# Properties

 Label 1014.2.b.c.337.1 Level $1014$ Weight $2$ Character 1014.337 Analytic conductor $8.097$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1014 = 2 \cdot 3 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1014.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$8.09683076496$$ 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 78) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 337.1 Root $$1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 1014.337 Dual form 1014.2.b.c.337.2

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} +3.00000i q^{5} -1.00000i q^{6} -2.00000i q^{7} +1.00000i q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} +3.00000i q^{5} -1.00000i q^{6} -2.00000i q^{7} +1.00000i q^{8} +1.00000 q^{9} +3.00000 q^{10} -6.00000i q^{11} -1.00000 q^{12} -2.00000 q^{14} +3.00000i q^{15} +1.00000 q^{16} +3.00000 q^{17} -1.00000i q^{18} +2.00000i q^{19} -3.00000i q^{20} -2.00000i q^{21} -6.00000 q^{22} +6.00000 q^{23} +1.00000i q^{24} -4.00000 q^{25} +1.00000 q^{27} +2.00000i q^{28} +3.00000 q^{29} +3.00000 q^{30} -4.00000i q^{31} -1.00000i q^{32} -6.00000i q^{33} -3.00000i q^{34} +6.00000 q^{35} -1.00000 q^{36} +7.00000i q^{37} +2.00000 q^{38} -3.00000 q^{40} -3.00000i q^{41} -2.00000 q^{42} +10.0000 q^{43} +6.00000i q^{44} +3.00000i q^{45} -6.00000i q^{46} -6.00000i q^{47} +1.00000 q^{48} +3.00000 q^{49} +4.00000i q^{50} +3.00000 q^{51} +3.00000 q^{53} -1.00000i q^{54} +18.0000 q^{55} +2.00000 q^{56} +2.00000i q^{57} -3.00000i q^{58} -3.00000i q^{60} -7.00000 q^{61} -4.00000 q^{62} -2.00000i q^{63} -1.00000 q^{64} -6.00000 q^{66} -10.0000i q^{67} -3.00000 q^{68} +6.00000 q^{69} -6.00000i q^{70} +6.00000i q^{71} +1.00000i q^{72} +13.0000i q^{73} +7.00000 q^{74} -4.00000 q^{75} -2.00000i q^{76} -12.0000 q^{77} -4.00000 q^{79} +3.00000i q^{80} +1.00000 q^{81} -3.00000 q^{82} -6.00000i q^{83} +2.00000i q^{84} +9.00000i q^{85} -10.0000i q^{86} +3.00000 q^{87} +6.00000 q^{88} -18.0000i q^{89} +3.00000 q^{90} -6.00000 q^{92} -4.00000i q^{93} -6.00000 q^{94} -6.00000 q^{95} -1.00000i q^{96} +14.0000i q^{97} -3.00000i q^{98} -6.00000i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{3} - 2 q^{4} + 2 q^{9}+O(q^{10})$$ 2 * q + 2 * q^3 - 2 * q^4 + 2 * q^9 $$2 q + 2 q^{3} - 2 q^{4} + 2 q^{9} + 6 q^{10} - 2 q^{12} - 4 q^{14} + 2 q^{16} + 6 q^{17} - 12 q^{22} + 12 q^{23} - 8 q^{25} + 2 q^{27} + 6 q^{29} + 6 q^{30} + 12 q^{35} - 2 q^{36} + 4 q^{38} - 6 q^{40} - 4 q^{42} + 20 q^{43} + 2 q^{48} + 6 q^{49} + 6 q^{51} + 6 q^{53} + 36 q^{55} + 4 q^{56} - 14 q^{61} - 8 q^{62} - 2 q^{64} - 12 q^{66} - 6 q^{68} + 12 q^{69} + 14 q^{74} - 8 q^{75} - 24 q^{77} - 8 q^{79} + 2 q^{81} - 6 q^{82} + 6 q^{87} + 12 q^{88} + 6 q^{90} - 12 q^{92} - 12 q^{94} - 12 q^{95}+O(q^{100})$$ 2 * q + 2 * q^3 - 2 * q^4 + 2 * q^9 + 6 * q^10 - 2 * q^12 - 4 * q^14 + 2 * q^16 + 6 * q^17 - 12 * q^22 + 12 * q^23 - 8 * q^25 + 2 * q^27 + 6 * q^29 + 6 * q^30 + 12 * q^35 - 2 * q^36 + 4 * q^38 - 6 * q^40 - 4 * q^42 + 20 * q^43 + 2 * q^48 + 6 * q^49 + 6 * q^51 + 6 * q^53 + 36 * q^55 + 4 * q^56 - 14 * q^61 - 8 * q^62 - 2 * q^64 - 12 * q^66 - 6 * q^68 + 12 * q^69 + 14 * q^74 - 8 * q^75 - 24 * q^77 - 8 * q^79 + 2 * q^81 - 6 * q^82 + 6 * q^87 + 12 * q^88 + 6 * q^90 - 12 * q^92 - 12 * q^94 - 12 * q^95

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1014\mathbb{Z}\right)^\times$$.

 $$n$$ $$677$$ $$847$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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
$$3$$ 1.00000 0.577350
$$4$$ −1.00000 −0.500000
$$5$$ 3.00000i 1.34164i 0.741620 + 0.670820i $$0.234058\pi$$
−0.741620 + 0.670820i $$0.765942\pi$$
$$6$$ − 1.00000i − 0.408248i
$$7$$ − 2.00000i − 0.755929i −0.925820 0.377964i $$-0.876624\pi$$
0.925820 0.377964i $$-0.123376\pi$$
$$8$$ 1.00000i 0.353553i
$$9$$ 1.00000 0.333333
$$10$$ 3.00000 0.948683
$$11$$ − 6.00000i − 1.80907i −0.426401 0.904534i $$-0.640219\pi$$
0.426401 0.904534i $$-0.359781\pi$$
$$12$$ −1.00000 −0.288675
$$13$$ 0 0
$$14$$ −2.00000 −0.534522
$$15$$ 3.00000i 0.774597i
$$16$$ 1.00000 0.250000
$$17$$ 3.00000 0.727607 0.363803 0.931476i $$-0.381478\pi$$
0.363803 + 0.931476i $$0.381478\pi$$
$$18$$ − 1.00000i − 0.235702i
$$19$$ 2.00000i 0.458831i 0.973329 + 0.229416i $$0.0736815\pi$$
−0.973329 + 0.229416i $$0.926318\pi$$
$$20$$ − 3.00000i − 0.670820i
$$21$$ − 2.00000i − 0.436436i
$$22$$ −6.00000 −1.27920
$$23$$ 6.00000 1.25109 0.625543 0.780189i $$-0.284877\pi$$
0.625543 + 0.780189i $$0.284877\pi$$
$$24$$ 1.00000i 0.204124i
$$25$$ −4.00000 −0.800000
$$26$$ 0 0
$$27$$ 1.00000 0.192450
$$28$$ 2.00000i 0.377964i
$$29$$ 3.00000 0.557086 0.278543 0.960424i $$-0.410149\pi$$
0.278543 + 0.960424i $$0.410149\pi$$
$$30$$ 3.00000 0.547723
$$31$$ − 4.00000i − 0.718421i −0.933257 0.359211i $$-0.883046\pi$$
0.933257 0.359211i $$-0.116954\pi$$
$$32$$ − 1.00000i − 0.176777i
$$33$$ − 6.00000i − 1.04447i
$$34$$ − 3.00000i − 0.514496i
$$35$$ 6.00000 1.01419
$$36$$ −1.00000 −0.166667
$$37$$ 7.00000i 1.15079i 0.817875 + 0.575396i $$0.195152\pi$$
−0.817875 + 0.575396i $$0.804848\pi$$
$$38$$ 2.00000 0.324443
$$39$$ 0 0
$$40$$ −3.00000 −0.474342
$$41$$ − 3.00000i − 0.468521i −0.972174 0.234261i $$-0.924733\pi$$
0.972174 0.234261i $$-0.0752669\pi$$
$$42$$ −2.00000 −0.308607
$$43$$ 10.0000 1.52499 0.762493 0.646997i $$-0.223975\pi$$
0.762493 + 0.646997i $$0.223975\pi$$
$$44$$ 6.00000i 0.904534i
$$45$$ 3.00000i 0.447214i
$$46$$ − 6.00000i − 0.884652i
$$47$$ − 6.00000i − 0.875190i −0.899172 0.437595i $$-0.855830\pi$$
0.899172 0.437595i $$-0.144170\pi$$
$$48$$ 1.00000 0.144338
$$49$$ 3.00000 0.428571
$$50$$ 4.00000i 0.565685i
$$51$$ 3.00000 0.420084
$$52$$ 0 0
$$53$$ 3.00000 0.412082 0.206041 0.978543i $$-0.433942\pi$$
0.206041 + 0.978543i $$0.433942\pi$$
$$54$$ − 1.00000i − 0.136083i
$$55$$ 18.0000 2.42712
$$56$$ 2.00000 0.267261
$$57$$ 2.00000i 0.264906i
$$58$$ − 3.00000i − 0.393919i
$$59$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$60$$ − 3.00000i − 0.387298i
$$61$$ −7.00000 −0.896258 −0.448129 0.893969i $$-0.647910\pi$$
−0.448129 + 0.893969i $$0.647910\pi$$
$$62$$ −4.00000 −0.508001
$$63$$ − 2.00000i − 0.251976i
$$64$$ −1.00000 −0.125000
$$65$$ 0 0
$$66$$ −6.00000 −0.738549
$$67$$ − 10.0000i − 1.22169i −0.791748 0.610847i $$-0.790829\pi$$
0.791748 0.610847i $$-0.209171\pi$$
$$68$$ −3.00000 −0.363803
$$69$$ 6.00000 0.722315
$$70$$ − 6.00000i − 0.717137i
$$71$$ 6.00000i 0.712069i 0.934473 + 0.356034i $$0.115871\pi$$
−0.934473 + 0.356034i $$0.884129\pi$$
$$72$$ 1.00000i 0.117851i
$$73$$ 13.0000i 1.52153i 0.649025 + 0.760767i $$0.275177\pi$$
−0.649025 + 0.760767i $$0.724823\pi$$
$$74$$ 7.00000 0.813733
$$75$$ −4.00000 −0.461880
$$76$$ − 2.00000i − 0.229416i
$$77$$ −12.0000 −1.36753
$$78$$ 0 0
$$79$$ −4.00000 −0.450035 −0.225018 0.974355i $$-0.572244\pi$$
−0.225018 + 0.974355i $$0.572244\pi$$
$$80$$ 3.00000i 0.335410i
$$81$$ 1.00000 0.111111
$$82$$ −3.00000 −0.331295
$$83$$ − 6.00000i − 0.658586i −0.944228 0.329293i $$-0.893190\pi$$
0.944228 0.329293i $$-0.106810\pi$$
$$84$$ 2.00000i 0.218218i
$$85$$ 9.00000i 0.976187i
$$86$$ − 10.0000i − 1.07833i
$$87$$ 3.00000 0.321634
$$88$$ 6.00000 0.639602
$$89$$ − 18.0000i − 1.90800i −0.299813 0.953998i $$-0.596924\pi$$
0.299813 0.953998i $$-0.403076\pi$$
$$90$$ 3.00000 0.316228
$$91$$ 0 0
$$92$$ −6.00000 −0.625543
$$93$$ − 4.00000i − 0.414781i
$$94$$ −6.00000 −0.618853
$$95$$ −6.00000 −0.615587
$$96$$ − 1.00000i − 0.102062i
$$97$$ 14.0000i 1.42148i 0.703452 + 0.710742i $$0.251641\pi$$
−0.703452 + 0.710742i $$0.748359\pi$$
$$98$$ − 3.00000i − 0.303046i
$$99$$ − 6.00000i − 0.603023i
$$100$$ 4.00000 0.400000
$$101$$ −15.0000 −1.49256 −0.746278 0.665635i $$-0.768161\pi$$
−0.746278 + 0.665635i $$0.768161\pi$$
$$102$$ − 3.00000i − 0.297044i
$$103$$ −14.0000 −1.37946 −0.689730 0.724066i $$-0.742271\pi$$
−0.689730 + 0.724066i $$0.742271\pi$$
$$104$$ 0 0
$$105$$ 6.00000 0.585540
$$106$$ − 3.00000i − 0.291386i
$$107$$ −6.00000 −0.580042 −0.290021 0.957020i $$-0.593662\pi$$
−0.290021 + 0.957020i $$0.593662\pi$$
$$108$$ −1.00000 −0.0962250
$$109$$ 14.0000i 1.34096i 0.741929 + 0.670478i $$0.233911\pi$$
−0.741929 + 0.670478i $$0.766089\pi$$
$$110$$ − 18.0000i − 1.71623i
$$111$$ 7.00000i 0.664411i
$$112$$ − 2.00000i − 0.188982i
$$113$$ −3.00000 −0.282216 −0.141108 0.989994i $$-0.545067\pi$$
−0.141108 + 0.989994i $$0.545067\pi$$
$$114$$ 2.00000 0.187317
$$115$$ 18.0000i 1.67851i
$$116$$ −3.00000 −0.278543
$$117$$ 0 0
$$118$$ 0 0
$$119$$ − 6.00000i − 0.550019i
$$120$$ −3.00000 −0.273861
$$121$$ −25.0000 −2.27273
$$122$$ 7.00000i 0.633750i
$$123$$ − 3.00000i − 0.270501i
$$124$$ 4.00000i 0.359211i
$$125$$ 3.00000i 0.268328i
$$126$$ −2.00000 −0.178174
$$127$$ 4.00000 0.354943 0.177471 0.984126i $$-0.443208\pi$$
0.177471 + 0.984126i $$0.443208\pi$$
$$128$$ 1.00000i 0.0883883i
$$129$$ 10.0000 0.880451
$$130$$ 0 0
$$131$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$132$$ 6.00000i 0.522233i
$$133$$ 4.00000 0.346844
$$134$$ −10.0000 −0.863868
$$135$$ 3.00000i 0.258199i
$$136$$ 3.00000i 0.257248i
$$137$$ − 9.00000i − 0.768922i −0.923141 0.384461i $$-0.874387\pi$$
0.923141 0.384461i $$-0.125613\pi$$
$$138$$ − 6.00000i − 0.510754i
$$139$$ −4.00000 −0.339276 −0.169638 0.985506i $$-0.554260\pi$$
−0.169638 + 0.985506i $$0.554260\pi$$
$$140$$ −6.00000 −0.507093
$$141$$ − 6.00000i − 0.505291i
$$142$$ 6.00000 0.503509
$$143$$ 0 0
$$144$$ 1.00000 0.0833333
$$145$$ 9.00000i 0.747409i
$$146$$ 13.0000 1.07589
$$147$$ 3.00000 0.247436
$$148$$ − 7.00000i − 0.575396i
$$149$$ − 9.00000i − 0.737309i −0.929567 0.368654i $$-0.879819\pi$$
0.929567 0.368654i $$-0.120181\pi$$
$$150$$ 4.00000i 0.326599i
$$151$$ 10.0000i 0.813788i 0.913475 + 0.406894i $$0.133388\pi$$
−0.913475 + 0.406894i $$0.866612\pi$$
$$152$$ −2.00000 −0.162221
$$153$$ 3.00000 0.242536
$$154$$ 12.0000i 0.966988i
$$155$$ 12.0000 0.963863
$$156$$ 0 0
$$157$$ 5.00000 0.399043 0.199522 0.979893i $$-0.436061\pi$$
0.199522 + 0.979893i $$0.436061\pi$$
$$158$$ 4.00000i 0.318223i
$$159$$ 3.00000 0.237915
$$160$$ 3.00000 0.237171
$$161$$ − 12.0000i − 0.945732i
$$162$$ − 1.00000i − 0.0785674i
$$163$$ 4.00000i 0.313304i 0.987654 + 0.156652i $$0.0500701\pi$$
−0.987654 + 0.156652i $$0.949930\pi$$
$$164$$ 3.00000i 0.234261i
$$165$$ 18.0000 1.40130
$$166$$ −6.00000 −0.465690
$$167$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$168$$ 2.00000 0.154303
$$169$$ 0 0
$$170$$ 9.00000 0.690268
$$171$$ 2.00000i 0.152944i
$$172$$ −10.0000 −0.762493
$$173$$ 6.00000 0.456172 0.228086 0.973641i $$-0.426753\pi$$
0.228086 + 0.973641i $$0.426753\pi$$
$$174$$ − 3.00000i − 0.227429i
$$175$$ 8.00000i 0.604743i
$$176$$ − 6.00000i − 0.452267i
$$177$$ 0 0
$$178$$ −18.0000 −1.34916
$$179$$ −6.00000 −0.448461 −0.224231 0.974536i $$-0.571987\pi$$
−0.224231 + 0.974536i $$0.571987\pi$$
$$180$$ − 3.00000i − 0.223607i
$$181$$ 7.00000 0.520306 0.260153 0.965567i $$-0.416227\pi$$
0.260153 + 0.965567i $$0.416227\pi$$
$$182$$ 0 0
$$183$$ −7.00000 −0.517455
$$184$$ 6.00000i 0.442326i
$$185$$ −21.0000 −1.54395
$$186$$ −4.00000 −0.293294
$$187$$ − 18.0000i − 1.31629i
$$188$$ 6.00000i 0.437595i
$$189$$ − 2.00000i − 0.145479i
$$190$$ 6.00000i 0.435286i
$$191$$ −12.0000 −0.868290 −0.434145 0.900843i $$-0.642949\pi$$
−0.434145 + 0.900843i $$0.642949\pi$$
$$192$$ −1.00000 −0.0721688
$$193$$ − 23.0000i − 1.65558i −0.561041 0.827788i $$-0.689599\pi$$
0.561041 0.827788i $$-0.310401\pi$$
$$194$$ 14.0000 1.00514
$$195$$ 0 0
$$196$$ −3.00000 −0.214286
$$197$$ 6.00000i 0.427482i 0.976890 + 0.213741i $$0.0685649\pi$$
−0.976890 + 0.213741i $$0.931435\pi$$
$$198$$ −6.00000 −0.426401
$$199$$ 10.0000 0.708881 0.354441 0.935079i $$-0.384671\pi$$
0.354441 + 0.935079i $$0.384671\pi$$
$$200$$ − 4.00000i − 0.282843i
$$201$$ − 10.0000i − 0.705346i
$$202$$ 15.0000i 1.05540i
$$203$$ − 6.00000i − 0.421117i
$$204$$ −3.00000 −0.210042
$$205$$ 9.00000 0.628587
$$206$$ 14.0000i 0.975426i
$$207$$ 6.00000 0.417029
$$208$$ 0 0
$$209$$ 12.0000 0.830057
$$210$$ − 6.00000i − 0.414039i
$$211$$ −16.0000 −1.10149 −0.550743 0.834675i $$-0.685655\pi$$
−0.550743 + 0.834675i $$0.685655\pi$$
$$212$$ −3.00000 −0.206041
$$213$$ 6.00000i 0.411113i
$$214$$ 6.00000i 0.410152i
$$215$$ 30.0000i 2.04598i
$$216$$ 1.00000i 0.0680414i
$$217$$ −8.00000 −0.543075
$$218$$ 14.0000 0.948200
$$219$$ 13.0000i 0.878459i
$$220$$ −18.0000 −1.21356
$$221$$ 0 0
$$222$$ 7.00000 0.469809
$$223$$ 8.00000i 0.535720i 0.963458 + 0.267860i $$0.0863164\pi$$
−0.963458 + 0.267860i $$0.913684\pi$$
$$224$$ −2.00000 −0.133631
$$225$$ −4.00000 −0.266667
$$226$$ 3.00000i 0.199557i
$$227$$ 18.0000i 1.19470i 0.801980 + 0.597351i $$0.203780\pi$$
−0.801980 + 0.597351i $$0.796220\pi$$
$$228$$ − 2.00000i − 0.132453i
$$229$$ 22.0000i 1.45380i 0.686743 + 0.726900i $$0.259040\pi$$
−0.686743 + 0.726900i $$0.740960\pi$$
$$230$$ 18.0000 1.18688
$$231$$ −12.0000 −0.789542
$$232$$ 3.00000i 0.196960i
$$233$$ 6.00000 0.393073 0.196537 0.980497i $$-0.437031\pi$$
0.196537 + 0.980497i $$0.437031\pi$$
$$234$$ 0 0
$$235$$ 18.0000 1.17419
$$236$$ 0 0
$$237$$ −4.00000 −0.259828
$$238$$ −6.00000 −0.388922
$$239$$ 6.00000i 0.388108i 0.980991 + 0.194054i $$0.0621637\pi$$
−0.980991 + 0.194054i $$0.937836\pi$$
$$240$$ 3.00000i 0.193649i
$$241$$ 1.00000i 0.0644157i 0.999481 + 0.0322078i $$0.0102538\pi$$
−0.999481 + 0.0322078i $$0.989746\pi$$
$$242$$ 25.0000i 1.60706i
$$243$$ 1.00000 0.0641500
$$244$$ 7.00000 0.448129
$$245$$ 9.00000i 0.574989i
$$246$$ −3.00000 −0.191273
$$247$$ 0 0
$$248$$ 4.00000 0.254000
$$249$$ − 6.00000i − 0.380235i
$$250$$ 3.00000 0.189737
$$251$$ 12.0000 0.757433 0.378717 0.925513i $$-0.376365\pi$$
0.378717 + 0.925513i $$0.376365\pi$$
$$252$$ 2.00000i 0.125988i
$$253$$ − 36.0000i − 2.26330i
$$254$$ − 4.00000i − 0.250982i
$$255$$ 9.00000i 0.563602i
$$256$$ 1.00000 0.0625000
$$257$$ 3.00000 0.187135 0.0935674 0.995613i $$-0.470173\pi$$
0.0935674 + 0.995613i $$0.470173\pi$$
$$258$$ − 10.0000i − 0.622573i
$$259$$ 14.0000 0.869918
$$260$$ 0 0
$$261$$ 3.00000 0.185695
$$262$$ 0 0
$$263$$ −6.00000 −0.369976 −0.184988 0.982741i $$-0.559225\pi$$
−0.184988 + 0.982741i $$0.559225\pi$$
$$264$$ 6.00000 0.369274
$$265$$ 9.00000i 0.552866i
$$266$$ − 4.00000i − 0.245256i
$$267$$ − 18.0000i − 1.10158i
$$268$$ 10.0000i 0.610847i
$$269$$ 18.0000 1.09748 0.548740 0.835993i $$-0.315108\pi$$
0.548740 + 0.835993i $$0.315108\pi$$
$$270$$ 3.00000 0.182574
$$271$$ 16.0000i 0.971931i 0.873978 + 0.485965i $$0.161532\pi$$
−0.873978 + 0.485965i $$0.838468\pi$$
$$272$$ 3.00000 0.181902
$$273$$ 0 0
$$274$$ −9.00000 −0.543710
$$275$$ 24.0000i 1.44725i
$$276$$ −6.00000 −0.361158
$$277$$ −17.0000 −1.02143 −0.510716 0.859750i $$-0.670619\pi$$
−0.510716 + 0.859750i $$0.670619\pi$$
$$278$$ 4.00000i 0.239904i
$$279$$ − 4.00000i − 0.239474i
$$280$$ 6.00000i 0.358569i
$$281$$ − 9.00000i − 0.536895i −0.963294 0.268447i $$-0.913489\pi$$
0.963294 0.268447i $$-0.0865106\pi$$
$$282$$ −6.00000 −0.357295
$$283$$ −14.0000 −0.832214 −0.416107 0.909316i $$-0.636606\pi$$
−0.416107 + 0.909316i $$0.636606\pi$$
$$284$$ − 6.00000i − 0.356034i
$$285$$ −6.00000 −0.355409
$$286$$ 0 0
$$287$$ −6.00000 −0.354169
$$288$$ − 1.00000i − 0.0589256i
$$289$$ −8.00000 −0.470588
$$290$$ 9.00000 0.528498
$$291$$ 14.0000i 0.820695i
$$292$$ − 13.0000i − 0.760767i
$$293$$ 21.0000i 1.22683i 0.789760 + 0.613417i $$0.210205\pi$$
−0.789760 + 0.613417i $$0.789795\pi$$
$$294$$ − 3.00000i − 0.174964i
$$295$$ 0 0
$$296$$ −7.00000 −0.406867
$$297$$ − 6.00000i − 0.348155i
$$298$$ −9.00000 −0.521356
$$299$$ 0 0
$$300$$ 4.00000 0.230940
$$301$$ − 20.0000i − 1.15278i
$$302$$ 10.0000 0.575435
$$303$$ −15.0000 −0.861727
$$304$$ 2.00000i 0.114708i
$$305$$ − 21.0000i − 1.20246i
$$306$$ − 3.00000i − 0.171499i
$$307$$ 10.0000i 0.570730i 0.958419 + 0.285365i $$0.0921148\pi$$
−0.958419 + 0.285365i $$0.907885\pi$$
$$308$$ 12.0000 0.683763
$$309$$ −14.0000 −0.796432
$$310$$ − 12.0000i − 0.681554i
$$311$$ 30.0000 1.70114 0.850572 0.525859i $$-0.176256\pi$$
0.850572 + 0.525859i $$0.176256\pi$$
$$312$$ 0 0
$$313$$ −10.0000 −0.565233 −0.282617 0.959233i $$-0.591202\pi$$
−0.282617 + 0.959233i $$0.591202\pi$$
$$314$$ − 5.00000i − 0.282166i
$$315$$ 6.00000 0.338062
$$316$$ 4.00000 0.225018
$$317$$ 3.00000i 0.168497i 0.996445 + 0.0842484i $$0.0268489\pi$$
−0.996445 + 0.0842484i $$0.973151\pi$$
$$318$$ − 3.00000i − 0.168232i
$$319$$ − 18.0000i − 1.00781i
$$320$$ − 3.00000i − 0.167705i
$$321$$ −6.00000 −0.334887
$$322$$ −12.0000 −0.668734
$$323$$ 6.00000i 0.333849i
$$324$$ −1.00000 −0.0555556
$$325$$ 0 0
$$326$$ 4.00000 0.221540
$$327$$ 14.0000i 0.774202i
$$328$$ 3.00000 0.165647
$$329$$ −12.0000 −0.661581
$$330$$ − 18.0000i − 0.990867i
$$331$$ − 4.00000i − 0.219860i −0.993939 0.109930i $$-0.964937\pi$$
0.993939 0.109930i $$-0.0350627\pi$$
$$332$$ 6.00000i 0.329293i
$$333$$ 7.00000i 0.383598i
$$334$$ 0 0
$$335$$ 30.0000 1.63908
$$336$$ − 2.00000i − 0.109109i
$$337$$ −23.0000 −1.25289 −0.626445 0.779466i $$-0.715491\pi$$
−0.626445 + 0.779466i $$0.715491\pi$$
$$338$$ 0 0
$$339$$ −3.00000 −0.162938
$$340$$ − 9.00000i − 0.488094i
$$341$$ −24.0000 −1.29967
$$342$$ 2.00000 0.108148
$$343$$ − 20.0000i − 1.07990i
$$344$$ 10.0000i 0.539164i
$$345$$ 18.0000i 0.969087i
$$346$$ − 6.00000i − 0.322562i
$$347$$ −30.0000 −1.61048 −0.805242 0.592946i $$-0.797965\pi$$
−0.805242 + 0.592946i $$0.797965\pi$$
$$348$$ −3.00000 −0.160817
$$349$$ 10.0000i 0.535288i 0.963518 + 0.267644i $$0.0862451\pi$$
−0.963518 + 0.267644i $$0.913755\pi$$
$$350$$ 8.00000 0.427618
$$351$$ 0 0
$$352$$ −6.00000 −0.319801
$$353$$ − 15.0000i − 0.798369i −0.916871 0.399185i $$-0.869293\pi$$
0.916871 0.399185i $$-0.130707\pi$$
$$354$$ 0 0
$$355$$ −18.0000 −0.955341
$$356$$ 18.0000i 0.953998i
$$357$$ − 6.00000i − 0.317554i
$$358$$ 6.00000i 0.317110i
$$359$$ − 6.00000i − 0.316668i −0.987386 0.158334i $$-0.949388\pi$$
0.987386 0.158334i $$-0.0506123\pi$$
$$360$$ −3.00000 −0.158114
$$361$$ 15.0000 0.789474
$$362$$ − 7.00000i − 0.367912i
$$363$$ −25.0000 −1.31216
$$364$$ 0 0
$$365$$ −39.0000 −2.04135
$$366$$ 7.00000i 0.365896i
$$367$$ 2.00000 0.104399 0.0521996 0.998637i $$-0.483377\pi$$
0.0521996 + 0.998637i $$0.483377\pi$$
$$368$$ 6.00000 0.312772
$$369$$ − 3.00000i − 0.156174i
$$370$$ 21.0000i 1.09174i
$$371$$ − 6.00000i − 0.311504i
$$372$$ 4.00000i 0.207390i
$$373$$ 29.0000 1.50156 0.750782 0.660551i $$-0.229677\pi$$
0.750782 + 0.660551i $$0.229677\pi$$
$$374$$ −18.0000 −0.930758
$$375$$ 3.00000i 0.154919i
$$376$$ 6.00000 0.309426
$$377$$ 0 0
$$378$$ −2.00000 −0.102869
$$379$$ 20.0000i 1.02733i 0.857991 + 0.513665i $$0.171713\pi$$
−0.857991 + 0.513665i $$0.828287\pi$$
$$380$$ 6.00000 0.307794
$$381$$ 4.00000 0.204926
$$382$$ 12.0000i 0.613973i
$$383$$ 24.0000i 1.22634i 0.789950 + 0.613171i $$0.210106\pi$$
−0.789950 + 0.613171i $$0.789894\pi$$
$$384$$ 1.00000i 0.0510310i
$$385$$ − 36.0000i − 1.83473i
$$386$$ −23.0000 −1.17067
$$387$$ 10.0000 0.508329
$$388$$ − 14.0000i − 0.710742i
$$389$$ −39.0000 −1.97738 −0.988689 0.149979i $$-0.952080\pi$$
−0.988689 + 0.149979i $$0.952080\pi$$
$$390$$ 0 0
$$391$$ 18.0000 0.910299
$$392$$ 3.00000i 0.151523i
$$393$$ 0 0
$$394$$ 6.00000 0.302276
$$395$$ − 12.0000i − 0.603786i
$$396$$ 6.00000i 0.301511i
$$397$$ − 14.0000i − 0.702640i −0.936255 0.351320i $$-0.885733\pi$$
0.936255 0.351320i $$-0.114267\pi$$
$$398$$ − 10.0000i − 0.501255i
$$399$$ 4.00000 0.200250
$$400$$ −4.00000 −0.200000
$$401$$ 3.00000i 0.149813i 0.997191 + 0.0749064i $$0.0238658\pi$$
−0.997191 + 0.0749064i $$0.976134\pi$$
$$402$$ −10.0000 −0.498755
$$403$$ 0 0
$$404$$ 15.0000 0.746278
$$405$$ 3.00000i 0.149071i
$$406$$ −6.00000 −0.297775
$$407$$ 42.0000 2.08186
$$408$$ 3.00000i 0.148522i
$$409$$ − 1.00000i − 0.0494468i −0.999694 0.0247234i $$-0.992129\pi$$
0.999694 0.0247234i $$-0.00787051\pi$$
$$410$$ − 9.00000i − 0.444478i
$$411$$ − 9.00000i − 0.443937i
$$412$$ 14.0000 0.689730
$$413$$ 0 0
$$414$$ − 6.00000i − 0.294884i
$$415$$ 18.0000 0.883585
$$416$$ 0 0
$$417$$ −4.00000 −0.195881
$$418$$ − 12.0000i − 0.586939i
$$419$$ 24.0000 1.17248 0.586238 0.810139i $$-0.300608\pi$$
0.586238 + 0.810139i $$0.300608\pi$$
$$420$$ −6.00000 −0.292770
$$421$$ 29.0000i 1.41337i 0.707527 + 0.706687i $$0.249811\pi$$
−0.707527 + 0.706687i $$0.750189\pi$$
$$422$$ 16.0000i 0.778868i
$$423$$ − 6.00000i − 0.291730i
$$424$$ 3.00000i 0.145693i
$$425$$ −12.0000 −0.582086
$$426$$ 6.00000 0.290701
$$427$$ 14.0000i 0.677507i
$$428$$ 6.00000 0.290021
$$429$$ 0 0
$$430$$ 30.0000 1.44673
$$431$$ − 6.00000i − 0.289010i −0.989504 0.144505i $$-0.953841\pi$$
0.989504 0.144505i $$-0.0461589\pi$$
$$432$$ 1.00000 0.0481125
$$433$$ 13.0000 0.624740 0.312370 0.949960i $$-0.398877\pi$$
0.312370 + 0.949960i $$0.398877\pi$$
$$434$$ 8.00000i 0.384012i
$$435$$ 9.00000i 0.431517i
$$436$$ − 14.0000i − 0.670478i
$$437$$ 12.0000i 0.574038i
$$438$$ 13.0000 0.621164
$$439$$ −14.0000 −0.668184 −0.334092 0.942541i $$-0.608430\pi$$
−0.334092 + 0.942541i $$0.608430\pi$$
$$440$$ 18.0000i 0.858116i
$$441$$ 3.00000 0.142857
$$442$$ 0 0
$$443$$ −36.0000 −1.71041 −0.855206 0.518289i $$-0.826569\pi$$
−0.855206 + 0.518289i $$0.826569\pi$$
$$444$$ − 7.00000i − 0.332205i
$$445$$ 54.0000 2.55985
$$446$$ 8.00000 0.378811
$$447$$ − 9.00000i − 0.425685i
$$448$$ 2.00000i 0.0944911i
$$449$$ − 18.0000i − 0.849473i −0.905317 0.424736i $$-0.860367\pi$$
0.905317 0.424736i $$-0.139633\pi$$
$$450$$ 4.00000i 0.188562i
$$451$$ −18.0000 −0.847587
$$452$$ 3.00000 0.141108
$$453$$ 10.0000i 0.469841i
$$454$$ 18.0000 0.844782
$$455$$ 0 0
$$456$$ −2.00000 −0.0936586
$$457$$ 11.0000i 0.514558i 0.966337 + 0.257279i $$0.0828260\pi$$
−0.966337 + 0.257279i $$0.917174\pi$$
$$458$$ 22.0000 1.02799
$$459$$ 3.00000 0.140028
$$460$$ − 18.0000i − 0.839254i
$$461$$ 15.0000i 0.698620i 0.937007 + 0.349310i $$0.113584\pi$$
−0.937007 + 0.349310i $$0.886416\pi$$
$$462$$ 12.0000i 0.558291i
$$463$$ − 38.0000i − 1.76601i −0.469364 0.883005i $$-0.655517\pi$$
0.469364 0.883005i $$-0.344483\pi$$
$$464$$ 3.00000 0.139272
$$465$$ 12.0000 0.556487
$$466$$ − 6.00000i − 0.277945i
$$467$$ 18.0000 0.832941 0.416470 0.909149i $$-0.363267\pi$$
0.416470 + 0.909149i $$0.363267\pi$$
$$468$$ 0 0
$$469$$ −20.0000 −0.923514
$$470$$ − 18.0000i − 0.830278i
$$471$$ 5.00000 0.230388
$$472$$ 0 0
$$473$$ − 60.0000i − 2.75880i
$$474$$ 4.00000i 0.183726i
$$475$$ − 8.00000i − 0.367065i
$$476$$ 6.00000i 0.275010i
$$477$$ 3.00000 0.137361
$$478$$ 6.00000 0.274434
$$479$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$480$$ 3.00000 0.136931
$$481$$ 0 0
$$482$$ 1.00000 0.0455488
$$483$$ − 12.0000i − 0.546019i
$$484$$ 25.0000 1.13636
$$485$$ −42.0000 −1.90712
$$486$$ − 1.00000i − 0.0453609i
$$487$$ 2.00000i 0.0906287i 0.998973 + 0.0453143i $$0.0144289\pi$$
−0.998973 + 0.0453143i $$0.985571\pi$$
$$488$$ − 7.00000i − 0.316875i
$$489$$ 4.00000i 0.180886i
$$490$$ 9.00000 0.406579
$$491$$ 18.0000 0.812329 0.406164 0.913800i $$-0.366866\pi$$
0.406164 + 0.913800i $$0.366866\pi$$
$$492$$ 3.00000i 0.135250i
$$493$$ 9.00000 0.405340
$$494$$ 0 0
$$495$$ 18.0000 0.809040
$$496$$ − 4.00000i − 0.179605i
$$497$$ 12.0000 0.538274
$$498$$ −6.00000 −0.268866
$$499$$ 32.0000i 1.43252i 0.697835 + 0.716258i $$0.254147\pi$$
−0.697835 + 0.716258i $$0.745853\pi$$
$$500$$ − 3.00000i − 0.134164i
$$501$$ 0 0
$$502$$ − 12.0000i − 0.535586i
$$503$$ 6.00000 0.267527 0.133763 0.991013i $$-0.457294\pi$$
0.133763 + 0.991013i $$0.457294\pi$$
$$504$$ 2.00000 0.0890871
$$505$$ − 45.0000i − 2.00247i
$$506$$ −36.0000 −1.60040
$$507$$ 0 0
$$508$$ −4.00000 −0.177471
$$509$$ 3.00000i 0.132973i 0.997787 + 0.0664863i $$0.0211789\pi$$
−0.997787 + 0.0664863i $$0.978821\pi$$
$$510$$ 9.00000 0.398527
$$511$$ 26.0000 1.15017
$$512$$ − 1.00000i − 0.0441942i
$$513$$ 2.00000i 0.0883022i
$$514$$ − 3.00000i − 0.132324i
$$515$$ − 42.0000i − 1.85074i
$$516$$ −10.0000 −0.440225
$$517$$ −36.0000 −1.58328
$$518$$ − 14.0000i − 0.615125i
$$519$$ 6.00000 0.263371
$$520$$ 0 0
$$521$$ 33.0000 1.44576 0.722878 0.690976i $$-0.242819\pi$$
0.722878 + 0.690976i $$0.242819\pi$$
$$522$$ − 3.00000i − 0.131306i
$$523$$ −34.0000 −1.48672 −0.743358 0.668894i $$-0.766768\pi$$
−0.743358 + 0.668894i $$0.766768\pi$$
$$524$$ 0 0
$$525$$ 8.00000i 0.349149i
$$526$$ 6.00000i 0.261612i
$$527$$ − 12.0000i − 0.522728i
$$528$$ − 6.00000i − 0.261116i
$$529$$ 13.0000 0.565217
$$530$$ 9.00000 0.390935
$$531$$ 0 0
$$532$$ −4.00000 −0.173422
$$533$$ 0 0
$$534$$ −18.0000 −0.778936
$$535$$ − 18.0000i − 0.778208i
$$536$$ 10.0000 0.431934
$$537$$ −6.00000 −0.258919
$$538$$ − 18.0000i − 0.776035i
$$539$$ − 18.0000i − 0.775315i
$$540$$ − 3.00000i − 0.129099i
$$541$$ − 29.0000i − 1.24681i −0.781900 0.623404i $$-0.785749\pi$$
0.781900 0.623404i $$-0.214251\pi$$
$$542$$ 16.0000 0.687259
$$543$$ 7.00000 0.300399
$$544$$ − 3.00000i − 0.128624i
$$545$$ −42.0000 −1.79908
$$546$$ 0 0
$$547$$ −34.0000 −1.45374 −0.726868 0.686778i $$-0.759025\pi$$
−0.726868 + 0.686778i $$0.759025\pi$$
$$548$$ 9.00000i 0.384461i
$$549$$ −7.00000 −0.298753
$$550$$ 24.0000 1.02336
$$551$$ 6.00000i 0.255609i
$$552$$ 6.00000i 0.255377i
$$553$$ 8.00000i 0.340195i
$$554$$ 17.0000i 0.722261i
$$555$$ −21.0000 −0.891400
$$556$$ 4.00000 0.169638
$$557$$ − 3.00000i − 0.127114i −0.997978 0.0635570i $$-0.979756\pi$$
0.997978 0.0635570i $$-0.0202445\pi$$
$$558$$ −4.00000 −0.169334
$$559$$ 0 0
$$560$$ 6.00000 0.253546
$$561$$ − 18.0000i − 0.759961i
$$562$$ −9.00000 −0.379642
$$563$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$564$$ 6.00000i 0.252646i
$$565$$ − 9.00000i − 0.378633i
$$566$$ 14.0000i 0.588464i
$$567$$ − 2.00000i − 0.0839921i
$$568$$ −6.00000 −0.251754
$$569$$ −6.00000 −0.251533 −0.125767 0.992060i $$-0.540139\pi$$
−0.125767 + 0.992060i $$0.540139\pi$$
$$570$$ 6.00000i 0.251312i
$$571$$ 22.0000 0.920671 0.460336 0.887745i $$-0.347729\pi$$
0.460336 + 0.887745i $$0.347729\pi$$
$$572$$ 0 0
$$573$$ −12.0000 −0.501307
$$574$$ 6.00000i 0.250435i
$$575$$ −24.0000 −1.00087
$$576$$ −1.00000 −0.0416667
$$577$$ 11.0000i 0.457936i 0.973434 + 0.228968i $$0.0735351\pi$$
−0.973434 + 0.228968i $$0.926465\pi$$
$$578$$ 8.00000i 0.332756i
$$579$$ − 23.0000i − 0.955847i
$$580$$ − 9.00000i − 0.373705i
$$581$$ −12.0000 −0.497844
$$582$$ 14.0000 0.580319
$$583$$ − 18.0000i − 0.745484i
$$584$$ −13.0000 −0.537944
$$585$$ 0 0
$$586$$ 21.0000 0.867502
$$587$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$588$$ −3.00000 −0.123718
$$589$$ 8.00000 0.329634
$$590$$ 0 0
$$591$$ 6.00000i 0.246807i
$$592$$ 7.00000i 0.287698i
$$593$$ − 9.00000i − 0.369586i −0.982777 0.184793i $$-0.940839\pi$$
0.982777 0.184793i $$-0.0591614\pi$$
$$594$$ −6.00000 −0.246183
$$595$$ 18.0000 0.737928
$$596$$ 9.00000i 0.368654i
$$597$$ 10.0000 0.409273
$$598$$ 0 0
$$599$$ 24.0000 0.980613 0.490307 0.871550i $$-0.336885\pi$$
0.490307 + 0.871550i $$0.336885\pi$$
$$600$$ − 4.00000i − 0.163299i
$$601$$ −37.0000 −1.50926 −0.754631 0.656150i $$-0.772184\pi$$
−0.754631 + 0.656150i $$0.772184\pi$$
$$602$$ −20.0000 −0.815139
$$603$$ − 10.0000i − 0.407231i
$$604$$ − 10.0000i − 0.406894i
$$605$$ − 75.0000i − 3.04918i
$$606$$ 15.0000i 0.609333i
$$607$$ 32.0000 1.29884 0.649420 0.760430i $$-0.275012\pi$$
0.649420 + 0.760430i $$0.275012\pi$$
$$608$$ 2.00000 0.0811107
$$609$$ − 6.00000i − 0.243132i
$$610$$ −21.0000 −0.850265
$$611$$ 0 0
$$612$$ −3.00000 −0.121268
$$613$$ − 31.0000i − 1.25208i −0.779792 0.626039i $$-0.784675\pi$$
0.779792 0.626039i $$-0.215325\pi$$
$$614$$ 10.0000 0.403567
$$615$$ 9.00000 0.362915
$$616$$ − 12.0000i − 0.483494i
$$617$$ − 15.0000i − 0.603877i −0.953327 0.301939i $$-0.902366\pi$$
0.953327 0.301939i $$-0.0976338\pi$$
$$618$$ 14.0000i 0.563163i
$$619$$ − 8.00000i − 0.321547i −0.986991 0.160774i $$-0.948601\pi$$
0.986991 0.160774i $$-0.0513989\pi$$
$$620$$ −12.0000 −0.481932
$$621$$ 6.00000 0.240772
$$622$$ − 30.0000i − 1.20289i
$$623$$ −36.0000 −1.44231
$$624$$ 0 0
$$625$$ −29.0000 −1.16000
$$626$$ 10.0000i 0.399680i
$$627$$ 12.0000 0.479234
$$628$$ −5.00000 −0.199522
$$629$$ 21.0000i 0.837325i
$$630$$ − 6.00000i − 0.239046i
$$631$$ − 20.0000i − 0.796187i −0.917345 0.398094i $$-0.869672\pi$$
0.917345 0.398094i $$-0.130328\pi$$
$$632$$ − 4.00000i − 0.159111i
$$633$$ −16.0000 −0.635943
$$634$$ 3.00000 0.119145
$$635$$ 12.0000i 0.476205i
$$636$$ −3.00000 −0.118958
$$637$$ 0 0
$$638$$ −18.0000 −0.712627
$$639$$ 6.00000i 0.237356i
$$640$$ −3.00000 −0.118585
$$641$$ 3.00000 0.118493 0.0592464 0.998243i $$-0.481130\pi$$
0.0592464 + 0.998243i $$0.481130\pi$$
$$642$$ 6.00000i 0.236801i
$$643$$ − 16.0000i − 0.630978i −0.948929 0.315489i $$-0.897831\pi$$
0.948929 0.315489i $$-0.102169\pi$$
$$644$$ 12.0000i 0.472866i
$$645$$ 30.0000i 1.18125i
$$646$$ 6.00000 0.236067
$$647$$ −24.0000 −0.943537 −0.471769 0.881722i $$-0.656384\pi$$
−0.471769 + 0.881722i $$0.656384\pi$$
$$648$$ 1.00000i 0.0392837i
$$649$$ 0 0
$$650$$ 0 0
$$651$$ −8.00000 −0.313545
$$652$$ − 4.00000i − 0.156652i
$$653$$ 42.0000 1.64359 0.821794 0.569785i $$-0.192974\pi$$
0.821794 + 0.569785i $$0.192974\pi$$
$$654$$ 14.0000 0.547443
$$655$$ 0 0
$$656$$ − 3.00000i − 0.117130i
$$657$$ 13.0000i 0.507178i
$$658$$ 12.0000i 0.467809i
$$659$$ 24.0000 0.934907 0.467454 0.884018i $$-0.345171\pi$$
0.467454 + 0.884018i $$0.345171\pi$$
$$660$$ −18.0000 −0.700649
$$661$$ − 5.00000i − 0.194477i −0.995261 0.0972387i $$-0.968999\pi$$
0.995261 0.0972387i $$-0.0310010\pi$$
$$662$$ −4.00000 −0.155464
$$663$$ 0 0
$$664$$ 6.00000 0.232845
$$665$$ 12.0000i 0.465340i
$$666$$ 7.00000 0.271244
$$667$$ 18.0000 0.696963
$$668$$ 0 0
$$669$$ 8.00000i 0.309298i
$$670$$ − 30.0000i − 1.15900i
$$671$$ 42.0000i 1.62139i
$$672$$ −2.00000 −0.0771517
$$673$$ 13.0000 0.501113 0.250557 0.968102i $$-0.419386\pi$$
0.250557 + 0.968102i $$0.419386\pi$$
$$674$$ 23.0000i 0.885927i
$$675$$ −4.00000 −0.153960
$$676$$ 0 0
$$677$$ 18.0000 0.691796 0.345898 0.938272i $$-0.387574\pi$$
0.345898 + 0.938272i $$0.387574\pi$$
$$678$$ 3.00000i 0.115214i
$$679$$ 28.0000 1.07454
$$680$$ −9.00000 −0.345134
$$681$$ 18.0000i 0.689761i
$$682$$ 24.0000i 0.919007i
$$683$$ 48.0000i 1.83667i 0.395805 + 0.918334i $$0.370466\pi$$
−0.395805 + 0.918334i $$0.629534\pi$$
$$684$$ − 2.00000i − 0.0764719i
$$685$$ 27.0000 1.03162
$$686$$ −20.0000 −0.763604
$$687$$ 22.0000i 0.839352i
$$688$$ 10.0000 0.381246
$$689$$ 0 0
$$690$$ 18.0000 0.685248
$$691$$ 26.0000i 0.989087i 0.869153 + 0.494543i $$0.164665\pi$$
−0.869153 + 0.494543i $$0.835335\pi$$
$$692$$ −6.00000 −0.228086
$$693$$ −12.0000 −0.455842
$$694$$ 30.0000i 1.13878i
$$695$$ − 12.0000i − 0.455186i
$$696$$ 3.00000i 0.113715i
$$697$$ − 9.00000i − 0.340899i
$$698$$ 10.0000 0.378506
$$699$$ 6.00000 0.226941
$$700$$ − 8.00000i − 0.302372i
$$701$$ 30.0000 1.13308 0.566542 0.824033i $$-0.308281\pi$$
0.566542 + 0.824033i $$0.308281\pi$$
$$702$$ 0 0
$$703$$ −14.0000 −0.528020
$$704$$ 6.00000i 0.226134i
$$705$$ 18.0000 0.677919
$$706$$ −15.0000 −0.564532
$$707$$ 30.0000i 1.12827i
$$708$$ 0 0
$$709$$ − 5.00000i − 0.187779i −0.995583 0.0938895i $$-0.970070\pi$$
0.995583 0.0938895i $$-0.0299300\pi$$
$$710$$ 18.0000i 0.675528i
$$711$$ −4.00000 −0.150012
$$712$$ 18.0000 0.674579
$$713$$ − 24.0000i − 0.898807i
$$714$$ −6.00000 −0.224544
$$715$$ 0 0
$$716$$ 6.00000 0.224231
$$717$$ 6.00000i 0.224074i
$$718$$ −6.00000 −0.223918
$$719$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$720$$ 3.00000i 0.111803i
$$721$$ 28.0000i 1.04277i
$$722$$ − 15.0000i − 0.558242i
$$723$$ 1.00000i 0.0371904i
$$724$$ −7.00000 −0.260153
$$725$$ −12.0000 −0.445669
$$726$$ 25.0000i 0.927837i
$$727$$ −14.0000 −0.519231 −0.259616 0.965712i $$-0.583596\pi$$
−0.259616 + 0.965712i $$0.583596\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 39.0000i 1.44345i
$$731$$ 30.0000 1.10959
$$732$$ 7.00000 0.258727
$$733$$ − 31.0000i − 1.14501i −0.819901 0.572506i $$-0.805971\pi$$
0.819901 0.572506i $$-0.194029\pi$$
$$734$$ − 2.00000i − 0.0738213i
$$735$$ 9.00000i 0.331970i
$$736$$ − 6.00000i − 0.221163i
$$737$$ −60.0000 −2.21013
$$738$$ −3.00000 −0.110432
$$739$$ 16.0000i 0.588570i 0.955718 + 0.294285i $$0.0950814\pi$$
−0.955718 + 0.294285i $$0.904919\pi$$
$$740$$ 21.0000 0.771975
$$741$$ 0 0
$$742$$ −6.00000 −0.220267
$$743$$ − 36.0000i − 1.32071i −0.750953 0.660356i $$-0.770405\pi$$
0.750953 0.660356i $$-0.229595\pi$$
$$744$$ 4.00000 0.146647
$$745$$ 27.0000 0.989203
$$746$$ − 29.0000i − 1.06177i
$$747$$ − 6.00000i − 0.219529i
$$748$$ 18.0000i 0.658145i
$$749$$ 12.0000i 0.438470i
$$750$$ 3.00000 0.109545
$$751$$ −14.0000 −0.510867 −0.255434 0.966827i $$-0.582218\pi$$
−0.255434 + 0.966827i $$0.582218\pi$$
$$752$$ − 6.00000i − 0.218797i
$$753$$ 12.0000 0.437304
$$754$$ 0 0
$$755$$ −30.0000 −1.09181
$$756$$ 2.00000i 0.0727393i
$$757$$ −34.0000 −1.23575 −0.617876 0.786276i $$-0.712006\pi$$
−0.617876 + 0.786276i $$0.712006\pi$$
$$758$$ 20.0000 0.726433
$$759$$ − 36.0000i − 1.30672i
$$760$$ − 6.00000i − 0.217643i
$$761$$ 30.0000i 1.08750i 0.839248 + 0.543750i $$0.182996\pi$$
−0.839248 + 0.543750i $$0.817004\pi$$
$$762$$ − 4.00000i − 0.144905i
$$763$$ 28.0000 1.01367
$$764$$ 12.0000 0.434145
$$765$$ 9.00000i 0.325396i
$$766$$ 24.0000 0.867155
$$767$$ 0 0
$$768$$ 1.00000 0.0360844
$$769$$ 14.0000i 0.504853i 0.967616 + 0.252426i $$0.0812286\pi$$
−0.967616 + 0.252426i $$0.918771\pi$$
$$770$$ −36.0000 −1.29735
$$771$$ 3.00000 0.108042
$$772$$ 23.0000i 0.827788i
$$773$$ − 30.0000i − 1.07903i −0.841978 0.539513i $$-0.818609\pi$$
0.841978 0.539513i $$-0.181391\pi$$
$$774$$ − 10.0000i − 0.359443i
$$775$$ 16.0000i 0.574737i
$$776$$ −14.0000 −0.502571
$$777$$ 14.0000 0.502247
$$778$$ 39.0000i 1.39822i
$$779$$ 6.00000 0.214972
$$780$$ 0 0
$$781$$ 36.0000 1.28818
$$782$$ − 18.0000i − 0.643679i
$$783$$ 3.00000 0.107211
$$784$$ 3.00000 0.107143
$$785$$ 15.0000i 0.535373i
$$786$$ 0 0
$$787$$ 28.0000i 0.998092i 0.866575 + 0.499046i $$0.166316\pi$$
−0.866575 + 0.499046i $$0.833684\pi$$
$$788$$ − 6.00000i − 0.213741i
$$789$$ −6.00000 −0.213606
$$790$$ −12.0000 −0.426941
$$791$$ 6.00000i 0.213335i
$$792$$ 6.00000 0.213201
$$793$$ 0 0
$$794$$ −14.0000 −0.496841
$$795$$ 9.00000i 0.319197i
$$796$$ −10.0000 −0.354441
$$797$$ −30.0000 −1.06265 −0.531327 0.847167i $$-0.678307\pi$$
−0.531327 + 0.847167i $$0.678307\pi$$
$$798$$ − 4.00000i − 0.141598i
$$799$$ − 18.0000i − 0.636794i
$$800$$ 4.00000i 0.141421i
$$801$$ − 18.0000i − 0.635999i
$$802$$ 3.00000 0.105934
$$803$$ 78.0000 2.75256
$$804$$ 10.0000i 0.352673i
$$805$$ 36.0000 1.26883
$$806$$ 0 0
$$807$$ 18.0000 0.633630
$$808$$ − 15.0000i − 0.527698i
$$809$$ −51.0000 −1.79306 −0.896532 0.442978i $$-0.853922\pi$$
−0.896532 + 0.442978i $$0.853922\pi$$
$$810$$ 3.00000 0.105409
$$811$$ − 4.00000i − 0.140459i −0.997531 0.0702295i $$-0.977627\pi$$
0.997531 0.0702295i $$-0.0223732\pi$$
$$812$$ 6.00000i 0.210559i
$$813$$ 16.0000i 0.561144i
$$814$$ − 42.0000i − 1.47210i
$$815$$ −12.0000 −0.420342
$$816$$ 3.00000 0.105021
$$817$$ 20.0000i 0.699711i
$$818$$ −1.00000 −0.0349642
$$819$$ 0 0
$$820$$ −9.00000 −0.314294
$$821$$ 18.0000i 0.628204i 0.949389 + 0.314102i $$0.101703\pi$$
−0.949389 + 0.314102i $$0.898297\pi$$
$$822$$ −9.00000 −0.313911
$$823$$ 40.0000 1.39431 0.697156 0.716919i $$-0.254448\pi$$
0.697156 + 0.716919i $$0.254448\pi$$
$$824$$ − 14.0000i − 0.487713i
$$825$$ 24.0000i 0.835573i
$$826$$ 0 0
$$827$$ 48.0000i 1.66912i 0.550914 + 0.834562i $$0.314279\pi$$
−0.550914 + 0.834562i $$0.685721\pi$$
$$828$$ −6.00000 −0.208514
$$829$$ −17.0000 −0.590434 −0.295217 0.955430i $$-0.595392\pi$$
−0.295217 + 0.955430i $$0.595392\pi$$
$$830$$ − 18.0000i − 0.624789i
$$831$$ −17.0000 −0.589723
$$832$$ 0 0
$$833$$ 9.00000 0.311832
$$834$$ 4.00000i 0.138509i
$$835$$ 0 0
$$836$$ −12.0000 −0.415029
$$837$$ − 4.00000i − 0.138260i
$$838$$ − 24.0000i − 0.829066i
$$839$$ 12.0000i 0.414286i 0.978311 + 0.207143i $$0.0664165\pi$$
−0.978311 + 0.207143i $$0.933583\pi$$
$$840$$ 6.00000i 0.207020i
$$841$$ −20.0000 −0.689655
$$842$$ 29.0000 0.999406
$$843$$ − 9.00000i − 0.309976i
$$844$$ 16.0000 0.550743
$$845$$ 0 0
$$846$$ −6.00000 −0.206284
$$847$$ 50.0000i 1.71802i
$$848$$ 3.00000 0.103020
$$849$$ −14.0000 −0.480479
$$850$$ 12.0000i 0.411597i
$$851$$ 42.0000i 1.43974i
$$852$$ − 6.00000i − 0.205557i
$$853$$ 19.0000i 0.650548i 0.945620 + 0.325274i $$0.105456\pi$$
−0.945620 + 0.325274i $$0.894544\pi$$
$$854$$ 14.0000 0.479070
$$855$$ −6.00000 −0.205196
$$856$$ − 6.00000i − 0.205076i
$$857$$ −21.0000 −0.717346 −0.358673 0.933463i $$-0.616771\pi$$
−0.358673 + 0.933463i $$0.616771\pi$$
$$858$$ 0 0
$$859$$ 26.0000 0.887109 0.443554 0.896248i $$-0.353717\pi$$
0.443554 + 0.896248i $$0.353717\pi$$
$$860$$ − 30.0000i − 1.02299i
$$861$$ −6.00000 −0.204479
$$862$$ −6.00000 −0.204361
$$863$$ 18.0000i 0.612727i 0.951915 + 0.306364i $$0.0991123\pi$$
−0.951915 + 0.306364i $$0.900888\pi$$
$$864$$ − 1.00000i − 0.0340207i
$$865$$ 18.0000i 0.612018i
$$866$$ − 13.0000i − 0.441758i
$$867$$ −8.00000 −0.271694
$$868$$ 8.00000 0.271538
$$869$$ 24.0000i 0.814144i
$$870$$ 9.00000 0.305129
$$871$$ 0 0
$$872$$ −14.0000 −0.474100
$$873$$ 14.0000i 0.473828i
$$874$$ 12.0000 0.405906
$$875$$ 6.00000 0.202837
$$876$$ − 13.0000i − 0.439229i
$$877$$ 41.0000i 1.38447i 0.721671 + 0.692236i $$0.243374\pi$$
−0.721671 + 0.692236i $$0.756626\pi$$
$$878$$ 14.0000i 0.472477i
$$879$$ 21.0000i 0.708312i
$$880$$ 18.0000 0.606780
$$881$$ −33.0000 −1.11180 −0.555899 0.831250i $$-0.687626\pi$$
−0.555899 + 0.831250i $$0.687626\pi$$
$$882$$ − 3.00000i − 0.101015i
$$883$$ −8.00000 −0.269221 −0.134611 0.990899i $$-0.542978\pi$$
−0.134611 + 0.990899i $$0.542978\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 36.0000i 1.20944i
$$887$$ −48.0000 −1.61168 −0.805841 0.592132i $$-0.798286\pi$$
−0.805841 + 0.592132i $$0.798286\pi$$
$$888$$ −7.00000 −0.234905
$$889$$ − 8.00000i − 0.268311i
$$890$$ − 54.0000i − 1.81008i
$$891$$ − 6.00000i − 0.201008i
$$892$$ − 8.00000i − 0.267860i
$$893$$ 12.0000 0.401565
$$894$$ −9.00000 −0.301005
$$895$$ − 18.0000i − 0.601674i
$$896$$ 2.00000 0.0668153
$$897$$ 0 0
$$898$$ −18.0000 −0.600668
$$899$$ − 12.0000i − 0.400222i
$$900$$ 4.00000 0.133333
$$901$$ 9.00000 0.299833
$$902$$ 18.0000i 0.599334i
$$903$$ − 20.0000i − 0.665558i
$$904$$ − 3.00000i − 0.0997785i
$$905$$ 21.0000i 0.698064i
$$906$$ 10.0000 0.332228
$$907$$ −44.0000 −1.46100 −0.730498 0.682915i $$-0.760712\pi$$
−0.730498 + 0.682915i $$0.760712\pi$$
$$908$$ − 18.0000i − 0.597351i
$$909$$ −15.0000 −0.497519
$$910$$ 0 0
$$911$$ −24.0000 −0.795155 −0.397578 0.917568i $$-0.630149\pi$$
−0.397578 + 0.917568i $$0.630149\pi$$
$$912$$ 2.00000i 0.0662266i
$$913$$ −36.0000 −1.19143
$$914$$ 11.0000 0.363848
$$915$$ − 21.0000i − 0.694239i
$$916$$ − 22.0000i − 0.726900i
$$917$$ 0 0
$$918$$ − 3.00000i − 0.0990148i
$$919$$ −16.0000 −0.527791 −0.263896 0.964551i $$-0.585007\pi$$
−0.263896 + 0.964551i $$0.585007\pi$$
$$920$$ −18.0000 −0.593442
$$921$$ 10.0000i 0.329511i
$$922$$ 15.0000 0.493999
$$923$$ 0 0
$$924$$ 12.0000 0.394771
$$925$$ − 28.0000i − 0.920634i
$$926$$ −38.0000 −1.24876
$$927$$ −14.0000 −0.459820
$$928$$ − 3.00000i − 0.0984798i
$$929$$ 33.0000i 1.08269i 0.840799 + 0.541347i $$0.182086\pi$$
−0.840799 + 0.541347i $$0.817914\pi$$
$$930$$ − 12.0000i − 0.393496i
$$931$$ 6.00000i 0.196642i
$$932$$ −6.00000 −0.196537
$$933$$ 30.0000 0.982156
$$934$$ − 18.0000i − 0.588978i
$$935$$ 54.0000 1.76599
$$936$$ 0 0
$$937$$ 47.0000 1.53542 0.767712 0.640796i $$-0.221395\pi$$
0.767712 + 0.640796i $$0.221395\pi$$
$$938$$ 20.0000i 0.653023i
$$939$$ −10.0000 −0.326338
$$940$$ −18.0000 −0.587095
$$941$$ 42.0000i 1.36916i 0.728937 + 0.684580i $$0.240015\pi$$
−0.728937 + 0.684580i $$0.759985\pi$$
$$942$$ − 5.00000i − 0.162909i
$$943$$ − 18.0000i − 0.586161i
$$944$$ 0 0
$$945$$ 6.00000 0.195180
$$946$$ −60.0000 −1.95077
$$947$$ − 24.0000i − 0.779895i −0.920837 0.389948i $$-0.872493\pi$$
0.920837 0.389948i $$-0.127507\pi$$
$$948$$ 4.00000 0.129914
$$949$$ 0 0
$$950$$ −8.00000 −0.259554
$$951$$ 3.00000i 0.0972817i
$$952$$ 6.00000 0.194461
$$953$$ −54.0000 −1.74923 −0.874616 0.484817i $$-0.838886\pi$$
−0.874616 + 0.484817i $$0.838886\pi$$
$$954$$ − 3.00000i − 0.0971286i
$$955$$ − 36.0000i − 1.16493i
$$956$$ − 6.00000i − 0.194054i
$$957$$ − 18.0000i − 0.581857i
$$958$$ 0 0
$$959$$ −18.0000 −0.581250
$$960$$ − 3.00000i − 0.0968246i
$$961$$ 15.0000 0.483871
$$962$$ 0 0
$$963$$ −6.00000 −0.193347
$$964$$ − 1.00000i − 0.0322078i
$$965$$ 69.0000 2.22119
$$966$$ −12.0000 −0.386094
$$967$$ − 22.0000i − 0.707472i −0.935345 0.353736i $$-0.884911\pi$$
0.935345 0.353736i $$-0.115089\pi$$
$$968$$ − 25.0000i − 0.803530i
$$969$$ 6.00000i 0.192748i
$$970$$ 42.0000i 1.34854i
$$971$$ 60.0000 1.92549 0.962746 0.270408i $$-0.0871586\pi$$
0.962746 + 0.270408i $$0.0871586\pi$$
$$972$$ −1.00000 −0.0320750
$$973$$ 8.00000i 0.256468i
$$974$$ 2.00000 0.0640841
$$975$$ 0 0
$$976$$ −7.00000 −0.224065
$$977$$ − 3.00000i − 0.0959785i −0.998848 0.0479893i $$-0.984719\pi$$
0.998848 0.0479893i $$-0.0152813\pi$$
$$978$$ 4.00000 0.127906
$$979$$ −108.000 −3.45169
$$980$$ − 9.00000i − 0.287494i
$$981$$ 14.0000i 0.446986i
$$982$$ − 18.0000i − 0.574403i
$$983$$ 36.0000i 1.14822i 0.818778 + 0.574111i $$0.194652\pi$$
−0.818778 + 0.574111i $$0.805348\pi$$
$$984$$ 3.00000 0.0956365
$$985$$ −18.0000 −0.573528
$$986$$ − 9.00000i − 0.286618i
$$987$$ −12.0000 −0.381964
$$988$$ 0 0
$$989$$ 60.0000 1.90789
$$990$$ − 18.0000i − 0.572078i
$$991$$ 38.0000 1.20711 0.603555 0.797321i $$-0.293750\pi$$
0.603555 + 0.797321i $$0.293750\pi$$
$$992$$ −4.00000 −0.127000
$$993$$ − 4.00000i − 0.126936i
$$994$$ − 12.0000i − 0.380617i
$$995$$ 30.0000i 0.951064i
$$996$$ 6.00000i 0.190117i
$$997$$ 5.00000 0.158352 0.0791758 0.996861i $$-0.474771\pi$$
0.0791758 + 0.996861i $$0.474771\pi$$
$$998$$ 32.0000 1.01294
$$999$$ 7.00000i 0.221470i
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 1014.2.b.c.337.1 2
3.2 odd 2 3042.2.b.h.1351.2 2
13.2 odd 12 78.2.e.a.61.1 yes 2
13.3 even 3 1014.2.i.b.823.2 4
13.4 even 6 1014.2.i.b.361.2 4
13.5 odd 4 1014.2.a.c.1.1 1
13.6 odd 12 78.2.e.a.55.1 2
13.7 odd 12 1014.2.e.a.991.1 2
13.8 odd 4 1014.2.a.f.1.1 1
13.9 even 3 1014.2.i.b.361.1 4
13.10 even 6 1014.2.i.b.823.1 4
13.11 odd 12 1014.2.e.a.529.1 2
13.12 even 2 inner 1014.2.b.c.337.2 2
39.2 even 12 234.2.h.a.217.1 2
39.5 even 4 3042.2.a.i.1.1 1
39.8 even 4 3042.2.a.h.1.1 1
39.32 even 12 234.2.h.a.55.1 2
39.38 odd 2 3042.2.b.h.1351.1 2
52.15 even 12 624.2.q.g.529.1 2
52.19 even 12 624.2.q.g.289.1 2
52.31 even 4 8112.2.a.m.1.1 1
52.47 even 4 8112.2.a.c.1.1 1
65.2 even 12 1950.2.z.g.1699.2 4
65.19 odd 12 1950.2.i.m.601.1 2
65.28 even 12 1950.2.z.g.1699.1 4
65.32 even 12 1950.2.z.g.1849.1 4
65.54 odd 12 1950.2.i.m.451.1 2
65.58 even 12 1950.2.z.g.1849.2 4
156.71 odd 12 1872.2.t.c.289.1 2
156.119 odd 12 1872.2.t.c.1153.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
78.2.e.a.55.1 2 13.6 odd 12
78.2.e.a.61.1 yes 2 13.2 odd 12
234.2.h.a.55.1 2 39.32 even 12
234.2.h.a.217.1 2 39.2 even 12
624.2.q.g.289.1 2 52.19 even 12
624.2.q.g.529.1 2 52.15 even 12
1014.2.a.c.1.1 1 13.5 odd 4
1014.2.a.f.1.1 1 13.8 odd 4
1014.2.b.c.337.1 2 1.1 even 1 trivial
1014.2.b.c.337.2 2 13.12 even 2 inner
1014.2.e.a.529.1 2 13.11 odd 12
1014.2.e.a.991.1 2 13.7 odd 12
1014.2.i.b.361.1 4 13.9 even 3
1014.2.i.b.361.2 4 13.4 even 6
1014.2.i.b.823.1 4 13.10 even 6
1014.2.i.b.823.2 4 13.3 even 3
1872.2.t.c.289.1 2 156.71 odd 12
1872.2.t.c.1153.1 2 156.119 odd 12
1950.2.i.m.451.1 2 65.54 odd 12
1950.2.i.m.601.1 2 65.19 odd 12
1950.2.z.g.1699.1 4 65.28 even 12
1950.2.z.g.1699.2 4 65.2 even 12
1950.2.z.g.1849.1 4 65.32 even 12
1950.2.z.g.1849.2 4 65.58 even 12
3042.2.a.h.1.1 1 39.8 even 4
3042.2.a.i.1.1 1 39.5 even 4
3042.2.b.h.1351.1 2 39.38 odd 2
3042.2.b.h.1351.2 2 3.2 odd 2
8112.2.a.c.1.1 1 52.47 even 4
8112.2.a.m.1.1 1 52.31 even 4