 54.0000i 1.72761i 0.503824 + 0.863807i $$0.331926\pi$$
−0.503824 + 0.863807i $$0.668074\pi$$
$$978$$ 0 0
$$979$$ 56.0000 1.78977
$$980$$ 0 0
$$981$$ 0 0
$$982$$ − 36.0000i − 1.14881i
$$983$$ 16.0000i 0.510321i 0.966899 + 0.255160i $$0.0821283\pi$$
−0.966899 + 0.255160i $$0.917872\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ −16.0000 −0.509544
$$987$$ 0 0
$$988$$ − 6.00000i − 0.190885i
$$989$$ 0 0
$$990$$ 0 0
$$991$$ −48.0000 −1.52477 −0.762385 0.647124i $$-0.775972\pi$$
−0.762385 + 0.647124i $$0.775972\pi$$
$$992$$ 50.0000i 1.58750i
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ − 58.0000i − 1.83688i −0.395562 0.918439i $$-0.629450\pi$$
0.395562 0.918439i $$-0.370550\pi$$
$$998$$ − 38.0000i − 1.20287i
$$999$$ 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 2925.2.c.g.2224.1 2
3.2 odd 2 2925.2.c.k.2224.2 2
5.2 odd 4 2925.2.a.m.1.1 1
5.3 odd 4 585.2.a.d.1.1 1
5.4 even 2 inner 2925.2.c.g.2224.2 2
15.2 even 4 2925.2.a.c.1.1 1
15.8 even 4 585.2.a.i.1.1 yes 1
15.14 odd 2 2925.2.c.k.2224.1 2
20.3 even 4 9360.2.a.i.1.1 1
60.23 odd 4 9360.2.a.be.1.1 1
65.38 odd 4 7605.2.a.q.1.1 1
195.38 even 4 7605.2.a.c.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
585.2.a.d.1.1 1 5.3 odd 4
585.2.a.i.1.1 yes 1 15.8 even 4
2925.2.a.c.1.1 1 15.2 even 4
2925.2.a.m.1.1 1 5.2 odd 4
2925.2.c.g.2224.1 2 1.1 even 1 trivial
2925.2.c.g.2224.2 2 5.4 even 2 inner
2925.2.c.k.2224.1 2 15.14 odd 2
2925.2.c.k.2224.2 2 3.2 odd 2
7605.2.a.c.1.1 1 195.38 even 4
7605.2.a.q.1.1 1 65.38 odd 4
9360.2.a.i.1.1 1 20.3 even 4
9360.2.a.be.1.1 1 60.23 odd 4