# Properties

 Label 1850.2.b.o.149.5 Level $1850$ Weight $2$ Character 1850.149 Analytic conductor $14.772$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1850 = 2 \cdot 5^{2} \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1850.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$14.7723243739$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.0.3182656.1 Defining polynomial: $$x^{6} - 2x^{3} + 25x^{2} - 10x + 2$$ x^6 - 2*x^3 + 25*x^2 - 10*x + 2 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 370) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 149.5 Root $$0.203364 + 0.203364i$$ of defining polynomial Character $$\chi$$ $$=$$ 1850.149 Dual form 1850.2.b.o.149.2

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000i q^{2} -0.406728i q^{3} -1.00000 q^{4} +0.406728 q^{6} +2.91729i q^{7} -1.00000i q^{8} +2.83457 q^{9} +O(q^{10})$$ $$q+1.00000i q^{2} -0.406728i q^{3} -1.00000 q^{4} +0.406728 q^{6} +2.91729i q^{7} -1.00000i q^{8} +2.83457 q^{9} +6.51056 q^{11} +0.406728i q^{12} +0.813457i q^{13} -2.91729 q^{14} +1.00000 q^{16} -2.51056i q^{17} +2.83457i q^{18} -0.406728 q^{19} +1.18654 q^{21} +6.51056i q^{22} -5.02112i q^{23} -0.406728 q^{24} -0.813457 q^{26} -2.37309i q^{27} -2.91729i q^{28} +5.32401 q^{29} -8.75186 q^{31} +1.00000i q^{32} -2.64803i q^{33} +2.51056 q^{34} -2.83457 q^{36} -1.00000i q^{37} -0.406728i q^{38} +0.330856 q^{39} +6.34513 q^{41} +1.18654i q^{42} +7.32401i q^{43} -6.51056 q^{44} +5.02112 q^{46} -5.42784i q^{47} -0.406728i q^{48} -1.51056 q^{49} -1.02112 q^{51} -0.813457i q^{52} +2.34513i q^{53} +2.37309 q^{54} +2.91729 q^{56} +0.165428i q^{57} +5.32401i q^{58} +1.42784 q^{59} -1.32401 q^{61} -8.75186i q^{62} +8.26926i q^{63} -1.00000 q^{64} +2.64803 q^{66} -5.42784i q^{67} +2.51056i q^{68} -2.04223 q^{69} +14.6480 q^{71} -2.83457i q^{72} +11.0211i q^{73} +1.00000 q^{74} +0.406728 q^{76} +18.9932i q^{77} +0.330856i q^{78} +1.75870 q^{79} +7.53851 q^{81} +6.34513i q^{82} +7.05476i q^{83} -1.18654 q^{84} -7.32401 q^{86} -2.16543i q^{87} -6.51056i q^{88} -6.00000 q^{89} -2.37309 q^{91} +5.02112i q^{92} +3.55963i q^{93} +5.42784 q^{94} +0.406728 q^{96} +2.34513i q^{97} -1.51056i q^{98} +18.4546 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 6 q^{4} - 22 q^{9}+O(q^{10})$$ 6 * q - 6 * q^4 - 22 * q^9 $$6 q - 6 q^{4} - 22 q^{9} + 22 q^{11} + 2 q^{14} + 6 q^{16} + 12 q^{21} + 10 q^{29} + 6 q^{31} - 2 q^{34} + 22 q^{36} + 80 q^{39} - 18 q^{41} - 22 q^{44} - 4 q^{46} + 8 q^{49} + 28 q^{51} + 24 q^{54} - 2 q^{56} - 28 q^{59} + 14 q^{61} - 6 q^{64} - 28 q^{66} + 56 q^{69} + 44 q^{71} + 6 q^{74} + 52 q^{79} + 94 q^{81} - 12 q^{84} - 22 q^{86} - 36 q^{89} - 24 q^{91} - 4 q^{94} - 38 q^{99}+O(q^{100})$$ 6 * q - 6 * q^4 - 22 * q^9 + 22 * q^11 + 2 * q^14 + 6 * q^16 + 12 * q^21 + 10 * q^29 + 6 * q^31 - 2 * q^34 + 22 * q^36 + 80 * q^39 - 18 * q^41 - 22 * q^44 - 4 * q^46 + 8 * q^49 + 28 * q^51 + 24 * q^54 - 2 * q^56 - 28 * q^59 + 14 * q^61 - 6 * q^64 - 28 * q^66 + 56 * q^69 + 44 * q^71 + 6 * q^74 + 52 * q^79 + 94 * q^81 - 12 * q^84 - 22 * q^86 - 36 * q^89 - 24 * q^91 - 4 * q^94 - 38 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1850\mathbb{Z}\right)^\times$$.

 $$n$$ $$1001$$ $$1777$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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
$$3$$ − 0.406728i − 0.234825i −0.993083 0.117412i $$-0.962540\pi$$
0.993083 0.117412i $$-0.0374599\pi$$
$$4$$ −1.00000 −0.500000
$$5$$ 0 0
$$6$$ 0.406728 0.166046
$$7$$ 2.91729i 1.10263i 0.834297 + 0.551315i $$0.185874\pi$$
−0.834297 + 0.551315i $$0.814126\pi$$
$$8$$ − 1.00000i − 0.353553i
$$9$$ 2.83457 0.944857
$$10$$ 0 0
$$11$$ 6.51056 1.96301 0.981503 0.191444i $$-0.0613171\pi$$
0.981503 + 0.191444i $$0.0613171\pi$$
$$12$$ 0.406728i 0.117412i
$$13$$ 0.813457i 0.225612i 0.993617 + 0.112806i $$0.0359839\pi$$
−0.993617 + 0.112806i $$0.964016\pi$$
$$14$$ −2.91729 −0.779677
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ − 2.51056i − 0.608900i −0.952528 0.304450i $$-0.901527\pi$$
0.952528 0.304450i $$-0.0984726\pi$$
$$18$$ 2.83457i 0.668115i
$$19$$ −0.406728 −0.0933099 −0.0466550 0.998911i $$-0.514856\pi$$
−0.0466550 + 0.998911i $$0.514856\pi$$
$$20$$ 0 0
$$21$$ 1.18654 0.258925
$$22$$ 6.51056i 1.38806i
$$23$$ − 5.02112i − 1.04697i −0.852033 0.523487i $$-0.824631\pi$$
0.852033 0.523487i $$-0.175369\pi$$
$$24$$ −0.406728 −0.0830231
$$25$$ 0 0
$$26$$ −0.813457 −0.159532
$$27$$ − 2.37309i − 0.456701i
$$28$$ − 2.91729i − 0.551315i
$$29$$ 5.32401 0.988645 0.494322 0.869279i $$-0.335416\pi$$
0.494322 + 0.869279i $$0.335416\pi$$
$$30$$ 0 0
$$31$$ −8.75186 −1.57188 −0.785940 0.618303i $$-0.787821\pi$$
−0.785940 + 0.618303i $$0.787821\pi$$
$$32$$ 1.00000i 0.176777i
$$33$$ − 2.64803i − 0.460963i
$$34$$ 2.51056 0.430557
$$35$$ 0 0
$$36$$ −2.83457 −0.472429
$$37$$ − 1.00000i − 0.164399i
$$38$$ − 0.406728i − 0.0659801i
$$39$$ 0.330856 0.0529794
$$40$$ 0 0
$$41$$ 6.34513 0.990943 0.495471 0.868624i $$-0.334995\pi$$
0.495471 + 0.868624i $$0.334995\pi$$
$$42$$ 1.18654i 0.183088i
$$43$$ 7.32401i 1.11690i 0.829538 + 0.558451i $$0.188604\pi$$
−0.829538 + 0.558451i $$0.811396\pi$$
$$44$$ −6.51056 −0.981503
$$45$$ 0 0
$$46$$ 5.02112 0.740323
$$47$$ − 5.42784i − 0.791732i −0.918308 0.395866i $$-0.870444\pi$$
0.918308 0.395866i $$-0.129556\pi$$
$$48$$ − 0.406728i − 0.0587062i
$$49$$ −1.51056 −0.215794
$$50$$ 0 0
$$51$$ −1.02112 −0.142985
$$52$$ − 0.813457i − 0.112806i
$$53$$ 2.34513i 0.322128i 0.986944 + 0.161064i $$0.0514926\pi$$
−0.986944 + 0.161064i $$0.948507\pi$$
$$54$$ 2.37309 0.322936
$$55$$ 0 0
$$56$$ 2.91729 0.389839
$$57$$ 0.165428i 0.0219115i
$$58$$ 5.32401i 0.699077i
$$59$$ 1.42784 0.185889 0.0929447 0.995671i $$-0.470372\pi$$
0.0929447 + 0.995671i $$0.470372\pi$$
$$60$$ 0 0
$$61$$ −1.32401 −0.169523 −0.0847613 0.996401i $$-0.527013\pi$$
−0.0847613 + 0.996401i $$0.527013\pi$$
$$62$$ − 8.75186i − 1.11149i
$$63$$ 8.26926i 1.04183i
$$64$$ −1.00000 −0.125000
$$65$$ 0 0
$$66$$ 2.64803 0.325950
$$67$$ − 5.42784i − 0.663117i −0.943435 0.331558i $$-0.892426\pi$$
0.943435 0.331558i $$-0.107574\pi$$
$$68$$ 2.51056i 0.304450i
$$69$$ −2.04223 −0.245856
$$70$$ 0 0
$$71$$ 14.6480 1.73840 0.869201 0.494460i $$-0.164634\pi$$
0.869201 + 0.494460i $$0.164634\pi$$
$$72$$ − 2.83457i − 0.334058i
$$73$$ 11.0211i 1.28992i 0.764215 + 0.644962i $$0.223127\pi$$
−0.764215 + 0.644962i $$0.776873\pi$$
$$74$$ 1.00000 0.116248
$$75$$ 0 0
$$76$$ 0.406728 0.0466550
$$77$$ 18.9932i 2.16447i
$$78$$ 0.330856i 0.0374621i
$$79$$ 1.75870 0.197869 0.0989346 0.995094i $$-0.468457\pi$$
0.0989346 + 0.995094i $$0.468457\pi$$
$$80$$ 0 0
$$81$$ 7.53851 0.837613
$$82$$ 6.34513i 0.700702i
$$83$$ 7.05476i 0.774360i 0.922004 + 0.387180i $$0.126551\pi$$
−0.922004 + 0.387180i $$0.873449\pi$$
$$84$$ −1.18654 −0.129462
$$85$$ 0 0
$$86$$ −7.32401 −0.789769
$$87$$ − 2.16543i − 0.232158i
$$88$$ − 6.51056i − 0.694028i
$$89$$ −6.00000 −0.635999 −0.317999 0.948091i $$-0.603011\pi$$
−0.317999 + 0.948091i $$0.603011\pi$$
$$90$$ 0 0
$$91$$ −2.37309 −0.248767
$$92$$ 5.02112i 0.523487i
$$93$$ 3.55963i 0.369116i
$$94$$ 5.42784 0.559839
$$95$$ 0 0
$$96$$ 0.406728 0.0415115
$$97$$ 2.34513i 0.238112i 0.992888 + 0.119056i $$0.0379868\pi$$
−0.992888 + 0.119056i $$0.962013\pi$$
$$98$$ − 1.51056i − 0.152589i
$$99$$ 18.4546 1.85476
$$100$$ 0 0
$$101$$ −8.20766 −0.816693 −0.408346 0.912827i $$-0.633894\pi$$
−0.408346 + 0.912827i $$0.633894\pi$$
$$102$$ − 1.02112i − 0.101105i
$$103$$ 8.81346i 0.868416i 0.900813 + 0.434208i $$0.142972\pi$$
−0.900813 + 0.434208i $$0.857028\pi$$
$$104$$ 0.813457 0.0797660
$$105$$ 0 0
$$106$$ −2.34513 −0.227779
$$107$$ 15.2624i 1.47547i 0.675089 + 0.737737i $$0.264105\pi$$
−0.675089 + 0.737737i $$0.735895\pi$$
$$108$$ 2.37309i 0.228350i
$$109$$ −4.30290 −0.412143 −0.206072 0.978537i $$-0.566068\pi$$
−0.206072 + 0.978537i $$0.566068\pi$$
$$110$$ 0 0
$$111$$ −0.406728 −0.0386050
$$112$$ 2.91729i 0.275658i
$$113$$ 9.15859i 0.861567i 0.902455 + 0.430784i $$0.141763\pi$$
−0.902455 + 0.430784i $$0.858237\pi$$
$$114$$ −0.165428 −0.0154938
$$115$$ 0 0
$$116$$ −5.32401 −0.494322
$$117$$ 2.30580i 0.213171i
$$118$$ 1.42784i 0.131444i
$$119$$ 7.32401 0.671391
$$120$$ 0 0
$$121$$ 31.3874 2.85340
$$122$$ − 1.32401i − 0.119871i
$$123$$ − 2.58074i − 0.232698i
$$124$$ 8.75186 0.785940
$$125$$ 0 0
$$126$$ −8.26926 −0.736684
$$127$$ − 8.61439i − 0.764403i −0.924079 0.382202i $$-0.875166\pi$$
0.924079 0.382202i $$-0.124834\pi$$
$$128$$ − 1.00000i − 0.0883883i
$$129$$ 2.97888 0.262276
$$130$$ 0 0
$$131$$ 13.4278 1.17320 0.586598 0.809878i $$-0.300467\pi$$
0.586598 + 0.809878i $$0.300467\pi$$
$$132$$ 2.64803i 0.230481i
$$133$$ − 1.18654i − 0.102886i
$$134$$ 5.42784 0.468894
$$135$$ 0 0
$$136$$ −2.51056 −0.215279
$$137$$ − 14.6903i − 1.25507i −0.778587 0.627537i $$-0.784063\pi$$
0.778587 0.627537i $$-0.215937\pi$$
$$138$$ − 2.04223i − 0.173846i
$$139$$ −17.3662 −1.47299 −0.736493 0.676445i $$-0.763519\pi$$
−0.736493 + 0.676445i $$0.763519\pi$$
$$140$$ 0 0
$$141$$ −2.20766 −0.185918
$$142$$ 14.6480i 1.22924i
$$143$$ 5.29606i 0.442879i
$$144$$ 2.83457 0.236214
$$145$$ 0 0
$$146$$ −11.0211 −0.912114
$$147$$ 0.614387i 0.0506738i
$$148$$ 1.00000i 0.0821995i
$$149$$ 0.207658 0.0170120 0.00850601 0.999964i $$-0.497292\pi$$
0.00850601 + 0.999964i $$0.497292\pi$$
$$150$$ 0 0
$$151$$ 0.813457 0.0661982 0.0330991 0.999452i $$-0.489462\pi$$
0.0330991 + 0.999452i $$0.489462\pi$$
$$152$$ 0.406728i 0.0329900i
$$153$$ − 7.11636i − 0.575323i
$$154$$ −18.9932 −1.53051
$$155$$ 0 0
$$156$$ −0.330856 −0.0264897
$$157$$ 5.32401i 0.424903i 0.977172 + 0.212451i $$0.0681447\pi$$
−0.977172 + 0.212451i $$0.931855\pi$$
$$158$$ 1.75870i 0.139915i
$$159$$ 0.953831 0.0756437
$$160$$ 0 0
$$161$$ 14.6480 1.15443
$$162$$ 7.53851i 0.592282i
$$163$$ − 15.2008i − 1.19062i −0.803496 0.595310i $$-0.797029\pi$$
0.803496 0.595310i $$-0.202971\pi$$
$$164$$ −6.34513 −0.495471
$$165$$ 0 0
$$166$$ −7.05476 −0.547555
$$167$$ − 12.4826i − 0.965933i −0.875639 0.482966i $$-0.839559\pi$$
0.875639 0.482966i $$-0.160441\pi$$
$$168$$ − 1.18654i − 0.0915438i
$$169$$ 12.3383 0.949099
$$170$$ 0 0
$$171$$ −1.15290 −0.0881645
$$172$$ − 7.32401i − 0.558451i
$$173$$ − 23.0354i − 1.75135i −0.482903 0.875674i $$-0.660417\pi$$
0.482903 0.875674i $$-0.339583\pi$$
$$174$$ 2.16543 0.164161
$$175$$ 0 0
$$176$$ 6.51056 0.490752
$$177$$ − 0.580745i − 0.0436514i
$$178$$ − 6.00000i − 0.449719i
$$179$$ −24.2835 −1.81504 −0.907518 0.420013i $$-0.862026\pi$$
−0.907518 + 0.420013i $$0.862026\pi$$
$$180$$ 0 0
$$181$$ 2.16543 0.160955 0.0804775 0.996756i $$-0.474355\pi$$
0.0804775 + 0.996756i $$0.474355\pi$$
$$182$$ − 2.37309i − 0.175905i
$$183$$ 0.538514i 0.0398081i
$$184$$ −5.02112 −0.370162
$$185$$ 0 0
$$186$$ −3.55963 −0.261005
$$187$$ − 16.3451i − 1.19527i
$$188$$ 5.42784i 0.395866i
$$189$$ 6.92297 0.503572
$$190$$ 0 0
$$191$$ −19.3326 −1.39886 −0.699429 0.714702i $$-0.746562\pi$$
−0.699429 + 0.714702i $$0.746562\pi$$
$$192$$ 0.406728i 0.0293531i
$$193$$ 20.0422i 1.44267i 0.692586 + 0.721336i $$0.256471\pi$$
−0.692586 + 0.721336i $$0.743529\pi$$
$$194$$ −2.34513 −0.168370
$$195$$ 0 0
$$196$$ 1.51056 0.107897
$$197$$ 15.6269i 1.11337i 0.830723 + 0.556686i $$0.187927\pi$$
−0.830723 + 0.556686i $$0.812073\pi$$
$$198$$ 18.4546i 1.31151i
$$199$$ −20.2835 −1.43786 −0.718931 0.695082i $$-0.755368\pi$$
−0.718931 + 0.695082i $$0.755368\pi$$
$$200$$ 0 0
$$201$$ −2.20766 −0.155716
$$202$$ − 8.20766i − 0.577489i
$$203$$ 15.5317i 1.09011i
$$204$$ 1.02112 0.0714924
$$205$$ 0 0
$$206$$ −8.81346 −0.614063
$$207$$ − 14.2327i − 0.989242i
$$208$$ 0.813457i 0.0564031i
$$209$$ −2.64803 −0.183168
$$210$$ 0 0
$$211$$ 18.7182 1.28862 0.644308 0.764766i $$-0.277146\pi$$
0.644308 + 0.764766i $$0.277146\pi$$
$$212$$ − 2.34513i − 0.161064i
$$213$$ − 5.95777i − 0.408220i
$$214$$ −15.2624 −1.04332
$$215$$ 0 0
$$216$$ −2.37309 −0.161468
$$217$$ − 25.5317i − 1.73320i
$$218$$ − 4.30290i − 0.291429i
$$219$$ 4.48260 0.302906
$$220$$ 0 0
$$221$$ 2.04223 0.137375
$$222$$ − 0.406728i − 0.0272978i
$$223$$ 15.7307i 1.05341i 0.850049 + 0.526704i $$0.176572\pi$$
−0.850049 + 0.526704i $$0.823428\pi$$
$$224$$ −2.91729 −0.194919
$$225$$ 0 0
$$226$$ −9.15859 −0.609220
$$227$$ 12.8836i 0.855117i 0.903988 + 0.427559i $$0.140626\pi$$
−0.903988 + 0.427559i $$0.859374\pi$$
$$228$$ − 0.165428i − 0.0109557i
$$229$$ −5.25383 −0.347183 −0.173591 0.984818i $$-0.555537\pi$$
−0.173591 + 0.984818i $$0.555537\pi$$
$$230$$ 0 0
$$231$$ 7.72506 0.508271
$$232$$ − 5.32401i − 0.349539i
$$233$$ − 28.3172i − 1.85512i −0.373675 0.927560i $$-0.621902\pi$$
0.373675 0.927560i $$-0.378098\pi$$
$$234$$ −2.30580 −0.150735
$$235$$ 0 0
$$236$$ −1.42784 −0.0929447
$$237$$ − 0.715313i − 0.0464646i
$$238$$ 7.32401i 0.474745i
$$239$$ −14.3788 −0.930085 −0.465043 0.885288i $$-0.653961\pi$$
−0.465043 + 0.885288i $$0.653961\pi$$
$$240$$ 0 0
$$241$$ 18.2749 1.17719 0.588596 0.808427i $$-0.299681\pi$$
0.588596 + 0.808427i $$0.299681\pi$$
$$242$$ 31.3874i 2.01766i
$$243$$ − 10.1854i − 0.653393i
$$244$$ 1.32401 0.0847613
$$245$$ 0 0
$$246$$ 2.58074 0.164542
$$247$$ − 0.330856i − 0.0210519i
$$248$$ 8.75186i 0.555744i
$$249$$ 2.86937 0.181839
$$250$$ 0 0
$$251$$ −1.22019 −0.0770174 −0.0385087 0.999258i $$-0.512261\pi$$
−0.0385087 + 0.999258i $$0.512261\pi$$
$$252$$ − 8.26926i − 0.520914i
$$253$$ − 32.6903i − 2.05522i
$$254$$ 8.61439 0.540515
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 27.1306i 1.69236i 0.532895 + 0.846181i $$0.321104\pi$$
−0.532895 + 0.846181i $$0.678896\pi$$
$$258$$ 2.97888i 0.185457i
$$259$$ 2.91729 0.181271
$$260$$ 0 0
$$261$$ 15.0913 0.934128
$$262$$ 13.4278i 0.829575i
$$263$$ 20.2133i 1.24641i 0.782059 + 0.623204i $$0.214169\pi$$
−0.782059 + 0.623204i $$0.785831\pi$$
$$264$$ −2.64803 −0.162975
$$265$$ 0 0
$$266$$ 1.18654 0.0727516
$$267$$ 2.44037i 0.149348i
$$268$$ 5.42784i 0.331558i
$$269$$ 0.207658 0.0126611 0.00633057 0.999980i $$-0.497985\pi$$
0.00633057 + 0.999980i $$0.497985\pi$$
$$270$$ 0 0
$$271$$ −30.6480 −1.86174 −0.930868 0.365357i $$-0.880947\pi$$
−0.930868 + 0.365357i $$0.880947\pi$$
$$272$$ − 2.51056i − 0.152225i
$$273$$ 0.965202i 0.0584167i
$$274$$ 14.6903 0.887471
$$275$$ 0 0
$$276$$ 2.04223 0.122928
$$277$$ − 16.8135i − 1.01022i −0.863054 0.505111i $$-0.831451\pi$$
0.863054 0.505111i $$-0.168549\pi$$
$$278$$ − 17.3662i − 1.04156i
$$279$$ −24.8078 −1.48520
$$280$$ 0 0
$$281$$ −14.2749 −0.851572 −0.425786 0.904824i $$-0.640002\pi$$
−0.425786 + 0.904824i $$0.640002\pi$$
$$282$$ − 2.20766i − 0.131464i
$$283$$ 13.2288i 0.786369i 0.919460 + 0.393184i $$0.128627\pi$$
−0.919460 + 0.393184i $$0.871373\pi$$
$$284$$ −14.6480 −0.869201
$$285$$ 0 0
$$286$$ −5.29606 −0.313162
$$287$$ 18.5106i 1.09264i
$$288$$ 2.83457i 0.167029i
$$289$$ 10.6971 0.629241
$$290$$ 0 0
$$291$$ 0.953831 0.0559146
$$292$$ − 11.0211i − 0.644962i
$$293$$ − 2.34513i − 0.137004i −0.997651 0.0685020i $$-0.978178\pi$$
0.997651 0.0685020i $$-0.0218219\pi$$
$$294$$ −0.614387 −0.0358318
$$295$$ 0 0
$$296$$ −1.00000 −0.0581238
$$297$$ − 15.4501i − 0.896507i
$$298$$ 0.207658i 0.0120293i
$$299$$ 4.08446 0.236210
$$300$$ 0 0
$$301$$ −21.3662 −1.23153
$$302$$ 0.813457i 0.0468092i
$$303$$ 3.33829i 0.191780i
$$304$$ −0.406728 −0.0233275
$$305$$ 0 0
$$306$$ 7.11636 0.406815
$$307$$ − 23.2624i − 1.32766i −0.747885 0.663828i $$-0.768931\pi$$
0.747885 0.663828i $$-0.231069\pi$$
$$308$$ − 18.9932i − 1.08224i
$$309$$ 3.58468 0.203926
$$310$$ 0 0
$$311$$ −31.3999 −1.78052 −0.890262 0.455449i $$-0.849479\pi$$
−0.890262 + 0.455449i $$0.849479\pi$$
$$312$$ − 0.330856i − 0.0187310i
$$313$$ 8.04223i 0.454574i 0.973828 + 0.227287i $$0.0729855\pi$$
−0.973828 + 0.227287i $$0.927014\pi$$
$$314$$ −5.32401 −0.300452
$$315$$ 0 0
$$316$$ −1.75870 −0.0989346
$$317$$ − 12.3029i − 0.691000i −0.938419 0.345500i $$-0.887709\pi$$
0.938419 0.345500i $$-0.112291\pi$$
$$318$$ 0.953831i 0.0534882i
$$319$$ 34.6623 1.94072
$$320$$ 0 0
$$321$$ 6.20766 0.346478
$$322$$ 14.6480i 0.816303i
$$323$$ 1.02112i 0.0568164i
$$324$$ −7.53851 −0.418806
$$325$$ 0 0
$$326$$ 15.2008 0.841895
$$327$$ 1.75011i 0.0967814i
$$328$$ − 6.34513i − 0.350351i
$$329$$ 15.8346 0.872988
$$330$$ 0 0
$$331$$ −6.77981 −0.372652 −0.186326 0.982488i $$-0.559658\pi$$
−0.186326 + 0.982488i $$0.559658\pi$$
$$332$$ − 7.05476i − 0.387180i
$$333$$ − 2.83457i − 0.155334i
$$334$$ 12.4826 0.683018
$$335$$ 0 0
$$336$$ 1.18654 0.0647312
$$337$$ 10.6903i 0.582336i 0.956672 + 0.291168i $$0.0940438\pi$$
−0.956672 + 0.291168i $$0.905956\pi$$
$$338$$ 12.3383i 0.671114i
$$339$$ 3.72506 0.202317
$$340$$ 0 0
$$341$$ −56.9795 −3.08561
$$342$$ − 1.15290i − 0.0623417i
$$343$$ 16.0143i 0.864689i
$$344$$ 7.32401 0.394884
$$345$$ 0 0
$$346$$ 23.0354 1.23839
$$347$$ 5.62691i 0.302069i 0.988529 + 0.151034i $$0.0482604\pi$$
−0.988529 + 0.151034i $$0.951740\pi$$
$$348$$ 2.16543i 0.116079i
$$349$$ −15.1306 −0.809924 −0.404962 0.914334i $$-0.632715\pi$$
−0.404962 + 0.914334i $$0.632715\pi$$
$$350$$ 0 0
$$351$$ 1.93040 0.103037
$$352$$ 6.51056i 0.347014i
$$353$$ − 35.7816i − 1.90446i −0.305381 0.952230i $$-0.598784\pi$$
0.305381 0.952230i $$-0.401216\pi$$
$$354$$ 0.580745 0.0308662
$$355$$ 0 0
$$356$$ 6.00000 0.317999
$$357$$ − 2.97888i − 0.157659i
$$358$$ − 24.2835i − 1.28342i
$$359$$ −4.48260 −0.236583 −0.118291 0.992979i $$-0.537742\pi$$
−0.118291 + 0.992979i $$0.537742\pi$$
$$360$$ 0 0
$$361$$ −18.8346 −0.991293
$$362$$ 2.16543i 0.113812i
$$363$$ − 12.7661i − 0.670048i
$$364$$ 2.37309 0.124384
$$365$$ 0 0
$$366$$ −0.538514 −0.0281486
$$367$$ − 21.5653i − 1.12570i −0.826559 0.562850i $$-0.809705\pi$$
0.826559 0.562850i $$-0.190295\pi$$
$$368$$ − 5.02112i − 0.261744i
$$369$$ 17.9857 0.936300
$$370$$ 0 0
$$371$$ −6.84141 −0.355188
$$372$$ − 3.55963i − 0.184558i
$$373$$ − 16.9230i − 0.876238i −0.898917 0.438119i $$-0.855645\pi$$
0.898917 0.438119i $$-0.144355\pi$$
$$374$$ 16.3451 0.845187
$$375$$ 0 0
$$376$$ −5.42784 −0.279920
$$377$$ 4.33086i 0.223050i
$$378$$ 6.92297i 0.356079i
$$379$$ −31.2710 −1.60628 −0.803142 0.595788i $$-0.796840\pi$$
−0.803142 + 0.595788i $$0.796840\pi$$
$$380$$ 0 0
$$381$$ −3.50372 −0.179501
$$382$$ − 19.3326i − 0.989142i
$$383$$ − 1.62691i − 0.0831314i −0.999136 0.0415657i $$-0.986765\pi$$
0.999136 0.0415657i $$-0.0132346\pi$$
$$384$$ −0.406728 −0.0207558
$$385$$ 0 0
$$386$$ −20.0422 −1.02012
$$387$$ 20.7604i 1.05531i
$$388$$ − 2.34513i − 0.119056i
$$389$$ 31.6412 1.60427 0.802136 0.597141i $$-0.203697\pi$$
0.802136 + 0.597141i $$0.203697\pi$$
$$390$$ 0 0
$$391$$ −12.6058 −0.637503
$$392$$ 1.51056i 0.0762947i
$$393$$ − 5.46149i − 0.275496i
$$394$$ −15.6269 −0.787273
$$395$$ 0 0
$$396$$ −18.4546 −0.927381
$$397$$ − 39.2961i − 1.97221i −0.166116 0.986106i $$-0.553122\pi$$
0.166116 0.986106i $$-0.446878\pi$$
$$398$$ − 20.2835i − 1.01672i
$$399$$ −0.482601 −0.0241603
$$400$$ 0 0
$$401$$ −9.66914 −0.482854 −0.241427 0.970419i $$-0.577615\pi$$
−0.241427 + 0.970419i $$0.577615\pi$$
$$402$$ − 2.20766i − 0.110108i
$$403$$ − 7.11926i − 0.354636i
$$404$$ 8.20766 0.408346
$$405$$ 0 0
$$406$$ −15.5317 −0.770824
$$407$$ − 6.51056i − 0.322716i
$$408$$ 1.02112i 0.0505527i
$$409$$ 26.2749 1.29921 0.649606 0.760271i $$-0.274934\pi$$
0.649606 + 0.760271i $$0.274934\pi$$
$$410$$ 0 0
$$411$$ −5.97495 −0.294722
$$412$$ − 8.81346i − 0.434208i
$$413$$ 4.16543i 0.204967i
$$414$$ 14.2327 0.699500
$$415$$ 0 0
$$416$$ −0.813457 −0.0398830
$$417$$ 7.06335i 0.345894i
$$418$$ − 2.64803i − 0.129519i
$$419$$ 30.1940 1.47507 0.737536 0.675308i $$-0.235989\pi$$
0.737536 + 0.675308i $$0.235989\pi$$
$$420$$ 0 0
$$421$$ 22.0000 1.07221 0.536107 0.844150i $$-0.319894\pi$$
0.536107 + 0.844150i $$0.319894\pi$$
$$422$$ 18.7182i 0.911188i
$$423$$ − 15.3856i − 0.748074i
$$424$$ 2.34513 0.113890
$$425$$ 0 0
$$426$$ 5.95777 0.288655
$$427$$ − 3.86253i − 0.186921i
$$428$$ − 15.2624i − 0.737737i
$$429$$ 2.15406 0.103999
$$430$$ 0 0
$$431$$ 25.5653 1.23144 0.615719 0.787966i $$-0.288866\pi$$
0.615719 + 0.787966i $$0.288866\pi$$
$$432$$ − 2.37309i − 0.114175i
$$433$$ − 32.0422i − 1.53985i −0.638134 0.769926i $$-0.720293\pi$$
0.638134 0.769926i $$-0.279707\pi$$
$$434$$ 25.5317 1.22556
$$435$$ 0 0
$$436$$ 4.30290 0.206072
$$437$$ 2.04223i 0.0976931i
$$438$$ 4.48260i 0.214187i
$$439$$ −3.93840 −0.187970 −0.0939848 0.995574i $$-0.529961\pi$$
−0.0939848 + 0.995574i $$0.529961\pi$$
$$440$$ 0 0
$$441$$ −4.28178 −0.203894
$$442$$ 2.04223i 0.0971390i
$$443$$ 0.0758724i 0.00360481i 0.999998 + 0.00180240i $$0.000573723\pi$$
−0.999998 + 0.00180240i $$0.999426\pi$$
$$444$$ 0.406728 0.0193025
$$445$$ 0 0
$$446$$ −15.7307 −0.744872
$$447$$ − 0.0844605i − 0.00399485i
$$448$$ − 2.91729i − 0.137829i
$$449$$ −19.2961 −0.910637 −0.455319 0.890329i $$-0.650475\pi$$
−0.455319 + 0.890329i $$0.650475\pi$$
$$450$$ 0 0
$$451$$ 41.3103 1.94523
$$452$$ − 9.15859i − 0.430784i
$$453$$ − 0.330856i − 0.0155450i
$$454$$ −12.8836 −0.604659
$$455$$ 0 0
$$456$$ 0.165428 0.00774688
$$457$$ 3.00684i 0.140654i 0.997524 + 0.0703271i $$0.0224043\pi$$
−0.997524 + 0.0703271i $$0.977596\pi$$
$$458$$ − 5.25383i − 0.245495i
$$459$$ −5.95777 −0.278085
$$460$$ 0 0
$$461$$ −0.633755 −0.0295169 −0.0147585 0.999891i $$-0.504698\pi$$
−0.0147585 + 0.999891i $$0.504698\pi$$
$$462$$ 7.72506i 0.359402i
$$463$$ − 18.0422i − 0.838494i −0.907872 0.419247i $$-0.862294\pi$$
0.907872 0.419247i $$-0.137706\pi$$
$$464$$ 5.32401 0.247161
$$465$$ 0 0
$$466$$ 28.3172 1.31177
$$467$$ − 23.4758i − 1.08633i −0.839626 0.543164i $$-0.817226\pi$$
0.839626 0.543164i $$-0.182774\pi$$
$$468$$ − 2.30580i − 0.106586i
$$469$$ 15.8346 0.731173
$$470$$ 0 0
$$471$$ 2.16543 0.0997777
$$472$$ − 1.42784i − 0.0657218i
$$473$$ 47.6834i 2.19249i
$$474$$ 0.715313 0.0328554
$$475$$ 0 0
$$476$$ −7.32401 −0.335696
$$477$$ 6.64744i 0.304365i
$$478$$ − 14.3788i − 0.657670i
$$479$$ 8.19907 0.374625 0.187313 0.982300i $$-0.440022\pi$$
0.187313 + 0.982300i $$0.440022\pi$$
$$480$$ 0 0
$$481$$ 0.813457 0.0370904
$$482$$ 18.2749i 0.832401i
$$483$$ − 5.95777i − 0.271088i
$$484$$ −31.3874 −1.42670
$$485$$ 0 0
$$486$$ 10.1854 0.462019
$$487$$ 0.953831i 0.0432222i 0.999766 + 0.0216111i $$0.00687956\pi$$
−0.999766 + 0.0216111i $$0.993120\pi$$
$$488$$ 1.32401i 0.0599353i
$$489$$ −6.18260 −0.279587
$$490$$ 0 0
$$491$$ 19.3383 0.872725 0.436362 0.899771i $$-0.356267\pi$$
0.436362 + 0.899771i $$0.356267\pi$$
$$492$$ 2.58074i 0.116349i
$$493$$ − 13.3662i − 0.601985i
$$494$$ 0.330856 0.0148859
$$495$$ 0 0
$$496$$ −8.75186 −0.392970
$$497$$ 42.7325i 1.91681i
$$498$$ 2.86937i 0.128580i
$$499$$ 5.63550 0.252280 0.126140 0.992012i $$-0.459741\pi$$
0.126140 + 0.992012i $$0.459741\pi$$
$$500$$ 0 0
$$501$$ −5.07703 −0.226825
$$502$$ − 1.22019i − 0.0544595i
$$503$$ 33.6269i 1.49935i 0.661806 + 0.749675i $$0.269790\pi$$
−0.661806 + 0.749675i $$0.730210\pi$$
$$504$$ 8.26926 0.368342
$$505$$ 0 0
$$506$$ 32.6903 1.45326
$$507$$ − 5.01833i − 0.222872i
$$508$$ 8.61439i 0.382202i
$$509$$ 18.6903 0.828431 0.414216 0.910179i $$-0.364056\pi$$
0.414216 + 0.910179i $$0.364056\pi$$
$$510$$ 0 0
$$511$$ −32.1517 −1.42231
$$512$$ 1.00000i 0.0441942i
$$513$$ 0.965202i 0.0426147i
$$514$$ −27.1306 −1.19668
$$515$$ 0 0
$$516$$ −2.97888 −0.131138
$$517$$ − 35.3383i − 1.55418i
$$518$$ 2.91729i 0.128178i
$$519$$ −9.36915 −0.411260
$$520$$ 0 0
$$521$$ 25.1586 1.10222 0.551109 0.834433i $$-0.314205\pi$$
0.551109 + 0.834433i $$0.314205\pi$$
$$522$$ 15.0913i 0.660528i
$$523$$ 7.25383i 0.317188i 0.987344 + 0.158594i $$0.0506960\pi$$
−0.987344 + 0.158594i $$0.949304\pi$$
$$524$$ −13.4278 −0.586598
$$525$$ 0 0
$$526$$ −20.2133 −0.881344
$$527$$ 21.9720i 0.957117i
$$528$$ − 2.64803i − 0.115241i
$$529$$ −2.21160 −0.0961564
$$530$$ 0 0
$$531$$ 4.04733 0.175639
$$532$$ 1.18654i 0.0514432i
$$533$$ 5.16149i 0.223569i
$$534$$ −2.44037 −0.105605
$$535$$ 0 0
$$536$$ −5.42784 −0.234447
$$537$$ 9.87680i 0.426215i
$$538$$ 0.207658i 0.00895278i
$$539$$ −9.83457 −0.423605
$$540$$ 0 0
$$541$$ 16.0422 0.689709 0.344855 0.938656i $$-0.387928\pi$$
0.344855 + 0.938656i $$0.387928\pi$$
$$542$$ − 30.6480i − 1.31645i
$$543$$ − 0.880741i − 0.0377962i
$$544$$ 2.51056 0.107639
$$545$$ 0 0
$$546$$ −0.965202 −0.0413068
$$547$$ 16.0143i 0.684721i 0.939569 + 0.342360i $$0.111226\pi$$
−0.939569 + 0.342360i $$0.888774\pi$$
$$548$$ 14.6903i 0.627537i
$$549$$ −3.75301 −0.160175
$$550$$ 0 0
$$551$$ −2.16543 −0.0922503
$$552$$ 2.04223i 0.0869231i
$$553$$ 5.13063i 0.218177i
$$554$$ 16.8135 0.714335
$$555$$ 0 0
$$556$$ 17.3662 0.736493
$$557$$ − 10.8557i − 0.459970i −0.973194 0.229985i $$-0.926132\pi$$
0.973194 0.229985i $$-0.0738678\pi$$
$$558$$ − 24.8078i − 1.05020i
$$559$$ −5.95777 −0.251987
$$560$$ 0 0
$$561$$ −6.64803 −0.280680
$$562$$ − 14.2749i − 0.602152i
$$563$$ − 31.0776i − 1.30977i −0.755731 0.654883i $$-0.772718\pi$$
0.755731 0.654883i $$-0.227282\pi$$
$$564$$ 2.20766 0.0929592
$$565$$ 0 0
$$566$$ −13.2288 −0.556047
$$567$$ 21.9920i 0.923577i
$$568$$ − 14.6480i − 0.614618i
$$569$$ −30.0845 −1.26121 −0.630603 0.776105i $$-0.717192\pi$$
−0.630603 + 0.776105i $$0.717192\pi$$
$$570$$ 0 0
$$571$$ 5.55673 0.232542 0.116271 0.993218i $$-0.462906\pi$$
0.116271 + 0.993218i $$0.462906\pi$$
$$572$$ − 5.29606i − 0.221439i
$$573$$ 7.86312i 0.328487i
$$574$$ −18.5106 −0.772616
$$575$$ 0 0
$$576$$ −2.83457 −0.118107
$$577$$ − 22.2499i − 0.926275i −0.886286 0.463137i $$-0.846724\pi$$
0.886286 0.463137i $$-0.153276\pi$$
$$578$$ 10.6971i 0.444941i
$$579$$ 8.15174 0.338775
$$580$$ 0 0
$$581$$ −20.5807 −0.853833
$$582$$ 0.953831i 0.0395376i
$$583$$ 15.2681i 0.632340i
$$584$$ 11.0211 0.456057
$$585$$ 0 0
$$586$$ 2.34513 0.0968764
$$587$$ 45.3103i 1.87016i 0.354440 + 0.935079i $$0.384672\pi$$
−0.354440 + 0.935079i $$0.615328\pi$$
$$588$$ − 0.614387i − 0.0253369i
$$589$$ 3.55963 0.146672
$$590$$ 0 0
$$591$$ 6.35591 0.261447
$$592$$ − 1.00000i − 0.0410997i
$$593$$ 0.232712i 0.00955635i 0.999989 + 0.00477817i $$0.00152095\pi$$
−0.999989 + 0.00477817i $$0.998479\pi$$
$$594$$ 15.4501 0.633926
$$595$$ 0 0
$$596$$ −0.207658 −0.00850601
$$597$$ 8.24989i 0.337645i
$$598$$ 4.08446i 0.167026i
$$599$$ −43.4056 −1.77350 −0.886752 0.462246i $$-0.847044\pi$$
−0.886752 + 0.462246i $$0.847044\pi$$
$$600$$ 0 0
$$601$$ 35.5317 1.44937 0.724684 0.689082i $$-0.241986\pi$$
0.724684 + 0.689082i $$0.241986\pi$$
$$602$$ − 21.3662i − 0.870823i
$$603$$ − 15.3856i − 0.626551i
$$604$$ −0.813457 −0.0330991
$$605$$ 0 0
$$606$$ −3.33829 −0.135609
$$607$$ 26.7325i 1.08504i 0.840043 + 0.542519i $$0.182529\pi$$
−0.840043 + 0.542519i $$0.817471\pi$$
$$608$$ − 0.406728i − 0.0164950i
$$609$$ 6.31717 0.255985
$$610$$ 0 0
$$611$$ 4.41532 0.178625
$$612$$ 7.11636i 0.287662i
$$613$$ − 48.0565i − 1.94098i −0.241134 0.970492i $$-0.577519\pi$$
0.241134 0.970492i $$-0.422481\pi$$
$$614$$ 23.2624 0.938795
$$615$$ 0 0
$$616$$ 18.9932 0.765256
$$617$$ 12.9789i 0.522510i 0.965270 + 0.261255i $$0.0841364\pi$$
−0.965270 + 0.261255i $$0.915864\pi$$
$$618$$ 3.58468i 0.144197i
$$619$$ −25.3662 −1.01956 −0.509778 0.860306i $$-0.670272\pi$$
−0.509778 + 0.860306i $$0.670272\pi$$
$$620$$ 0 0
$$621$$ −11.9155 −0.478154
$$622$$ − 31.3999i − 1.25902i
$$623$$ − 17.5037i − 0.701272i
$$624$$ 0.330856 0.0132448
$$625$$ 0 0
$$626$$ −8.04223 −0.321432
$$627$$ 1.07703i 0.0430124i
$$628$$ − 5.32401i − 0.212451i
$$629$$ −2.51056 −0.100102
$$630$$ 0 0
$$631$$ −31.6075 −1.25828 −0.629138 0.777293i $$-0.716592\pi$$
−0.629138 + 0.777293i $$0.716592\pi$$
$$632$$ − 1.75870i − 0.0699573i
$$633$$ − 7.61323i − 0.302599i
$$634$$ 12.3029 0.488611
$$635$$ 0 0
$$636$$ −0.953831 −0.0378219
$$637$$ − 1.22877i − 0.0486858i
$$638$$ 34.6623i 1.37229i
$$639$$ 41.5209 1.64254
$$640$$ 0 0
$$641$$ −23.8625 −0.942513 −0.471257 0.881996i $$-0.656199\pi$$
−0.471257 + 0.881996i $$0.656199\pi$$
$$642$$ 6.20766i 0.244997i
$$643$$ − 32.8277i − 1.29460i −0.762236 0.647300i $$-0.775898\pi$$
0.762236 0.647300i $$-0.224102\pi$$
$$644$$ −14.6480 −0.577213
$$645$$ 0 0
$$646$$ −1.02112 −0.0401752
$$647$$ − 13.9578i − 0.548737i −0.961625 0.274368i $$-0.911531\pi$$
0.961625 0.274368i $$-0.0884687\pi$$
$$648$$ − 7.53851i − 0.296141i
$$649$$ 9.29606 0.364902
$$650$$ 0 0
$$651$$ −10.3845 −0.406999
$$652$$ 15.2008i 0.595310i
$$653$$ − 28.5921i − 1.11890i −0.828865 0.559448i $$-0.811013\pi$$
0.828865 0.559448i $$-0.188987\pi$$
$$654$$ −1.75011 −0.0684348
$$655$$ 0 0
$$656$$ 6.34513 0.247736
$$657$$ 31.2401i 1.21879i
$$658$$ 15.8346i 0.617296i
$$659$$ 5.62691 0.219193 0.109597 0.993976i $$-0.465044\pi$$
0.109597 + 0.993976i $$0.465044\pi$$
$$660$$ 0 0
$$661$$ 13.3240 0.518244 0.259122 0.965845i $$-0.416567\pi$$
0.259122 + 0.965845i $$0.416567\pi$$
$$662$$ − 6.77981i − 0.263505i
$$663$$ − 0.830633i − 0.0322591i
$$664$$ 7.05476 0.273778
$$665$$ 0 0
$$666$$ 2.83457 0.109837
$$667$$ − 26.7325i − 1.03509i
$$668$$ 12.4826i 0.482966i
$$669$$ 6.39814 0.247366
$$670$$ 0 0
$$671$$ −8.62007 −0.332774
$$672$$ 1.18654i 0.0457719i
$$673$$ − 29.6691i − 1.14366i −0.820372 0.571831i $$-0.806233\pi$$
0.820372 0.571831i $$-0.193767\pi$$
$$674$$ −10.6903 −0.411773
$$675$$ 0 0
$$676$$ −12.3383 −0.474550
$$677$$ − 1.25383i − 0.0481885i −0.999710 0.0240943i $$-0.992330\pi$$
0.999710 0.0240943i $$-0.00767018\pi$$
$$678$$ 3.72506i 0.143060i
$$679$$ −6.84141 −0.262549
$$680$$ 0 0
$$681$$ 5.24014 0.200803
$$682$$ − 56.9795i − 2.18186i
$$683$$ 4.76045i 0.182153i 0.995844 + 0.0910767i $$0.0290308\pi$$
−0.995844 + 0.0910767i $$0.970969\pi$$
$$684$$ 1.15290 0.0440823
$$685$$ 0 0
$$686$$ −16.0143 −0.611428
$$687$$ 2.13688i 0.0815271i
$$688$$ 7.32401i 0.279225i
$$689$$ −1.90766 −0.0726761
$$690$$ 0 0
$$691$$ −20.2219 −0.769279 −0.384639 0.923067i $$-0.625674\pi$$
−0.384639 + 0.923067i $$0.625674\pi$$
$$692$$ 23.0354i 0.875674i
$$693$$ 53.8375i 2.04512i
$$694$$ −5.62691 −0.213595
$$695$$ 0 0
$$696$$ −2.16543 −0.0820803
$$697$$ − 15.9298i − 0.603385i
$$698$$ − 15.1306i − 0.572703i
$$699$$ −11.5174 −0.435628
$$700$$ 0 0
$$701$$ −22.0845 −0.834119 −0.417059 0.908879i $$-0.636939\pi$$
−0.417059 + 0.908879i $$0.636939\pi$$
$$702$$ 1.93040i 0.0728584i
$$703$$ 0.406728i 0.0153401i
$$704$$ −6.51056 −0.245376
$$705$$ 0 0
$$706$$ 35.7816 1.34666
$$707$$ − 23.9441i − 0.900510i
$$708$$ 0.580745i 0.0218257i
$$709$$ −5.59896 −0.210273 −0.105137 0.994458i $$-0.533528\pi$$
−0.105137 + 0.994458i $$0.533528\pi$$
$$710$$ 0 0
$$711$$ 4.98516 0.186958
$$712$$ 6.00000i 0.224860i
$$713$$ 43.9441i 1.64572i
$$714$$ 2.97888 0.111482
$$715$$ 0 0
$$716$$ 24.2835 0.907518
$$717$$ 5.84826i 0.218407i
$$718$$ − 4.48260i − 0.167289i
$$719$$ −40.2921 −1.50264 −0.751321 0.659937i $$-0.770583\pi$$
−0.751321 + 0.659937i $$0.770583\pi$$
$$720$$ 0 0
$$721$$ −25.7114 −0.957542
$$722$$ − 18.8346i − 0.700950i
$$723$$ − 7.43294i − 0.276434i
$$724$$ −2.16543 −0.0804775
$$725$$ 0 0
$$726$$ 12.7661 0.473796
$$727$$ 11.2538i 0.417381i 0.977982 + 0.208691i $$0.0669202\pi$$
−0.977982 + 0.208691i $$0.933080\pi$$
$$728$$ 2.37309i 0.0879524i
$$729$$ 18.4729 0.684180
$$730$$ 0 0
$$731$$ 18.3874 0.680081
$$732$$ − 0.538514i − 0.0199041i
$$733$$ 42.2892i 1.56199i 0.624539 + 0.780994i $$0.285287\pi$$
−0.624539 + 0.780994i $$0.714713\pi$$
$$734$$ 21.5653 0.795990
$$735$$ 0 0
$$736$$ 5.02112 0.185081
$$737$$ − 35.3383i − 1.30170i
$$738$$ 17.9857i 0.662064i
$$739$$ 29.3103 1.07820 0.539099 0.842242i $$-0.318765\pi$$
0.539099 + 0.842242i $$0.318765\pi$$
$$740$$ 0 0
$$741$$ −0.134569 −0.00494350
$$742$$ − 6.84141i − 0.251156i
$$743$$ 4.39245i 0.161144i 0.996749 + 0.0805718i $$0.0256746\pi$$
−0.996749 + 0.0805718i $$0.974325\pi$$
$$744$$ 3.55963 0.130502
$$745$$ 0 0
$$746$$ 16.9230 0.619594
$$747$$ 19.9972i 0.731660i
$$748$$ 16.3451i 0.597637i
$$749$$ −44.5248 −1.62690
$$750$$ 0 0
$$751$$ 2.58074 0.0941727 0.0470864 0.998891i $$-0.485006\pi$$
0.0470864 + 0.998891i $$0.485006\pi$$
$$752$$ − 5.42784i − 0.197933i
$$753$$ 0.496284i 0.0180856i
$$754$$ −4.33086 −0.157720
$$755$$ 0 0
$$756$$ −6.92297 −0.251786
$$757$$ 27.3805i 0.995162i 0.867418 + 0.497581i $$0.165778\pi$$
−0.867418 + 0.497581i $$0.834222\pi$$
$$758$$ − 31.2710i − 1.13581i
$$759$$ −13.2961 −0.482616
$$760$$ 0 0
$$761$$ −42.9509 −1.55697 −0.778485 0.627663i $$-0.784011\pi$$
−0.778485 + 0.627663i $$0.784011\pi$$
$$762$$ − 3.50372i − 0.126926i
$$763$$ − 12.5528i − 0.454441i
$$764$$ 19.3326 0.699429
$$765$$ 0 0
$$766$$ 1.62691 0.0587828
$$767$$ 1.16149i 0.0419389i
$$768$$ − 0.406728i − 0.0146765i
$$769$$ 13.0633 0.471076 0.235538 0.971865i $$-0.424315\pi$$
0.235538 + 0.971865i $$0.424315\pi$$
$$770$$ 0 0
$$771$$ 11.0348 0.397409
$$772$$ − 20.0422i − 0.721336i
$$773$$ 15.3662i 0.552685i 0.961059 + 0.276343i $$0.0891225\pi$$
−0.961059 + 0.276343i $$0.910878\pi$$
$$774$$ −20.7604 −0.746219
$$775$$ 0 0
$$776$$ 2.34513 0.0841852
$$777$$ − 1.18654i − 0.0425670i
$$778$$ 31.6412i 1.13439i
$$779$$ −2.58074 −0.0924648
$$780$$ 0 0
$$781$$ 95.3668 3.41249
$$782$$ − 12.6058i − 0.450782i
$$783$$ − 12.6343i − 0.451515i
$$784$$ −1.51056 −0.0539485
$$785$$ 0 0
$$786$$ 5.46149 0.194805
$$787$$ 10.9316i 0.389668i 0.980836 + 0.194834i $$0.0624168\pi$$
−0.980836 + 0.194834i $$0.937583\pi$$
$$788$$ − 15.6269i − 0.556686i
$$789$$ 8.22134 0.292688
$$790$$ 0 0
$$791$$ −26.7182 −0.949990
$$792$$ − 18.4546i − 0.655757i
$$793$$ − 1.07703i − 0.0382464i
$$794$$ 39.2961 1.39456
$$795$$ 0 0
$$796$$ 20.2835 0.718931
$$797$$ − 24.2921i − 0.860471i −0.902717 0.430235i $$-0.858431\pi$$
0.902717 0.430235i $$-0.141569\pi$$
$$798$$ − 0.482601i − 0.0170839i
$$799$$ −13.6269 −0.482086
$$800$$ 0 0
$$801$$ −17.0074 −0.600928
$$802$$ − 9.66914i − 0.341429i
$$803$$ 71.7536i 2.53213i
$$804$$ 2.20766 0.0778581
$$805$$ 0 0
$$806$$ 7.11926 0.250765
$$807$$ − 0.0844605i − 0.00297315i
$$808$$ 8.20766i 0.288744i
$$809$$ 40.0422 1.40781 0.703905 0.710294i $$-0.251438\pi$$
0.703905 + 0.710294i $$0.251438\pi$$
$$810$$ 0 0
$$811$$ −40.6343 −1.42686 −0.713432 0.700724i $$-0.752860\pi$$
−0.713432 + 0.700724i $$0.752860\pi$$
$$812$$ − 15.5317i − 0.545055i
$$813$$ 12.4654i 0.437182i
$$814$$ 6.51056 0.228195
$$815$$ 0 0
$$816$$ −1.02112 −0.0357462
$$817$$ − 2.97888i − 0.104218i
$$818$$ 26.2749i 0.918682i
$$819$$ −6.72668 −0.235049
$$820$$ 0 0
$$821$$ −41.4787 −1.44762 −0.723808 0.690002i $$-0.757610\pi$$
−0.723808 + 0.690002i $$0.757610\pi$$
$$822$$ − 5.97495i − 0.208400i
$$823$$ 5.27610i 0.183913i 0.995763 + 0.0919566i $$0.0293121\pi$$
−0.995763 + 0.0919566i $$0.970688\pi$$
$$824$$ 8.81346 0.307031
$$825$$ 0 0
$$826$$ −4.16543 −0.144934
$$827$$ 1.76438i 0.0613537i 0.999529 + 0.0306768i $$0.00976627\pi$$
−0.999529 + 0.0306768i $$0.990234\pi$$
$$828$$ 14.2327i 0.494621i
$$829$$ −16.3029 −0.566223 −0.283112 0.959087i $$-0.591367\pi$$
−0.283112 + 0.959087i $$0.591367\pi$$
$$830$$ 0 0
$$831$$ −6.83851 −0.237225
$$832$$ − 0.813457i − 0.0282015i
$$833$$ 3.79234i 0.131397i
$$834$$ −7.06335 −0.244584
$$835$$ 0 0
$$836$$ 2.64803 0.0915840
$$837$$ 20.7689i 0.717879i
$$838$$ 30.1940i 1.04303i
$$839$$ −14.3731 −0.496214 −0.248107 0.968733i $$-0.579808\pi$$
−0.248107 + 0.968733i $$0.579808\pi$$
$$840$$ 0 0
$$841$$ −0.654870 −0.0225817
$$842$$ 22.0000i 0.758170i
$$843$$ 5.80602i 0.199970i
$$844$$ −18.7182 −0.644308
$$845$$ 0 0
$$846$$ 15.3856 0.528968
$$847$$ 91.5659i 3.14624i
$$848$$ 2.34513i 0.0805321i
$$849$$ 5.38052 0.184659
$$850$$ 0 0
$$851$$ −5.02112 −0.172122
$$852$$ 5.95777i 0.204110i
$$853$$ − 43.2961i − 1.48243i −0.671268 0.741214i $$-0.734250\pi$$
0.671268 0.741214i $$-0.265750\pi$$
$$854$$ 3.86253 0.132173
$$855$$ 0 0
$$856$$ 15.2624 0.521659
$$857$$ − 49.0776i − 1.67646i −0.545317 0.838230i $$-0.683591\pi$$
0.545317 0.838230i $$-0.316409\pi$$
$$858$$ 2.15406i 0.0735383i
$$859$$ −15.5374 −0.530128 −0.265064 0.964231i $$-0.585393\pi$$
−0.265064 + 0.964231i $$0.585393\pi$$
$$860$$ 0 0
$$861$$ 7.52877 0.256580
$$862$$ 25.5653i 0.870758i
$$863$$ 36.0901i 1.22852i 0.789103 + 0.614261i $$0.210546\pi$$
−0.789103 + 0.614261i $$0.789454\pi$$
$$864$$ 2.37309 0.0807340
$$865$$ 0 0
$$866$$ 32.0422 1.08884
$$867$$ − 4.35081i − 0.147761i
$$868$$ 25.5317i 0.866601i
$$869$$ 11.4501 0.388419
$$870$$ 0 0
$$871$$ 4.41532 0.149607
$$872$$ 4.30290i 0.145715i
$$873$$ 6.64744i 0.224982i
$$874$$ −2.04223 −0.0690795
$$875$$ 0 0
$$876$$ −4.48260 −0.151453
$$877$$ 29.3240i 0.990202i 0.868836 + 0.495101i $$0.164869\pi$$
−0.868836 + 0.495101i $$0.835131\pi$$
$$878$$ − 3.93840i − 0.132915i
$$879$$ −0.953831 −0.0321719
$$880$$ 0 0
$$881$$ 23.8066 0.802065 0.401033 0.916064i $$-0.368651\pi$$
0.401033 + 0.916064i $$0.368651\pi$$
$$882$$ − 4.28178i − 0.144175i
$$883$$ 3.94699i 0.132827i 0.997792 + 0.0664134i $$0.0211556\pi$$
−0.997792 + 0.0664134i $$0.978844\pi$$
$$884$$ −2.04223 −0.0686876
$$885$$ 0 0
$$886$$ −0.0758724 −0.00254898
$$887$$ 1.96346i 0.0659264i 0.999457 + 0.0329632i $$0.0104944\pi$$
−0.999457 + 0.0329632i $$0.989506\pi$$
$$888$$ 0.406728i 0.0136489i
$$889$$ 25.1306 0.842854
$$890$$ 0 0
$$891$$ 49.0799 1.64424
$$892$$ − 15.7307i − 0.526704i
$$893$$ 2.20766i 0.0738765i
$$894$$ 0.0844605 0.00282478
$$895$$ 0 0
$$896$$ 2.91729 0.0974597
$$897$$ − 1.66127i − 0.0554681i
$$898$$ − 19.2961i − 0.643918i
$$899$$ −46.5950 −1.55403
$$900$$ 0 0
$$901$$ 5.88758 0.196144
$$902$$ 41.3103i 1.37548i
$$903$$ 8.69026i 0.289194i
$$904$$ 9.15859 0.304610
$$905$$ 0 0
$$906$$ 0.330856 0.0109920
$$907$$ 5.89049i 0.195590i 0.995207 + 0.0977952i $$0.0311790\pi$$
−0.995207 + 0.0977952i $$0.968821\pi$$
$$908$$ − 12.8836i − 0.427559i
$$909$$ −23.2652 −0.771658
$$910$$ 0 0
$$911$$ −27.9527 −0.926113 −0.463057 0.886329i $$-0.653247\pi$$
−0.463057 + 0.886329i $$0.653247\pi$$
$$912$$ 0.165428i 0.00547787i
$$913$$ 45.9304i 1.52007i
$$914$$ −3.00684 −0.0994575
$$915$$ 0 0
$$916$$ 5.25383 0.173591
$$917$$ 39.1729i 1.29360i
$$918$$ − 5.95777i − 0.196636i
$$919$$ −17.7587 −0.585805 −0.292903 0.956142i $$-0.594621\pi$$
−0.292903 + 0.956142i $$0.594621\pi$$
$$920$$ 0 0
$$921$$ −9.46149 −0.311767
$$922$$ − 0.633755i − 0.0208716i
$$923$$ 11.9155i 0.392205i
$$924$$ −7.72506 −0.254136
$$925$$ 0 0
$$926$$ 18.0422 0.592904
$$927$$ 24.9824i 0.820529i
$$928$$ 5.32401i 0.174769i
$$929$$ −34.2892 −1.12499 −0.562496 0.826800i $$-0.690159\pi$$
−0.562496 + 0.826800i $$0.690159\pi$$
$$930$$ 0 0
$$931$$ 0.614387 0.0201357
$$932$$ 28.3172i 0.927560i
$$933$$ 12.7712i 0.418111i
$$934$$ 23.4758 0.768150
$$935$$ 0 0
$$936$$ 2.30580 0.0753675
$$937$$ − 14.0845i − 0.460119i −0.973177 0.230060i $$-0.926108\pi$$
0.973177 0.230060i $$-0.0738921\pi$$
$$938$$ 15.8346i 0.517017i
$$939$$ 3.27100 0.106745
$$940$$ 0 0
$$941$$ 14.2499 0.464533 0.232267 0.972652i $$-0.425386\pi$$
0.232267 + 0.972652i $$0.425386\pi$$
$$942$$ 2.16543i 0.0705535i
$$943$$ − 31.8596i − 1.03749i
$$944$$ 1.42784 0.0464723
$$945$$ 0 0
$$946$$ −47.6834 −1.55032
$$947$$ − 29.8203i − 0.969029i −0.874783 0.484515i $$-0.838996\pi$$
0.874783 0.484515i $$-0.161004\pi$$
$$948$$ 0.715313i 0.0232323i
$$949$$ −8.96520 −0.291023
$$950$$ 0 0
$$951$$ −5.00394 −0.162264
$$952$$ − 7.32401i − 0.237373i
$$953$$ − 8.64803i − 0.280137i −0.990142 0.140069i $$-0.955268\pi$$
0.990142 0.140069i $$-0.0447323\pi$$
$$954$$ −6.64744 −0.215219
$$955$$ 0 0
$$956$$ 14.3788 0.465043
$$957$$ − 14.0981i − 0.455728i
$$958$$ 8.19907i 0.264900i
$$959$$ 42.8557 1.38388
$$960$$ 0 0
$$961$$ 45.5950 1.47081
$$962$$ 0.813457i 0.0262269i
$$963$$ 43.2624i 1.39411i
$$964$$ −18.2749 −0.588596
$$965$$ 0 0
$$966$$ 5.95777 0.191688
$$967$$ − 45.0325i − 1.44815i −0.689723 0.724074i $$-0.742268\pi$$
0.689723 0.724074i $$-0.257732\pi$$
$$968$$ − 31.3874i − 1.00883i
$$969$$ 0.415317 0.0133419
$$970$$ 0 0
$$971$$ 26.3029 0.844100 0.422050 0.906573i $$-0.361311\pi$$
0.422050 + 0.906573i $$0.361311\pi$$
$$972$$ 10.1854i 0.326696i
$$973$$ − 50.6623i − 1.62416i
$$974$$ −0.953831 −0.0305627
$$975$$ 0 0
$$976$$ −1.32401 −0.0423807
$$977$$ 25.6549i 0.820772i 0.911912 + 0.410386i $$0.134606\pi$$
−0.911912 + 0.410386i $$0.865394\pi$$
$$978$$ − 6.18260i − 0.197698i
$$979$$ −39.0633 −1.24847
$$980$$ 0 0
$$981$$ −12.1969 −0.389416
$$982$$ 19.3383i 0.617110i
$$983$$ 55.3611i 1.76575i 0.469611 + 0.882873i $$0.344394\pi$$
−0.469611 + 0.882873i $$0.655606\pi$$
$$984$$ −2.58074 −0.0822711
$$985$$ 0 0
$$986$$ 13.3662 0.425668
$$987$$ − 6.44037i − 0.204999i
$$988$$ 0.330856i 0.0105259i
$$989$$ 36.7747 1.16937
$$990$$ 0 0
$$991$$ −3.24814 −0.103181 −0.0515903 0.998668i $$-0.516429\pi$$
−0.0515903 + 0.998668i $$0.516429\pi$$
$$992$$ − 8.75186i − 0.277872i
$$993$$ 2.75754i 0.0875080i
$$994$$ −42.7325 −1.35539
$$995$$ 0 0
$$996$$ −2.86937 −0.0909195
$$997$$ − 12.6480i − 0.400567i −0.979738 0.200284i $$-0.935814\pi$$
0.979738 0.200284i $$-0.0641863\pi$$
$$998$$ 5.63550i 0.178389i
$$999$$ −2.37309 −0.0750811
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 1850.2.b.o.149.5 6
5.2 odd 4 1850.2.a.z.1.2 3
5.3 odd 4 370.2.a.g.1.2 3
5.4 even 2 inner 1850.2.b.o.149.2 6
15.8 even 4 3330.2.a.bg.1.3 3
20.3 even 4 2960.2.a.u.1.2 3

By twisted newform
Twist Min Dim Char Parity Ord Type
370.2.a.g.1.2 3 5.3 odd 4
1850.2.a.z.1.2 3 5.2 odd 4
1850.2.b.o.149.2 6 5.4 even 2 inner
1850.2.b.o.149.5 6 1.1 even 1 trivial
2960.2.a.u.1.2 3 20.3 even 4
3330.2.a.bg.1.3 3 15.8 even 4