# Properties

 Label 3450.2.d.l.2899.1 Level $3450$ Weight $2$ Character 3450.2899 Analytic conductor $27.548$ Analytic rank $1$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3450 = 2 \cdot 3 \cdot 5^{2} \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3450.d (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$27.5483886973$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 2899.1 Root $$-1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 3450.2899 Dual form 3450.2.d.l.2899.2

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -5.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$$ $$q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -5.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} -3.00000 q^{11} -1.00000i q^{12} +5.00000i q^{13} -5.00000 q^{14} +1.00000 q^{16} -6.00000i q^{17} +1.00000i q^{18} +1.00000 q^{19} +5.00000 q^{21} +3.00000i q^{22} +1.00000i q^{23} -1.00000 q^{24} +5.00000 q^{26} -1.00000i q^{27} +5.00000i q^{28} +5.00000 q^{29} -8.00000 q^{31} -1.00000i q^{32} -3.00000i q^{33} -6.00000 q^{34} +1.00000 q^{36} -4.00000i q^{37} -1.00000i q^{38} -5.00000 q^{39} -7.00000 q^{41} -5.00000i q^{42} -7.00000i q^{43} +3.00000 q^{44} +1.00000 q^{46} +6.00000i q^{47} +1.00000i q^{48} -18.0000 q^{49} +6.00000 q^{51} -5.00000i q^{52} +8.00000i q^{53} -1.00000 q^{54} +5.00000 q^{56} +1.00000i q^{57} -5.00000i q^{58} +10.0000 q^{59} -12.0000 q^{61} +8.00000i q^{62} +5.00000i q^{63} -1.00000 q^{64} -3.00000 q^{66} +12.0000i q^{67} +6.00000i q^{68} -1.00000 q^{69} +10.0000 q^{71} -1.00000i q^{72} +15.0000i q^{73} -4.00000 q^{74} -1.00000 q^{76} +15.0000i q^{77} +5.00000i q^{78} +5.00000 q^{79} +1.00000 q^{81} +7.00000i q^{82} -9.00000i q^{83} -5.00000 q^{84} -7.00000 q^{86} +5.00000i q^{87} -3.00000i q^{88} -14.0000 q^{89} +25.0000 q^{91} -1.00000i q^{92} -8.00000i q^{93} +6.00000 q^{94} +1.00000 q^{96} -16.0000i q^{97} +18.0000i q^{98} +3.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{4} + 2q^{6} - 2q^{9} + O(q^{10})$$ $$2q - 2q^{4} + 2q^{6} - 2q^{9} - 6q^{11} - 10q^{14} + 2q^{16} + 2q^{19} + 10q^{21} - 2q^{24} + 10q^{26} + 10q^{29} - 16q^{31} - 12q^{34} + 2q^{36} - 10q^{39} - 14q^{41} + 6q^{44} + 2q^{46} - 36q^{49} + 12q^{51} - 2q^{54} + 10q^{56} + 20q^{59} - 24q^{61} - 2q^{64} - 6q^{66} - 2q^{69} + 20q^{71} - 8q^{74} - 2q^{76} + 10q^{79} + 2q^{81} - 10q^{84} - 14q^{86} - 28q^{89} + 50q^{91} + 12q^{94} + 2q^{96} + 6q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3450\mathbb{Z}\right)^\times$$.

 $$n$$ $$277$$ $$1151$$ $$1201$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

## 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
$$3$$ 1.00000i 0.577350i
$$4$$ −1.00000 −0.500000
$$5$$ 0 0
$$6$$ 1.00000 0.408248
$$7$$ − 5.00000i − 1.88982i −0.327327 0.944911i $$-0.606148\pi$$
0.327327 0.944911i $$-0.393852\pi$$
$$8$$ 1.00000i 0.353553i
$$9$$ −1.00000 −0.333333
$$10$$ 0 0
$$11$$ −3.00000 −0.904534 −0.452267 0.891883i $$-0.649385\pi$$
−0.452267 + 0.891883i $$0.649385\pi$$
$$12$$ − 1.00000i − 0.288675i
$$13$$ 5.00000i 1.38675i 0.720577 + 0.693375i $$0.243877\pi$$
−0.720577 + 0.693375i $$0.756123\pi$$
$$14$$ −5.00000 −1.33631
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ − 6.00000i − 1.45521i −0.685994 0.727607i $$-0.740633\pi$$
0.685994 0.727607i $$-0.259367\pi$$
$$18$$ 1.00000i 0.235702i
$$19$$ 1.00000 0.229416 0.114708 0.993399i $$-0.463407\pi$$
0.114708 + 0.993399i $$0.463407\pi$$
$$20$$ 0 0
$$21$$ 5.00000 1.09109
$$22$$ 3.00000i 0.639602i
$$23$$ 1.00000i 0.208514i
$$24$$ −1.00000 −0.204124
$$25$$ 0 0
$$26$$ 5.00000 0.980581
$$27$$ − 1.00000i − 0.192450i
$$28$$ 5.00000i 0.944911i
$$29$$ 5.00000 0.928477 0.464238 0.885710i $$-0.346328\pi$$
0.464238 + 0.885710i $$0.346328\pi$$
$$30$$ 0 0
$$31$$ −8.00000 −1.43684 −0.718421 0.695608i $$-0.755135\pi$$
−0.718421 + 0.695608i $$0.755135\pi$$
$$32$$ − 1.00000i − 0.176777i
$$33$$ − 3.00000i − 0.522233i
$$34$$ −6.00000 −1.02899
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ − 4.00000i − 0.657596i −0.944400 0.328798i $$-0.893356\pi$$
0.944400 0.328798i $$-0.106644\pi$$
$$38$$ − 1.00000i − 0.162221i
$$39$$ −5.00000 −0.800641
$$40$$ 0 0
$$41$$ −7.00000 −1.09322 −0.546608 0.837389i $$-0.684081\pi$$
−0.546608 + 0.837389i $$0.684081\pi$$
$$42$$ − 5.00000i − 0.771517i
$$43$$ − 7.00000i − 1.06749i −0.845645 0.533745i $$-0.820784\pi$$
0.845645 0.533745i $$-0.179216\pi$$
$$44$$ 3.00000 0.452267
$$45$$ 0 0
$$46$$ 1.00000 0.147442
$$47$$ 6.00000i 0.875190i 0.899172 + 0.437595i $$0.144170\pi$$
−0.899172 + 0.437595i $$0.855830\pi$$
$$48$$ 1.00000i 0.144338i
$$49$$ −18.0000 −2.57143
$$50$$ 0 0
$$51$$ 6.00000 0.840168
$$52$$ − 5.00000i − 0.693375i
$$53$$ 8.00000i 1.09888i 0.835532 + 0.549442i $$0.185160\pi$$
−0.835532 + 0.549442i $$0.814840\pi$$
$$54$$ −1.00000 −0.136083
$$55$$ 0 0
$$56$$ 5.00000 0.668153
$$57$$ 1.00000i 0.132453i
$$58$$ − 5.00000i − 0.656532i
$$59$$ 10.0000 1.30189 0.650945 0.759125i $$-0.274373\pi$$
0.650945 + 0.759125i $$0.274373\pi$$
$$60$$ 0 0
$$61$$ −12.0000 −1.53644 −0.768221 0.640184i $$-0.778858\pi$$
−0.768221 + 0.640184i $$0.778858\pi$$
$$62$$ 8.00000i 1.01600i
$$63$$ 5.00000i 0.629941i
$$64$$ −1.00000 −0.125000
$$65$$ 0 0
$$66$$ −3.00000 −0.369274
$$67$$ 12.0000i 1.46603i 0.680211 + 0.733017i $$0.261888\pi$$
−0.680211 + 0.733017i $$0.738112\pi$$
$$68$$ 6.00000i 0.727607i
$$69$$ −1.00000 −0.120386
$$70$$ 0 0
$$71$$ 10.0000 1.18678 0.593391 0.804914i $$-0.297789\pi$$
0.593391 + 0.804914i $$0.297789\pi$$
$$72$$ − 1.00000i − 0.117851i
$$73$$ 15.0000i 1.75562i 0.479012 + 0.877809i $$0.340995\pi$$
−0.479012 + 0.877809i $$0.659005\pi$$
$$74$$ −4.00000 −0.464991
$$75$$ 0 0
$$76$$ −1.00000 −0.114708
$$77$$ 15.0000i 1.70941i
$$78$$ 5.00000i 0.566139i
$$79$$ 5.00000 0.562544 0.281272 0.959628i $$-0.409244\pi$$
0.281272 + 0.959628i $$0.409244\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 7.00000i 0.773021i
$$83$$ − 9.00000i − 0.987878i −0.869496 0.493939i $$-0.835557\pi$$
0.869496 0.493939i $$-0.164443\pi$$
$$84$$ −5.00000 −0.545545
$$85$$ 0 0
$$86$$ −7.00000 −0.754829
$$87$$ 5.00000i 0.536056i
$$88$$ − 3.00000i − 0.319801i
$$89$$ −14.0000 −1.48400 −0.741999 0.670402i $$-0.766122\pi$$
−0.741999 + 0.670402i $$0.766122\pi$$
$$90$$ 0 0
$$91$$ 25.0000 2.62071
$$92$$ − 1.00000i − 0.104257i
$$93$$ − 8.00000i − 0.829561i
$$94$$ 6.00000 0.618853
$$95$$ 0 0
$$96$$ 1.00000 0.102062
$$97$$ − 16.0000i − 1.62455i −0.583272 0.812277i $$-0.698228\pi$$
0.583272 0.812277i $$-0.301772\pi$$
$$98$$ 18.0000i 1.81827i
$$99$$ 3.00000 0.301511
$$100$$ 0 0
$$101$$ −10.0000 −0.995037 −0.497519 0.867453i $$-0.665755\pi$$
−0.497519 + 0.867453i $$0.665755\pi$$
$$102$$ − 6.00000i − 0.594089i
$$103$$ 19.0000i 1.87213i 0.351833 + 0.936063i $$0.385559\pi$$
−0.351833 + 0.936063i $$0.614441\pi$$
$$104$$ −5.00000 −0.490290
$$105$$ 0 0
$$106$$ 8.00000 0.777029
$$107$$ 12.0000i 1.16008i 0.814587 + 0.580042i $$0.196964\pi$$
−0.814587 + 0.580042i $$0.803036\pi$$
$$108$$ 1.00000i 0.0962250i
$$109$$ −12.0000 −1.14939 −0.574696 0.818367i $$-0.694880\pi$$
−0.574696 + 0.818367i $$0.694880\pi$$
$$110$$ 0 0
$$111$$ 4.00000 0.379663
$$112$$ − 5.00000i − 0.472456i
$$113$$ − 10.0000i − 0.940721i −0.882474 0.470360i $$-0.844124\pi$$
0.882474 0.470360i $$-0.155876\pi$$
$$114$$ 1.00000 0.0936586
$$115$$ 0 0
$$116$$ −5.00000 −0.464238
$$117$$ − 5.00000i − 0.462250i
$$118$$ − 10.0000i − 0.920575i
$$119$$ −30.0000 −2.75010
$$120$$ 0 0
$$121$$ −2.00000 −0.181818
$$122$$ 12.0000i 1.08643i
$$123$$ − 7.00000i − 0.631169i
$$124$$ 8.00000 0.718421
$$125$$ 0 0
$$126$$ 5.00000 0.445435
$$127$$ − 6.00000i − 0.532414i −0.963916 0.266207i $$-0.914230\pi$$
0.963916 0.266207i $$-0.0857705\pi$$
$$128$$ 1.00000i 0.0883883i
$$129$$ 7.00000 0.616316
$$130$$ 0 0
$$131$$ −4.00000 −0.349482 −0.174741 0.984614i $$-0.555909\pi$$
−0.174741 + 0.984614i $$0.555909\pi$$
$$132$$ 3.00000i 0.261116i
$$133$$ − 5.00000i − 0.433555i
$$134$$ 12.0000 1.03664
$$135$$ 0 0
$$136$$ 6.00000 0.514496
$$137$$ 8.00000i 0.683486i 0.939793 + 0.341743i $$0.111017\pi$$
−0.939793 + 0.341743i $$0.888983\pi$$
$$138$$ 1.00000i 0.0851257i
$$139$$ −2.00000 −0.169638 −0.0848189 0.996396i $$-0.527031\pi$$
−0.0848189 + 0.996396i $$0.527031\pi$$
$$140$$ 0 0
$$141$$ −6.00000 −0.505291
$$142$$ − 10.0000i − 0.839181i
$$143$$ − 15.0000i − 1.25436i
$$144$$ −1.00000 −0.0833333
$$145$$ 0 0
$$146$$ 15.0000 1.24141
$$147$$ − 18.0000i − 1.48461i
$$148$$ 4.00000i 0.328798i
$$149$$ 20.0000 1.63846 0.819232 0.573462i $$-0.194400\pi$$
0.819232 + 0.573462i $$0.194400\pi$$
$$150$$ 0 0
$$151$$ −10.0000 −0.813788 −0.406894 0.913475i $$-0.633388\pi$$
−0.406894 + 0.913475i $$0.633388\pi$$
$$152$$ 1.00000i 0.0811107i
$$153$$ 6.00000i 0.485071i
$$154$$ 15.0000 1.20873
$$155$$ 0 0
$$156$$ 5.00000 0.400320
$$157$$ − 4.00000i − 0.319235i −0.987179 0.159617i $$-0.948974\pi$$
0.987179 0.159617i $$-0.0510260\pi$$
$$158$$ − 5.00000i − 0.397779i
$$159$$ −8.00000 −0.634441
$$160$$ 0 0
$$161$$ 5.00000 0.394055
$$162$$ − 1.00000i − 0.0785674i
$$163$$ 12.0000i 0.939913i 0.882690 + 0.469956i $$0.155730\pi$$
−0.882690 + 0.469956i $$0.844270\pi$$
$$164$$ 7.00000 0.546608
$$165$$ 0 0
$$166$$ −9.00000 −0.698535
$$167$$ 12.0000i 0.928588i 0.885681 + 0.464294i $$0.153692\pi$$
−0.885681 + 0.464294i $$0.846308\pi$$
$$168$$ 5.00000i 0.385758i
$$169$$ −12.0000 −0.923077
$$170$$ 0 0
$$171$$ −1.00000 −0.0764719
$$172$$ 7.00000i 0.533745i
$$173$$ 21.0000i 1.59660i 0.602260 + 0.798300i $$0.294267\pi$$
−0.602260 + 0.798300i $$0.705733\pi$$
$$174$$ 5.00000 0.379049
$$175$$ 0 0
$$176$$ −3.00000 −0.226134
$$177$$ 10.0000i 0.751646i
$$178$$ 14.0000i 1.04934i
$$179$$ −6.00000 −0.448461 −0.224231 0.974536i $$-0.571987\pi$$
−0.224231 + 0.974536i $$0.571987\pi$$
$$180$$ 0 0
$$181$$ −14.0000 −1.04061 −0.520306 0.853980i $$-0.674182\pi$$
−0.520306 + 0.853980i $$0.674182\pi$$
$$182$$ − 25.0000i − 1.85312i
$$183$$ − 12.0000i − 0.887066i
$$184$$ −1.00000 −0.0737210
$$185$$ 0 0
$$186$$ −8.00000 −0.586588
$$187$$ 18.0000i 1.31629i
$$188$$ − 6.00000i − 0.437595i
$$189$$ −5.00000 −0.363696
$$190$$ 0 0
$$191$$ −13.0000 −0.940647 −0.470323 0.882494i $$-0.655863\pi$$
−0.470323 + 0.882494i $$0.655863\pi$$
$$192$$ − 1.00000i − 0.0721688i
$$193$$ 6.00000i 0.431889i 0.976406 + 0.215945i $$0.0692831\pi$$
−0.976406 + 0.215945i $$0.930717\pi$$
$$194$$ −16.0000 −1.14873
$$195$$ 0 0
$$196$$ 18.0000 1.28571
$$197$$ 11.0000i 0.783718i 0.920025 + 0.391859i $$0.128168\pi$$
−0.920025 + 0.391859i $$0.871832\pi$$
$$198$$ − 3.00000i − 0.213201i
$$199$$ 1.00000 0.0708881 0.0354441 0.999372i $$-0.488715\pi$$
0.0354441 + 0.999372i $$0.488715\pi$$
$$200$$ 0 0
$$201$$ −12.0000 −0.846415
$$202$$ 10.0000i 0.703598i
$$203$$ − 25.0000i − 1.75466i
$$204$$ −6.00000 −0.420084
$$205$$ 0 0
$$206$$ 19.0000 1.32379
$$207$$ − 1.00000i − 0.0695048i
$$208$$ 5.00000i 0.346688i
$$209$$ −3.00000 −0.207514
$$210$$ 0 0
$$211$$ 10.0000 0.688428 0.344214 0.938891i $$-0.388145\pi$$
0.344214 + 0.938891i $$0.388145\pi$$
$$212$$ − 8.00000i − 0.549442i
$$213$$ 10.0000i 0.685189i
$$214$$ 12.0000 0.820303
$$215$$ 0 0
$$216$$ 1.00000 0.0680414
$$217$$ 40.0000i 2.71538i
$$218$$ 12.0000i 0.812743i
$$219$$ −15.0000 −1.01361
$$220$$ 0 0
$$221$$ 30.0000 2.01802
$$222$$ − 4.00000i − 0.268462i
$$223$$ − 14.0000i − 0.937509i −0.883328 0.468755i $$-0.844703\pi$$
0.883328 0.468755i $$-0.155297\pi$$
$$224$$ −5.00000 −0.334077
$$225$$ 0 0
$$226$$ −10.0000 −0.665190
$$227$$ − 8.00000i − 0.530979i −0.964114 0.265489i $$-0.914466\pi$$
0.964114 0.265489i $$-0.0855335\pi$$
$$228$$ − 1.00000i − 0.0662266i
$$229$$ −16.0000 −1.05731 −0.528655 0.848837i $$-0.677303\pi$$
−0.528655 + 0.848837i $$0.677303\pi$$
$$230$$ 0 0
$$231$$ −15.0000 −0.986928
$$232$$ 5.00000i 0.328266i
$$233$$ 7.00000i 0.458585i 0.973358 + 0.229293i $$0.0736413\pi$$
−0.973358 + 0.229293i $$0.926359\pi$$
$$234$$ −5.00000 −0.326860
$$235$$ 0 0
$$236$$ −10.0000 −0.650945
$$237$$ 5.00000i 0.324785i
$$238$$ 30.0000i 1.94461i
$$239$$ −24.0000 −1.55243 −0.776215 0.630468i $$-0.782863\pi$$
−0.776215 + 0.630468i $$0.782863\pi$$
$$240$$ 0 0
$$241$$ −18.0000 −1.15948 −0.579741 0.814801i $$-0.696846\pi$$
−0.579741 + 0.814801i $$0.696846\pi$$
$$242$$ 2.00000i 0.128565i
$$243$$ 1.00000i 0.0641500i
$$244$$ 12.0000 0.768221
$$245$$ 0 0
$$246$$ −7.00000 −0.446304
$$247$$ 5.00000i 0.318142i
$$248$$ − 8.00000i − 0.508001i
$$249$$ 9.00000 0.570352
$$250$$ 0 0
$$251$$ 12.0000 0.757433 0.378717 0.925513i $$-0.376365\pi$$
0.378717 + 0.925513i $$0.376365\pi$$
$$252$$ − 5.00000i − 0.314970i
$$253$$ − 3.00000i − 0.188608i
$$254$$ −6.00000 −0.376473
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ − 6.00000i − 0.374270i −0.982334 0.187135i $$-0.940080\pi$$
0.982334 0.187135i $$-0.0599201\pi$$
$$258$$ − 7.00000i − 0.435801i
$$259$$ −20.0000 −1.24274
$$260$$ 0 0
$$261$$ −5.00000 −0.309492
$$262$$ 4.00000i 0.247121i
$$263$$ − 4.00000i − 0.246651i −0.992366 0.123325i $$-0.960644\pi$$
0.992366 0.123325i $$-0.0393559\pi$$
$$264$$ 3.00000 0.184637
$$265$$ 0 0
$$266$$ −5.00000 −0.306570
$$267$$ − 14.0000i − 0.856786i
$$268$$ − 12.0000i − 0.733017i
$$269$$ −3.00000 −0.182913 −0.0914566 0.995809i $$-0.529152\pi$$
−0.0914566 + 0.995809i $$0.529152\pi$$
$$270$$ 0 0
$$271$$ −28.0000 −1.70088 −0.850439 0.526073i $$-0.823664\pi$$
−0.850439 + 0.526073i $$0.823664\pi$$
$$272$$ − 6.00000i − 0.363803i
$$273$$ 25.0000i 1.51307i
$$274$$ 8.00000 0.483298
$$275$$ 0 0
$$276$$ 1.00000 0.0601929
$$277$$ 1.00000i 0.0600842i 0.999549 + 0.0300421i $$0.00956413\pi$$
−0.999549 + 0.0300421i $$0.990436\pi$$
$$278$$ 2.00000i 0.119952i
$$279$$ 8.00000 0.478947
$$280$$ 0 0
$$281$$ 2.00000 0.119310 0.0596550 0.998219i $$-0.481000\pi$$
0.0596550 + 0.998219i $$0.481000\pi$$
$$282$$ 6.00000i 0.357295i
$$283$$ 20.0000i 1.18888i 0.804141 + 0.594438i $$0.202626\pi$$
−0.804141 + 0.594438i $$0.797374\pi$$
$$284$$ −10.0000 −0.593391
$$285$$ 0 0
$$286$$ −15.0000 −0.886969
$$287$$ 35.0000i 2.06598i
$$288$$ 1.00000i 0.0589256i
$$289$$ −19.0000 −1.11765
$$290$$ 0 0
$$291$$ 16.0000 0.937937
$$292$$ − 15.0000i − 0.877809i
$$293$$ − 28.0000i − 1.63578i −0.575376 0.817889i $$-0.695144\pi$$
0.575376 0.817889i $$-0.304856\pi$$
$$294$$ −18.0000 −1.04978
$$295$$ 0 0
$$296$$ 4.00000 0.232495
$$297$$ 3.00000i 0.174078i
$$298$$ − 20.0000i − 1.15857i
$$299$$ −5.00000 −0.289157
$$300$$ 0 0
$$301$$ −35.0000 −2.01737
$$302$$ 10.0000i 0.575435i
$$303$$ − 10.0000i − 0.574485i
$$304$$ 1.00000 0.0573539
$$305$$ 0 0
$$306$$ 6.00000 0.342997
$$307$$ − 12.0000i − 0.684876i −0.939540 0.342438i $$-0.888747\pi$$
0.939540 0.342438i $$-0.111253\pi$$
$$308$$ − 15.0000i − 0.854704i
$$309$$ −19.0000 −1.08087
$$310$$ 0 0
$$311$$ 28.0000 1.58773 0.793867 0.608091i $$-0.208065\pi$$
0.793867 + 0.608091i $$0.208065\pi$$
$$312$$ − 5.00000i − 0.283069i
$$313$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$314$$ −4.00000 −0.225733
$$315$$ 0 0
$$316$$ −5.00000 −0.281272
$$317$$ − 9.00000i − 0.505490i −0.967533 0.252745i $$-0.918667\pi$$
0.967533 0.252745i $$-0.0813334\pi$$
$$318$$ 8.00000i 0.448618i
$$319$$ −15.0000 −0.839839
$$320$$ 0 0
$$321$$ −12.0000 −0.669775
$$322$$ − 5.00000i − 0.278639i
$$323$$ − 6.00000i − 0.333849i
$$324$$ −1.00000 −0.0555556
$$325$$ 0 0
$$326$$ 12.0000 0.664619
$$327$$ − 12.0000i − 0.663602i
$$328$$ − 7.00000i − 0.386510i
$$329$$ 30.0000 1.65395
$$330$$ 0 0
$$331$$ 4.00000 0.219860 0.109930 0.993939i $$-0.464937\pi$$
0.109930 + 0.993939i $$0.464937\pi$$
$$332$$ 9.00000i 0.493939i
$$333$$ 4.00000i 0.219199i
$$334$$ 12.0000 0.656611
$$335$$ 0 0
$$336$$ 5.00000 0.272772
$$337$$ − 8.00000i − 0.435788i −0.975972 0.217894i $$-0.930081\pi$$
0.975972 0.217894i $$-0.0699187\pi$$
$$338$$ 12.0000i 0.652714i
$$339$$ 10.0000 0.543125
$$340$$ 0 0
$$341$$ 24.0000 1.29967
$$342$$ 1.00000i 0.0540738i
$$343$$ 55.0000i 2.96972i
$$344$$ 7.00000 0.377415
$$345$$ 0 0
$$346$$ 21.0000 1.12897
$$347$$ 4.00000i 0.214731i 0.994220 + 0.107366i $$0.0342415\pi$$
−0.994220 + 0.107366i $$0.965758\pi$$
$$348$$ − 5.00000i − 0.268028i
$$349$$ −1.00000 −0.0535288 −0.0267644 0.999642i $$-0.508520\pi$$
−0.0267644 + 0.999642i $$0.508520\pi$$
$$350$$ 0 0
$$351$$ 5.00000 0.266880
$$352$$ 3.00000i 0.159901i
$$353$$ 9.00000i 0.479022i 0.970894 + 0.239511i $$0.0769871\pi$$
−0.970894 + 0.239511i $$0.923013\pi$$
$$354$$ 10.0000 0.531494
$$355$$ 0 0
$$356$$ 14.0000 0.741999
$$357$$ − 30.0000i − 1.58777i
$$358$$ 6.00000i 0.317110i
$$359$$ 7.00000 0.369446 0.184723 0.982791i $$-0.440861\pi$$
0.184723 + 0.982791i $$0.440861\pi$$
$$360$$ 0 0
$$361$$ −18.0000 −0.947368
$$362$$ 14.0000i 0.735824i
$$363$$ − 2.00000i − 0.104973i
$$364$$ −25.0000 −1.31036
$$365$$ 0 0
$$366$$ −12.0000 −0.627250
$$367$$ − 1.00000i − 0.0521996i −0.999659 0.0260998i $$-0.991691\pi$$
0.999659 0.0260998i $$-0.00830876\pi$$
$$368$$ 1.00000i 0.0521286i
$$369$$ 7.00000 0.364405
$$370$$ 0 0
$$371$$ 40.0000 2.07670
$$372$$ 8.00000i 0.414781i
$$373$$ − 26.0000i − 1.34623i −0.739538 0.673114i $$-0.764956\pi$$
0.739538 0.673114i $$-0.235044\pi$$
$$374$$ 18.0000 0.930758
$$375$$ 0 0
$$376$$ −6.00000 −0.309426
$$377$$ 25.0000i 1.28757i
$$378$$ 5.00000i 0.257172i
$$379$$ −12.0000 −0.616399 −0.308199 0.951322i $$-0.599726\pi$$
−0.308199 + 0.951322i $$0.599726\pi$$
$$380$$ 0 0
$$381$$ 6.00000 0.307389
$$382$$ 13.0000i 0.665138i
$$383$$ − 9.00000i − 0.459879i −0.973205 0.229939i $$-0.926147\pi$$
0.973205 0.229939i $$-0.0738528\pi$$
$$384$$ −1.00000 −0.0510310
$$385$$ 0 0
$$386$$ 6.00000 0.305392
$$387$$ 7.00000i 0.355830i
$$388$$ 16.0000i 0.812277i
$$389$$ 38.0000 1.92668 0.963338 0.268290i $$-0.0864585\pi$$
0.963338 + 0.268290i $$0.0864585\pi$$
$$390$$ 0 0
$$391$$ 6.00000 0.303433
$$392$$ − 18.0000i − 0.909137i
$$393$$ − 4.00000i − 0.201773i
$$394$$ 11.0000 0.554172
$$395$$ 0 0
$$396$$ −3.00000 −0.150756
$$397$$ 2.00000i 0.100377i 0.998740 + 0.0501886i $$0.0159822\pi$$
−0.998740 + 0.0501886i $$0.984018\pi$$
$$398$$ − 1.00000i − 0.0501255i
$$399$$ 5.00000 0.250313
$$400$$ 0 0
$$401$$ 20.0000 0.998752 0.499376 0.866385i $$-0.333563\pi$$
0.499376 + 0.866385i $$0.333563\pi$$
$$402$$ 12.0000i 0.598506i
$$403$$ − 40.0000i − 1.99254i
$$404$$ 10.0000 0.497519
$$405$$ 0 0
$$406$$ −25.0000 −1.24073
$$407$$ 12.0000i 0.594818i
$$408$$ 6.00000i 0.297044i
$$409$$ −17.0000 −0.840596 −0.420298 0.907386i $$-0.638074\pi$$
−0.420298 + 0.907386i $$0.638074\pi$$
$$410$$ 0 0
$$411$$ −8.00000 −0.394611
$$412$$ − 19.0000i − 0.936063i
$$413$$ − 50.0000i − 2.46034i
$$414$$ −1.00000 −0.0491473
$$415$$ 0 0
$$416$$ 5.00000 0.245145
$$417$$ − 2.00000i − 0.0979404i
$$418$$ 3.00000i 0.146735i
$$419$$ 1.00000 0.0488532 0.0244266 0.999702i $$-0.492224\pi$$
0.0244266 + 0.999702i $$0.492224\pi$$
$$420$$ 0 0
$$421$$ −18.0000 −0.877266 −0.438633 0.898666i $$-0.644537\pi$$
−0.438633 + 0.898666i $$0.644537\pi$$
$$422$$ − 10.0000i − 0.486792i
$$423$$ − 6.00000i − 0.291730i
$$424$$ −8.00000 −0.388514
$$425$$ 0 0
$$426$$ 10.0000 0.484502
$$427$$ 60.0000i 2.90360i
$$428$$ − 12.0000i − 0.580042i
$$429$$ 15.0000 0.724207
$$430$$ 0 0
$$431$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$432$$ − 1.00000i − 0.0481125i
$$433$$ − 28.0000i − 1.34559i −0.739827 0.672797i $$-0.765093\pi$$
0.739827 0.672797i $$-0.234907\pi$$
$$434$$ 40.0000 1.92006
$$435$$ 0 0
$$436$$ 12.0000 0.574696
$$437$$ 1.00000i 0.0478365i
$$438$$ 15.0000i 0.716728i
$$439$$ 12.0000 0.572729 0.286364 0.958121i $$-0.407553\pi$$
0.286364 + 0.958121i $$0.407553\pi$$
$$440$$ 0 0
$$441$$ 18.0000 0.857143
$$442$$ − 30.0000i − 1.42695i
$$443$$ − 4.00000i − 0.190046i −0.995475 0.0950229i $$-0.969708\pi$$
0.995475 0.0950229i $$-0.0302924\pi$$
$$444$$ −4.00000 −0.189832
$$445$$ 0 0
$$446$$ −14.0000 −0.662919
$$447$$ 20.0000i 0.945968i
$$448$$ 5.00000i 0.236228i
$$449$$ 2.00000 0.0943858 0.0471929 0.998886i $$-0.484972\pi$$
0.0471929 + 0.998886i $$0.484972\pi$$
$$450$$ 0 0
$$451$$ 21.0000 0.988851
$$452$$ 10.0000i 0.470360i
$$453$$ − 10.0000i − 0.469841i
$$454$$ −8.00000 −0.375459
$$455$$ 0 0
$$456$$ −1.00000 −0.0468293
$$457$$ 18.0000i 0.842004i 0.907060 + 0.421002i $$0.138322\pi$$
−0.907060 + 0.421002i $$0.861678\pi$$
$$458$$ 16.0000i 0.747631i
$$459$$ −6.00000 −0.280056
$$460$$ 0 0
$$461$$ 7.00000 0.326023 0.163011 0.986624i $$-0.447879\pi$$
0.163011 + 0.986624i $$0.447879\pi$$
$$462$$ 15.0000i 0.697863i
$$463$$ 4.00000i 0.185896i 0.995671 + 0.0929479i $$0.0296290\pi$$
−0.995671 + 0.0929479i $$0.970371\pi$$
$$464$$ 5.00000 0.232119
$$465$$ 0 0
$$466$$ 7.00000 0.324269
$$467$$ − 21.0000i − 0.971764i −0.874024 0.485882i $$-0.838498\pi$$
0.874024 0.485882i $$-0.161502\pi$$
$$468$$ 5.00000i 0.231125i
$$469$$ 60.0000 2.77054
$$470$$ 0 0
$$471$$ 4.00000 0.184310
$$472$$ 10.0000i 0.460287i
$$473$$ 21.0000i 0.965581i
$$474$$ 5.00000 0.229658
$$475$$ 0 0
$$476$$ 30.0000 1.37505
$$477$$ − 8.00000i − 0.366295i
$$478$$ 24.0000i 1.09773i
$$479$$ 25.0000 1.14228 0.571140 0.820853i $$-0.306501\pi$$
0.571140 + 0.820853i $$0.306501\pi$$
$$480$$ 0 0
$$481$$ 20.0000 0.911922
$$482$$ 18.0000i 0.819878i
$$483$$ 5.00000i 0.227508i
$$484$$ 2.00000 0.0909091
$$485$$ 0 0
$$486$$ 1.00000 0.0453609
$$487$$ − 2.00000i − 0.0906287i −0.998973 0.0453143i $$-0.985571\pi$$
0.998973 0.0453143i $$-0.0144289\pi$$
$$488$$ − 12.0000i − 0.543214i
$$489$$ −12.0000 −0.542659
$$490$$ 0 0
$$491$$ −32.0000 −1.44414 −0.722070 0.691820i $$-0.756809\pi$$
−0.722070 + 0.691820i $$0.756809\pi$$
$$492$$ 7.00000i 0.315584i
$$493$$ − 30.0000i − 1.35113i
$$494$$ 5.00000 0.224961
$$495$$ 0 0
$$496$$ −8.00000 −0.359211
$$497$$ − 50.0000i − 2.24281i
$$498$$ − 9.00000i − 0.403300i
$$499$$ 6.00000 0.268597 0.134298 0.990941i $$-0.457122\pi$$
0.134298 + 0.990941i $$0.457122\pi$$
$$500$$ 0 0
$$501$$ −12.0000 −0.536120
$$502$$ − 12.0000i − 0.535586i
$$503$$ − 19.0000i − 0.847168i −0.905857 0.423584i $$-0.860772\pi$$
0.905857 0.423584i $$-0.139228\pi$$
$$504$$ −5.00000 −0.222718
$$505$$ 0 0
$$506$$ −3.00000 −0.133366
$$507$$ − 12.0000i − 0.532939i
$$508$$ 6.00000i 0.266207i
$$509$$ −30.0000 −1.32973 −0.664863 0.746965i $$-0.731510\pi$$
−0.664863 + 0.746965i $$0.731510\pi$$
$$510$$ 0 0
$$511$$ 75.0000 3.31780
$$512$$ − 1.00000i − 0.0441942i
$$513$$ − 1.00000i − 0.0441511i
$$514$$ −6.00000 −0.264649
$$515$$ 0 0
$$516$$ −7.00000 −0.308158
$$517$$ − 18.0000i − 0.791639i
$$518$$ 20.0000i 0.878750i
$$519$$ −21.0000 −0.921798
$$520$$ 0 0
$$521$$ −4.00000 −0.175243 −0.0876216 0.996154i $$-0.527927\pi$$
−0.0876216 + 0.996154i $$0.527927\pi$$
$$522$$ 5.00000i 0.218844i
$$523$$ − 1.00000i − 0.0437269i −0.999761 0.0218635i $$-0.993040\pi$$
0.999761 0.0218635i $$-0.00695991\pi$$
$$524$$ 4.00000 0.174741
$$525$$ 0 0
$$526$$ −4.00000 −0.174408
$$527$$ 48.0000i 2.09091i
$$528$$ − 3.00000i − 0.130558i
$$529$$ −1.00000 −0.0434783
$$530$$ 0 0
$$531$$ −10.0000 −0.433963
$$532$$ 5.00000i 0.216777i
$$533$$ − 35.0000i − 1.51602i
$$534$$ −14.0000 −0.605839
$$535$$ 0 0
$$536$$ −12.0000 −0.518321
$$537$$ − 6.00000i − 0.258919i
$$538$$ 3.00000i 0.129339i
$$539$$ 54.0000 2.32594
$$540$$ 0 0
$$541$$ −39.0000 −1.67674 −0.838370 0.545101i $$-0.816491\pi$$
−0.838370 + 0.545101i $$0.816491\pi$$
$$542$$ 28.0000i 1.20270i
$$543$$ − 14.0000i − 0.600798i
$$544$$ −6.00000 −0.257248
$$545$$ 0 0
$$546$$ 25.0000 1.06990
$$547$$ − 30.0000i − 1.28271i −0.767245 0.641354i $$-0.778373\pi$$
0.767245 0.641354i $$-0.221627\pi$$
$$548$$ − 8.00000i − 0.341743i
$$549$$ 12.0000 0.512148
$$550$$ 0 0
$$551$$ 5.00000 0.213007
$$552$$ − 1.00000i − 0.0425628i
$$553$$ − 25.0000i − 1.06311i
$$554$$ 1.00000 0.0424859
$$555$$ 0 0
$$556$$ 2.00000 0.0848189
$$557$$ 10.0000i 0.423714i 0.977301 + 0.211857i $$0.0679510\pi$$
−0.977301 + 0.211857i $$0.932049\pi$$
$$558$$ − 8.00000i − 0.338667i
$$559$$ 35.0000 1.48034
$$560$$ 0 0
$$561$$ −18.0000 −0.759961
$$562$$ − 2.00000i − 0.0843649i
$$563$$ − 21.0000i − 0.885044i −0.896758 0.442522i $$-0.854084\pi$$
0.896758 0.442522i $$-0.145916\pi$$
$$564$$ 6.00000 0.252646
$$565$$ 0 0
$$566$$ 20.0000 0.840663
$$567$$ − 5.00000i − 0.209980i
$$568$$ 10.0000i 0.419591i
$$569$$ 30.0000 1.25767 0.628833 0.777541i $$-0.283533\pi$$
0.628833 + 0.777541i $$0.283533\pi$$
$$570$$ 0 0
$$571$$ 4.00000 0.167395 0.0836974 0.996491i $$-0.473327\pi$$
0.0836974 + 0.996491i $$0.473327\pi$$
$$572$$ 15.0000i 0.627182i
$$573$$ − 13.0000i − 0.543083i
$$574$$ 35.0000 1.46087
$$575$$ 0 0
$$576$$ 1.00000 0.0416667
$$577$$ − 23.0000i − 0.957503i −0.877951 0.478751i $$-0.841090\pi$$
0.877951 0.478751i $$-0.158910\pi$$
$$578$$ 19.0000i 0.790296i
$$579$$ −6.00000 −0.249351
$$580$$ 0 0
$$581$$ −45.0000 −1.86691
$$582$$ − 16.0000i − 0.663221i
$$583$$ − 24.0000i − 0.993978i
$$584$$ −15.0000 −0.620704
$$585$$ 0 0
$$586$$ −28.0000 −1.15667
$$587$$ − 42.0000i − 1.73353i −0.498721 0.866763i $$-0.666197\pi$$
0.498721 0.866763i $$-0.333803\pi$$
$$588$$ 18.0000i 0.742307i
$$589$$ −8.00000 −0.329634
$$590$$ 0 0
$$591$$ −11.0000 −0.452480
$$592$$ − 4.00000i − 0.164399i
$$593$$ 3.00000i 0.123195i 0.998101 + 0.0615976i $$0.0196196\pi$$
−0.998101 + 0.0615976i $$0.980380\pi$$
$$594$$ 3.00000 0.123091
$$595$$ 0 0
$$596$$ −20.0000 −0.819232
$$597$$ 1.00000i 0.0409273i
$$598$$ 5.00000i 0.204465i
$$599$$ 44.0000 1.79779 0.898896 0.438163i $$-0.144371\pi$$
0.898896 + 0.438163i $$0.144371\pi$$
$$600$$ 0 0
$$601$$ −38.0000 −1.55005 −0.775026 0.631929i $$-0.782263\pi$$
−0.775026 + 0.631929i $$0.782263\pi$$
$$602$$ 35.0000i 1.42649i
$$603$$ − 12.0000i − 0.488678i
$$604$$ 10.0000 0.406894
$$605$$ 0 0
$$606$$ −10.0000 −0.406222
$$607$$ 22.0000i 0.892952i 0.894795 + 0.446476i $$0.147321\pi$$
−0.894795 + 0.446476i $$0.852679\pi$$
$$608$$ − 1.00000i − 0.0405554i
$$609$$ 25.0000 1.01305
$$610$$ 0 0
$$611$$ −30.0000 −1.21367
$$612$$ − 6.00000i − 0.242536i
$$613$$ 10.0000i 0.403896i 0.979396 + 0.201948i $$0.0647272\pi$$
−0.979396 + 0.201948i $$0.935273\pi$$
$$614$$ −12.0000 −0.484281
$$615$$ 0 0
$$616$$ −15.0000 −0.604367
$$617$$ 2.00000i 0.0805170i 0.999189 + 0.0402585i $$0.0128181\pi$$
−0.999189 + 0.0402585i $$0.987182\pi$$
$$618$$ 19.0000i 0.764292i
$$619$$ 20.0000 0.803868 0.401934 0.915669i $$-0.368338\pi$$
0.401934 + 0.915669i $$0.368338\pi$$
$$620$$ 0 0
$$621$$ 1.00000 0.0401286
$$622$$ − 28.0000i − 1.12270i
$$623$$ 70.0000i 2.80449i
$$624$$ −5.00000 −0.200160
$$625$$ 0 0
$$626$$ 0 0
$$627$$ − 3.00000i − 0.119808i
$$628$$ 4.00000i 0.159617i
$$629$$ −24.0000 −0.956943
$$630$$ 0 0
$$631$$ 29.0000 1.15447 0.577236 0.816577i $$-0.304131\pi$$
0.577236 + 0.816577i $$0.304131\pi$$
$$632$$ 5.00000i 0.198889i
$$633$$ 10.0000i 0.397464i
$$634$$ −9.00000 −0.357436
$$635$$ 0 0
$$636$$ 8.00000 0.317221
$$637$$ − 90.0000i − 3.56593i
$$638$$ 15.0000i 0.593856i
$$639$$ −10.0000 −0.395594
$$640$$ 0 0
$$641$$ −36.0000 −1.42191 −0.710957 0.703235i $$-0.751738\pi$$
−0.710957 + 0.703235i $$0.751738\pi$$
$$642$$ 12.0000i 0.473602i
$$643$$ 1.00000i 0.0394362i 0.999806 + 0.0197181i $$0.00627687\pi$$
−0.999806 + 0.0197181i $$0.993723\pi$$
$$644$$ −5.00000 −0.197028
$$645$$ 0 0
$$646$$ −6.00000 −0.236067
$$647$$ − 24.0000i − 0.943537i −0.881722 0.471769i $$-0.843616\pi$$
0.881722 0.471769i $$-0.156384\pi$$
$$648$$ 1.00000i 0.0392837i
$$649$$ −30.0000 −1.17760
$$650$$ 0 0
$$651$$ −40.0000 −1.56772
$$652$$ − 12.0000i − 0.469956i
$$653$$ 11.0000i 0.430463i 0.976563 + 0.215232i $$0.0690506\pi$$
−0.976563 + 0.215232i $$0.930949\pi$$
$$654$$ −12.0000 −0.469237
$$655$$ 0 0
$$656$$ −7.00000 −0.273304
$$657$$ − 15.0000i − 0.585206i
$$658$$ − 30.0000i − 1.16952i
$$659$$ −3.00000 −0.116863 −0.0584317 0.998291i $$-0.518610\pi$$
−0.0584317 + 0.998291i $$0.518610\pi$$
$$660$$ 0 0
$$661$$ 40.0000 1.55582 0.777910 0.628376i $$-0.216280\pi$$
0.777910 + 0.628376i $$0.216280\pi$$
$$662$$ − 4.00000i − 0.155464i
$$663$$ 30.0000i 1.16510i
$$664$$ 9.00000 0.349268
$$665$$ 0 0
$$666$$ 4.00000 0.154997
$$667$$ 5.00000i 0.193601i
$$668$$ − 12.0000i − 0.464294i
$$669$$ 14.0000 0.541271
$$670$$ 0 0
$$671$$ 36.0000 1.38976
$$672$$ − 5.00000i − 0.192879i
$$673$$ 45.0000i 1.73462i 0.497766 + 0.867311i $$0.334154\pi$$
−0.497766 + 0.867311i $$0.665846\pi$$
$$674$$ −8.00000 −0.308148
$$675$$ 0 0
$$676$$ 12.0000 0.461538
$$677$$ − 24.0000i − 0.922395i −0.887298 0.461197i $$-0.847420\pi$$
0.887298 0.461197i $$-0.152580\pi$$
$$678$$ − 10.0000i − 0.384048i
$$679$$ −80.0000 −3.07012
$$680$$ 0 0
$$681$$ 8.00000 0.306561
$$682$$ − 24.0000i − 0.919007i
$$683$$ − 10.0000i − 0.382639i −0.981528 0.191320i $$-0.938723\pi$$
0.981528 0.191320i $$-0.0612767\pi$$
$$684$$ 1.00000 0.0382360
$$685$$ 0 0
$$686$$ 55.0000 2.09991
$$687$$ − 16.0000i − 0.610438i
$$688$$ − 7.00000i − 0.266872i
$$689$$ −40.0000 −1.52388
$$690$$ 0 0
$$691$$ −10.0000 −0.380418 −0.190209 0.981744i $$-0.560917\pi$$
−0.190209 + 0.981744i $$0.560917\pi$$
$$692$$ − 21.0000i − 0.798300i
$$693$$ − 15.0000i − 0.569803i
$$694$$ 4.00000 0.151838
$$695$$ 0 0
$$696$$ −5.00000 −0.189525
$$697$$ 42.0000i 1.59086i
$$698$$ 1.00000i 0.0378506i
$$699$$ −7.00000 −0.264764
$$700$$ 0 0
$$701$$ 12.0000 0.453234 0.226617 0.973984i $$-0.427233\pi$$
0.226617 + 0.973984i $$0.427233\pi$$
$$702$$ − 5.00000i − 0.188713i
$$703$$ − 4.00000i − 0.150863i
$$704$$ 3.00000 0.113067
$$705$$ 0 0
$$706$$ 9.00000 0.338719
$$707$$ 50.0000i 1.88044i
$$708$$ − 10.0000i − 0.375823i
$$709$$ −12.0000 −0.450669 −0.225335 0.974281i $$-0.572348\pi$$
−0.225335 + 0.974281i $$0.572348\pi$$
$$710$$ 0 0
$$711$$ −5.00000 −0.187515
$$712$$ − 14.0000i − 0.524672i
$$713$$ − 8.00000i − 0.299602i
$$714$$ −30.0000 −1.12272
$$715$$ 0 0
$$716$$ 6.00000 0.224231
$$717$$ − 24.0000i − 0.896296i
$$718$$ − 7.00000i − 0.261238i
$$719$$ 22.0000 0.820462 0.410231 0.911982i $$-0.365448\pi$$
0.410231 + 0.911982i $$0.365448\pi$$
$$720$$ 0 0
$$721$$ 95.0000 3.53798
$$722$$ 18.0000i 0.669891i
$$723$$ − 18.0000i − 0.669427i
$$724$$ 14.0000 0.520306
$$725$$ 0 0
$$726$$ −2.00000 −0.0742270
$$727$$ − 32.0000i − 1.18681i −0.804902 0.593407i $$-0.797782\pi$$
0.804902 0.593407i $$-0.202218\pi$$
$$728$$ 25.0000i 0.926562i
$$729$$ −1.00000 −0.0370370
$$730$$ 0 0
$$731$$ −42.0000 −1.55343
$$732$$ 12.0000i 0.443533i
$$733$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$734$$ −1.00000 −0.0369107
$$735$$ 0 0
$$736$$ 1.00000 0.0368605
$$737$$ − 36.0000i − 1.32608i
$$738$$ − 7.00000i − 0.257674i
$$739$$ 24.0000 0.882854 0.441427 0.897297i $$-0.354472\pi$$
0.441427 + 0.897297i $$0.354472\pi$$
$$740$$ 0 0
$$741$$ −5.00000 −0.183680
$$742$$ − 40.0000i − 1.46845i
$$743$$ 11.0000i 0.403551i 0.979432 + 0.201775i $$0.0646711\pi$$
−0.979432 + 0.201775i $$0.935329\pi$$
$$744$$ 8.00000 0.293294
$$745$$ 0 0
$$746$$ −26.0000 −0.951928
$$747$$ 9.00000i 0.329293i
$$748$$ − 18.0000i − 0.658145i
$$749$$ 60.0000 2.19235
$$750$$ 0 0
$$751$$ −43.0000 −1.56909 −0.784546 0.620070i $$-0.787104\pi$$
−0.784546 + 0.620070i $$0.787104\pi$$
$$752$$ 6.00000i 0.218797i
$$753$$ 12.0000i 0.437304i
$$754$$ 25.0000 0.910446
$$755$$ 0 0
$$756$$ 5.00000 0.181848
$$757$$ − 28.0000i − 1.01768i −0.860862 0.508839i $$-0.830075\pi$$
0.860862 0.508839i $$-0.169925\pi$$
$$758$$ 12.0000i 0.435860i
$$759$$ 3.00000 0.108893
$$760$$ 0 0
$$761$$ 51.0000 1.84875 0.924374 0.381487i $$-0.124588\pi$$
0.924374 + 0.381487i $$0.124588\pi$$
$$762$$ − 6.00000i − 0.217357i
$$763$$ 60.0000i 2.17215i
$$764$$ 13.0000 0.470323
$$765$$ 0 0
$$766$$ −9.00000 −0.325183
$$767$$ 50.0000i 1.80540i
$$768$$ 1.00000i 0.0360844i
$$769$$ 2.00000 0.0721218 0.0360609 0.999350i $$-0.488519\pi$$
0.0360609 + 0.999350i $$0.488519\pi$$
$$770$$ 0 0
$$771$$ 6.00000 0.216085
$$772$$ − 6.00000i − 0.215945i
$$773$$ − 12.0000i − 0.431610i −0.976436 0.215805i $$-0.930762\pi$$
0.976436 0.215805i $$-0.0692376\pi$$
$$774$$ 7.00000 0.251610
$$775$$ 0 0
$$776$$ 16.0000 0.574367
$$777$$ − 20.0000i − 0.717496i
$$778$$ − 38.0000i − 1.36237i
$$779$$ −7.00000 −0.250801
$$780$$ 0 0
$$781$$ −30.0000 −1.07348
$$782$$ − 6.00000i − 0.214560i
$$783$$ − 5.00000i − 0.178685i
$$784$$ −18.0000 −0.642857
$$785$$ 0 0
$$786$$ −4.00000 −0.142675
$$787$$ 17.0000i 0.605985i 0.952993 + 0.302992i $$0.0979856\pi$$
−0.952993 + 0.302992i $$0.902014\pi$$
$$788$$ − 11.0000i − 0.391859i
$$789$$ 4.00000 0.142404
$$790$$ 0 0
$$791$$ −50.0000 −1.77780
$$792$$ 3.00000i 0.106600i
$$793$$ − 60.0000i − 2.13066i
$$794$$ 2.00000 0.0709773
$$795$$ 0 0
$$796$$ −1.00000 −0.0354441
$$797$$ − 42.0000i − 1.48772i −0.668338 0.743858i $$-0.732994\pi$$
0.668338 0.743858i $$-0.267006\pi$$
$$798$$ − 5.00000i − 0.176998i
$$799$$ 36.0000 1.27359
$$800$$ 0 0
$$801$$ 14.0000 0.494666
$$802$$ − 20.0000i − 0.706225i
$$803$$ − 45.0000i − 1.58802i
$$804$$ 12.0000 0.423207
$$805$$ 0 0
$$806$$ −40.0000 −1.40894
$$807$$ − 3.00000i − 0.105605i
$$808$$ − 10.0000i − 0.351799i
$$809$$ −19.0000 −0.668004 −0.334002 0.942572i $$-0.608399\pi$$
−0.334002 + 0.942572i $$0.608399\pi$$
$$810$$ 0 0
$$811$$ −30.0000 −1.05344 −0.526721 0.850038i $$-0.676579\pi$$
−0.526721 + 0.850038i $$0.676579\pi$$
$$812$$ 25.0000i 0.877328i
$$813$$ − 28.0000i − 0.982003i
$$814$$ 12.0000 0.420600
$$815$$ 0 0
$$816$$ 6.00000 0.210042
$$817$$ − 7.00000i − 0.244899i
$$818$$ 17.0000i 0.594391i
$$819$$ −25.0000 −0.873571
$$820$$ 0 0
$$821$$ 9.00000 0.314102 0.157051 0.987590i $$-0.449801\pi$$
0.157051 + 0.987590i $$0.449801\pi$$
$$822$$ 8.00000i 0.279032i
$$823$$ 50.0000i 1.74289i 0.490493 + 0.871445i $$0.336817\pi$$
−0.490493 + 0.871445i $$0.663183\pi$$
$$824$$ −19.0000 −0.661896
$$825$$ 0 0
$$826$$ −50.0000 −1.73972
$$827$$ 27.0000i 0.938882i 0.882964 + 0.469441i $$0.155545\pi$$
−0.882964 + 0.469441i $$0.844455\pi$$
$$828$$ 1.00000i 0.0347524i
$$829$$ 31.0000 1.07667 0.538337 0.842729i $$-0.319053\pi$$
0.538337 + 0.842729i $$0.319053\pi$$
$$830$$ 0 0
$$831$$ −1.00000 −0.0346896
$$832$$ − 5.00000i − 0.173344i
$$833$$ 108.000i 3.74198i
$$834$$ −2.00000 −0.0692543
$$835$$ 0 0
$$836$$ 3.00000 0.103757
$$837$$ 8.00000i 0.276520i
$$838$$ − 1.00000i − 0.0345444i
$$839$$ −7.00000 −0.241667 −0.120833 0.992673i $$-0.538557\pi$$
−0.120833 + 0.992673i $$0.538557\pi$$
$$840$$ 0 0
$$841$$ −4.00000 −0.137931
$$842$$ 18.0000i 0.620321i
$$843$$ 2.00000i 0.0688837i
$$844$$ −10.0000 −0.344214
$$845$$ 0 0
$$846$$ −6.00000 −0.206284
$$847$$ 10.0000i 0.343604i
$$848$$ 8.00000i 0.274721i
$$849$$ −20.0000 −0.686398
$$850$$ 0 0
$$851$$ 4.00000 0.137118
$$852$$ − 10.0000i − 0.342594i
$$853$$ 9.00000i 0.308154i 0.988059 + 0.154077i $$0.0492404\pi$$
−0.988059 + 0.154077i $$0.950760\pi$$
$$854$$ 60.0000 2.05316
$$855$$ 0 0
$$856$$ −12.0000 −0.410152
$$857$$ 22.0000i 0.751506i 0.926720 + 0.375753i $$0.122616\pi$$
−0.926720 + 0.375753i $$0.877384\pi$$
$$858$$ − 15.0000i − 0.512092i
$$859$$ 30.0000 1.02359 0.511793 0.859109i $$-0.328981\pi$$
0.511793 + 0.859109i $$0.328981\pi$$
$$860$$ 0 0
$$861$$ −35.0000 −1.19280
$$862$$ 0 0
$$863$$ 18.0000i 0.612727i 0.951915 + 0.306364i $$0.0991123\pi$$
−0.951915 + 0.306364i $$0.900888\pi$$
$$864$$ −1.00000 −0.0340207
$$865$$ 0 0
$$866$$ −28.0000 −0.951479
$$867$$ − 19.0000i − 0.645274i
$$868$$ − 40.0000i − 1.35769i
$$869$$ −15.0000 −0.508840
$$870$$ 0 0
$$871$$ −60.0000 −2.03302
$$872$$ − 12.0000i − 0.406371i
$$873$$ 16.0000i 0.541518i
$$874$$ 1.00000 0.0338255
$$875$$ 0 0
$$876$$ 15.0000 0.506803
$$877$$ − 42.0000i − 1.41824i −0.705088 0.709120i $$-0.749093\pi$$
0.705088 0.709120i $$-0.250907\pi$$
$$878$$ − 12.0000i − 0.404980i
$$879$$ 28.0000 0.944417
$$880$$ 0 0
$$881$$ −24.0000 −0.808581 −0.404290 0.914631i $$-0.632481\pi$$
−0.404290 + 0.914631i $$0.632481\pi$$
$$882$$ − 18.0000i − 0.606092i
$$883$$ 8.00000i 0.269221i 0.990899 + 0.134611i $$0.0429784\pi$$
−0.990899 + 0.134611i $$0.957022\pi$$
$$884$$ −30.0000 −1.00901
$$885$$ 0 0
$$886$$ −4.00000 −0.134383
$$887$$ − 22.0000i − 0.738688i −0.929293 0.369344i $$-0.879582\pi$$
0.929293 0.369344i $$-0.120418\pi$$
$$888$$ 4.00000i 0.134231i
$$889$$ −30.0000 −1.00617
$$890$$ 0 0
$$891$$ −3.00000 −0.100504
$$892$$ 14.0000i 0.468755i
$$893$$ 6.00000i 0.200782i
$$894$$ 20.0000 0.668900
$$895$$ 0 0
$$896$$ 5.00000 0.167038
$$897$$ − 5.00000i − 0.166945i
$$898$$ − 2.00000i − 0.0667409i
$$899$$ −40.0000 −1.33407
$$900$$ 0 0
$$901$$ 48.0000 1.59911
$$902$$ − 21.0000i − 0.699224i
$$903$$ − 35.0000i − 1.16473i
$$904$$ 10.0000 0.332595
$$905$$ 0 0
$$906$$ −10.0000 −0.332228
$$907$$ 37.0000i 1.22856i 0.789086 + 0.614282i $$0.210554\pi$$
−0.789086 + 0.614282i $$0.789446\pi$$
$$908$$ 8.00000i 0.265489i
$$909$$ 10.0000 0.331679
$$910$$ 0 0
$$911$$ −15.0000 −0.496972 −0.248486 0.968635i $$-0.579933\pi$$
−0.248486 + 0.968635i $$0.579933\pi$$
$$912$$ 1.00000i 0.0331133i
$$913$$ 27.0000i 0.893570i
$$914$$ 18.0000 0.595387
$$915$$ 0 0
$$916$$ 16.0000 0.528655
$$917$$ 20.0000i 0.660458i
$$918$$ 6.00000i 0.198030i
$$919$$ 12.0000 0.395843 0.197922 0.980218i $$-0.436581\pi$$
0.197922 + 0.980218i $$0.436581\pi$$
$$920$$ 0 0
$$921$$ 12.0000 0.395413
$$922$$ − 7.00000i − 0.230533i
$$923$$ 50.0000i 1.64577i
$$924$$ 15.0000 0.493464
$$925$$ 0 0
$$926$$ 4.00000 0.131448
$$927$$ − 19.0000i − 0.624042i
$$928$$ − 5.00000i − 0.164133i
$$929$$ 43.0000 1.41078 0.705392 0.708817i $$-0.250771\pi$$
0.705392 + 0.708817i $$0.250771\pi$$
$$930$$ 0 0
$$931$$ −18.0000 −0.589926
$$932$$ − 7.00000i − 0.229293i
$$933$$ 28.0000i 0.916679i
$$934$$ −21.0000 −0.687141
$$935$$ 0 0
$$936$$ 5.00000 0.163430
$$937$$ 20.0000i 0.653372i 0.945133 + 0.326686i $$0.105932\pi$$
−0.945133 + 0.326686i $$0.894068\pi$$
$$938$$ − 60.0000i − 1.95907i
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −48.0000 −1.56476 −0.782378 0.622804i $$-0.785993\pi$$
−0.782378 + 0.622804i $$0.785993\pi$$
$$942$$ − 4.00000i − 0.130327i
$$943$$ − 7.00000i − 0.227951i
$$944$$ 10.0000 0.325472
$$945$$ 0 0
$$946$$ 21.0000 0.682769
$$947$$ − 32.0000i − 1.03986i −0.854209 0.519930i $$-0.825958\pi$$
0.854209 0.519930i $$-0.174042\pi$$
$$948$$ − 5.00000i − 0.162392i
$$949$$ −75.0000 −2.43460
$$950$$ 0 0
$$951$$ 9.00000 0.291845
$$952$$ − 30.0000i − 0.972306i
$$953$$ − 12.0000i − 0.388718i −0.980930 0.194359i $$-0.937737\pi$$
0.980930 0.194359i $$-0.0622627\pi$$
$$954$$ −8.00000 −0.259010
$$955$$ 0 0
$$956$$ 24.0000 0.776215
$$957$$ − 15.0000i − 0.484881i
$$958$$ − 25.0000i − 0.807713i
$$959$$ 40.0000 1.29167
$$960$$ 0 0
$$961$$ 33.0000 1.06452
$$962$$ − 20.0000i − 0.644826i
$$963$$ − 12.0000i − 0.386695i
$$964$$ 18.0000 0.579741
$$965$$ 0 0
$$966$$ 5.00000 0.160872
$$967$$ 8.00000i 0.257263i 0.991692 + 0.128631i $$0.0410584\pi$$
−0.991692 + 0.128631i $$0.958942\pi$$
$$968$$ − 2.00000i − 0.0642824i
$$969$$ 6.00000 0.192748
$$970$$ 0 0
$$971$$ −49.0000 −1.57248 −0.786242 0.617918i $$-0.787976\pi$$
−0.786242 + 0.617918i $$0.787976\pi$$
$$972$$ − 1.00000i − 0.0320750i
$$973$$ 10.0000i 0.320585i
$$974$$ −2.00000 −0.0640841
$$975$$ 0 0
$$976$$ −12.0000 −0.384111
$$977$$ − 8.00000i − 0.255943i −0.991778 0.127971i $$-0.959153\pi$$
0.991778 0.127971i $$-0.0408466\pi$$
$$978$$ 12.0000i 0.383718i
$$979$$ 42.0000 1.34233
$$980$$ 0 0
$$981$$ 12.0000 0.383131
$$982$$ 32.0000i 1.02116i
$$983$$ − 31.0000i − 0.988746i −0.869250 0.494373i $$-0.835398\pi$$
0.869250 0.494373i $$-0.164602\pi$$
$$984$$ 7.00000 0.223152
$$985$$ 0 0
$$986$$ −30.0000 −0.955395
$$987$$ 30.0000i 0.954911i
$$988$$ − 5.00000i − 0.159071i
$$989$$ 7.00000 0.222587
$$990$$ 0 0
$$991$$ 4.00000 0.127064 0.0635321 0.997980i $$-0.479763\pi$$
0.0635321 + 0.997980i $$0.479763\pi$$
$$992$$ 8.00000i 0.254000i
$$993$$ 4.00000i 0.126936i
$$994$$ −50.0000 −1.58590
$$995$$ 0 0
$$996$$ −9.00000 −0.285176
$$997$$ 1.00000i 0.0316703i 0.999875 + 0.0158352i $$0.00504070\pi$$
−0.999875 + 0.0158352i $$0.994959\pi$$
$$998$$ − 6.00000i − 0.189927i
$$999$$ −4.00000 −0.126554
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 3450.2.d.l.2899.1 2
5.2 odd 4 3450.2.a.bb.1.1 yes 1
5.3 odd 4 3450.2.a.a.1.1 1
5.4 even 2 inner 3450.2.d.l.2899.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
3450.2.a.a.1.1 1 5.3 odd 4
3450.2.a.bb.1.1 yes 1 5.2 odd 4
3450.2.d.l.2899.1 2 1.1 even 1 trivial
3450.2.d.l.2899.2 2 5.4 even 2 inner