Properties

 Label 5070.2.b.k.1351.2 Level $5070$ Weight $2$ Character 5070.1351 Analytic conductor $40.484$ Analytic rank $1$ Dimension $2$ CM no Inner twists $2$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5070.b (of order $$2$$, degree $$1$$, not minimal)

Newform invariants

 Self dual: no Analytic conductor: $$40.4841538248$$ 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: no (minimal twist has level 30) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

 Embedding label 1351.2 Root $$-1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 5070.1351 Dual form 5070.2.b.k.1351.1

$q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} +1.00000i q^{6} -4.00000i q^{7} -1.00000i q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} +1.00000i q^{6} -4.00000i q^{7} -1.00000i q^{8} +1.00000 q^{9} -1.00000 q^{10} -1.00000 q^{12} +4.00000 q^{14} +1.00000i q^{15} +1.00000 q^{16} -6.00000 q^{17} +1.00000i q^{18} +4.00000i q^{19} -1.00000i q^{20} -4.00000i q^{21} -1.00000i q^{24} -1.00000 q^{25} +1.00000 q^{27} +4.00000i q^{28} -6.00000 q^{29} -1.00000 q^{30} -8.00000i q^{31} +1.00000i q^{32} -6.00000i q^{34} +4.00000 q^{35} -1.00000 q^{36} +2.00000i q^{37} -4.00000 q^{38} +1.00000 q^{40} +6.00000i q^{41} +4.00000 q^{42} +4.00000 q^{43} +1.00000i q^{45} +1.00000 q^{48} -9.00000 q^{49} -1.00000i q^{50} -6.00000 q^{51} -6.00000 q^{53} +1.00000i q^{54} -4.00000 q^{56} +4.00000i q^{57} -6.00000i q^{58} -1.00000i q^{60} -10.0000 q^{61} +8.00000 q^{62} -4.00000i q^{63} -1.00000 q^{64} +4.00000i q^{67} +6.00000 q^{68} +4.00000i q^{70} -1.00000i q^{72} +2.00000i q^{73} -2.00000 q^{74} -1.00000 q^{75} -4.00000i q^{76} +8.00000 q^{79} +1.00000i q^{80} +1.00000 q^{81} -6.00000 q^{82} -12.0000i q^{83} +4.00000i q^{84} -6.00000i q^{85} +4.00000i q^{86} -6.00000 q^{87} +18.0000i q^{89} -1.00000 q^{90} -8.00000i q^{93} -4.00000 q^{95} +1.00000i q^{96} -2.00000i q^{97} -9.00000i q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{3} - 2q^{4} + 2q^{9} + O(q^{10})$$ $$2q + 2q^{3} - 2q^{4} + 2q^{9} - 2q^{10} - 2q^{12} + 8q^{14} + 2q^{16} - 12q^{17} - 2q^{25} + 2q^{27} - 12q^{29} - 2q^{30} + 8q^{35} - 2q^{36} - 8q^{38} + 2q^{40} + 8q^{42} + 8q^{43} + 2q^{48} - 18q^{49} - 12q^{51} - 12q^{53} - 8q^{56} - 20q^{61} + 16q^{62} - 2q^{64} + 12q^{68} - 4q^{74} - 2q^{75} + 16q^{79} + 2q^{81} - 12q^{82} - 12q^{87} - 2q^{90} - 8q^{95} + O(q^{100})$$

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/5070\mathbb{Z}\right)^\times$$.

 $$n$$ $$1691$$ $$1861$$ $$4057$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

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
$$3$$ 1.00000 0.577350
$$4$$ −1.00000 −0.500000
$$5$$ 1.00000i 0.447214i
$$6$$ 1.00000i 0.408248i
$$7$$ − 4.00000i − 1.51186i −0.654654 0.755929i $$-0.727186\pi$$
0.654654 0.755929i $$-0.272814\pi$$
$$8$$ − 1.00000i − 0.353553i
$$9$$ 1.00000 0.333333
$$10$$ −1.00000 −0.316228
$$11$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$12$$ −1.00000 −0.288675
$$13$$ 0 0
$$14$$ 4.00000 1.06904
$$15$$ 1.00000i 0.258199i
$$16$$ 1.00000 0.250000
$$17$$ −6.00000 −1.45521 −0.727607 0.685994i $$-0.759367\pi$$
−0.727607 + 0.685994i $$0.759367\pi$$
$$18$$ 1.00000i 0.235702i
$$19$$ 4.00000i 0.917663i 0.888523 + 0.458831i $$0.151732\pi$$
−0.888523 + 0.458831i $$0.848268\pi$$
$$20$$ − 1.00000i − 0.223607i
$$21$$ − 4.00000i − 0.872872i
$$22$$ 0 0
$$23$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$24$$ − 1.00000i − 0.204124i
$$25$$ −1.00000 −0.200000
$$26$$ 0 0
$$27$$ 1.00000 0.192450
$$28$$ 4.00000i 0.755929i
$$29$$ −6.00000 −1.11417 −0.557086 0.830455i $$-0.688081\pi$$
−0.557086 + 0.830455i $$0.688081\pi$$
$$30$$ −1.00000 −0.182574
$$31$$ − 8.00000i − 1.43684i −0.695608 0.718421i $$-0.744865\pi$$
0.695608 0.718421i $$-0.255135\pi$$
$$32$$ 1.00000i 0.176777i
$$33$$ 0 0
$$34$$ − 6.00000i − 1.02899i
$$35$$ 4.00000 0.676123
$$36$$ −1.00000 −0.166667
$$37$$ 2.00000i 0.328798i 0.986394 + 0.164399i $$0.0525685\pi$$
−0.986394 + 0.164399i $$0.947432\pi$$
$$38$$ −4.00000 −0.648886
$$39$$ 0 0
$$40$$ 1.00000 0.158114
$$41$$ 6.00000i 0.937043i 0.883452 + 0.468521i $$0.155213\pi$$
−0.883452 + 0.468521i $$0.844787\pi$$
$$42$$ 4.00000 0.617213
$$43$$ 4.00000 0.609994 0.304997 0.952353i $$-0.401344\pi$$
0.304997 + 0.952353i $$0.401344\pi$$
$$44$$ 0 0
$$45$$ 1.00000i 0.149071i
$$46$$ 0 0
$$47$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$48$$ 1.00000 0.144338
$$49$$ −9.00000 −1.28571
$$50$$ − 1.00000i − 0.141421i
$$51$$ −6.00000 −0.840168
$$52$$ 0 0
$$53$$ −6.00000 −0.824163 −0.412082 0.911147i $$-0.635198\pi$$
−0.412082 + 0.911147i $$0.635198\pi$$
$$54$$ 1.00000i 0.136083i
$$55$$ 0 0
$$56$$ −4.00000 −0.534522
$$57$$ 4.00000i 0.529813i
$$58$$ − 6.00000i − 0.787839i
$$59$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$60$$ − 1.00000i − 0.129099i
$$61$$ −10.0000 −1.28037 −0.640184 0.768221i $$-0.721142\pi$$
−0.640184 + 0.768221i $$0.721142\pi$$
$$62$$ 8.00000 1.01600
$$63$$ − 4.00000i − 0.503953i
$$64$$ −1.00000 −0.125000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 4.00000i 0.488678i 0.969690 + 0.244339i $$0.0785709\pi$$
−0.969690 + 0.244339i $$0.921429\pi$$
$$68$$ 6.00000 0.727607
$$69$$ 0 0
$$70$$ 4.00000i 0.478091i
$$71$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$72$$ − 1.00000i − 0.117851i
$$73$$ 2.00000i 0.234082i 0.993127 + 0.117041i $$0.0373409\pi$$
−0.993127 + 0.117041i $$0.962659\pi$$
$$74$$ −2.00000 −0.232495
$$75$$ −1.00000 −0.115470
$$76$$ − 4.00000i − 0.458831i
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 8.00000 0.900070 0.450035 0.893011i $$-0.351411\pi$$
0.450035 + 0.893011i $$0.351411\pi$$
$$80$$ 1.00000i 0.111803i
$$81$$ 1.00000 0.111111
$$82$$ −6.00000 −0.662589
$$83$$ − 12.0000i − 1.31717i −0.752506 0.658586i $$-0.771155\pi$$
0.752506 0.658586i $$-0.228845\pi$$
$$84$$ 4.00000i 0.436436i
$$85$$ − 6.00000i − 0.650791i
$$86$$ 4.00000i 0.431331i
$$87$$ −6.00000 −0.643268
$$88$$ 0 0
$$89$$ 18.0000i 1.90800i 0.299813 + 0.953998i $$0.403076\pi$$
−0.299813 + 0.953998i $$0.596924\pi$$
$$90$$ −1.00000 −0.105409
$$91$$ 0 0
$$92$$ 0 0
$$93$$ − 8.00000i − 0.829561i
$$94$$ 0 0
$$95$$ −4.00000 −0.410391
$$96$$ 1.00000i 0.102062i
$$97$$ − 2.00000i − 0.203069i −0.994832 0.101535i $$-0.967625\pi$$
0.994832 0.101535i $$-0.0323753\pi$$
$$98$$ − 9.00000i − 0.909137i
$$99$$ 0 0
$$100$$ 1.00000 0.100000
$$101$$ −18.0000 −1.79107 −0.895533 0.444994i $$-0.853206\pi$$
−0.895533 + 0.444994i $$0.853206\pi$$
$$102$$ − 6.00000i − 0.594089i
$$103$$ 4.00000 0.394132 0.197066 0.980390i $$-0.436859\pi$$
0.197066 + 0.980390i $$0.436859\pi$$
$$104$$ 0 0
$$105$$ 4.00000 0.390360
$$106$$ − 6.00000i − 0.582772i
$$107$$ −12.0000 −1.16008 −0.580042 0.814587i $$-0.696964\pi$$
−0.580042 + 0.814587i $$0.696964\pi$$
$$108$$ −1.00000 −0.0962250
$$109$$ 10.0000i 0.957826i 0.877862 + 0.478913i $$0.158969\pi$$
−0.877862 + 0.478913i $$0.841031\pi$$
$$110$$ 0 0
$$111$$ 2.00000i 0.189832i
$$112$$ − 4.00000i − 0.377964i
$$113$$ −18.0000 −1.69330 −0.846649 0.532152i $$-0.821383\pi$$
−0.846649 + 0.532152i $$0.821383\pi$$
$$114$$ −4.00000 −0.374634
$$115$$ 0 0
$$116$$ 6.00000 0.557086
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 24.0000i 2.20008i
$$120$$ 1.00000 0.0912871
$$121$$ 11.0000 1.00000
$$122$$ − 10.0000i − 0.905357i
$$123$$ 6.00000i 0.541002i
$$124$$ 8.00000i 0.718421i
$$125$$ − 1.00000i − 0.0894427i
$$126$$ 4.00000 0.356348
$$127$$ −20.0000 −1.77471 −0.887357 0.461084i $$-0.847461\pi$$
−0.887357 + 0.461084i $$0.847461\pi$$
$$128$$ − 1.00000i − 0.0883883i
$$129$$ 4.00000 0.352180
$$130$$ 0 0
$$131$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$132$$ 0 0
$$133$$ 16.0000 1.38738
$$134$$ −4.00000 −0.345547
$$135$$ 1.00000i 0.0860663i
$$136$$ 6.00000i 0.514496i
$$137$$ 6.00000i 0.512615i 0.966595 + 0.256307i $$0.0825059\pi$$
−0.966595 + 0.256307i $$0.917494\pi$$
$$138$$ 0 0
$$139$$ −4.00000 −0.339276 −0.169638 0.985506i $$-0.554260\pi$$
−0.169638 + 0.985506i $$0.554260\pi$$
$$140$$ −4.00000 −0.338062
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 1.00000 0.0833333
$$145$$ − 6.00000i − 0.498273i
$$146$$ −2.00000 −0.165521
$$147$$ −9.00000 −0.742307
$$148$$ − 2.00000i − 0.164399i
$$149$$ 6.00000i 0.491539i 0.969328 + 0.245770i $$0.0790407\pi$$
−0.969328 + 0.245770i $$0.920959\pi$$
$$150$$ − 1.00000i − 0.0816497i
$$151$$ 8.00000i 0.651031i 0.945537 + 0.325515i $$0.105538\pi$$
−0.945537 + 0.325515i $$0.894462\pi$$
$$152$$ 4.00000 0.324443
$$153$$ −6.00000 −0.485071
$$154$$ 0 0
$$155$$ 8.00000 0.642575
$$156$$ 0 0
$$157$$ 2.00000 0.159617 0.0798087 0.996810i $$-0.474569\pi$$
0.0798087 + 0.996810i $$0.474569\pi$$
$$158$$ 8.00000i 0.636446i
$$159$$ −6.00000 −0.475831
$$160$$ −1.00000 −0.0790569
$$161$$ 0 0
$$162$$ 1.00000i 0.0785674i
$$163$$ − 4.00000i − 0.313304i −0.987654 0.156652i $$-0.949930\pi$$
0.987654 0.156652i $$-0.0500701\pi$$
$$164$$ − 6.00000i − 0.468521i
$$165$$ 0 0
$$166$$ 12.0000 0.931381
$$167$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$168$$ −4.00000 −0.308607
$$169$$ 0 0
$$170$$ 6.00000 0.460179
$$171$$ 4.00000i 0.305888i
$$172$$ −4.00000 −0.304997
$$173$$ −18.0000 −1.36851 −0.684257 0.729241i $$-0.739873\pi$$
−0.684257 + 0.729241i $$0.739873\pi$$
$$174$$ − 6.00000i − 0.454859i
$$175$$ 4.00000i 0.302372i
$$176$$ 0 0
$$177$$ 0 0
$$178$$ −18.0000 −1.34916
$$179$$ −24.0000 −1.79384 −0.896922 0.442189i $$-0.854202\pi$$
−0.896922 + 0.442189i $$0.854202\pi$$
$$180$$ − 1.00000i − 0.0745356i
$$181$$ −14.0000 −1.04061 −0.520306 0.853980i $$-0.674182\pi$$
−0.520306 + 0.853980i $$0.674182\pi$$
$$182$$ 0 0
$$183$$ −10.0000 −0.739221
$$184$$ 0 0
$$185$$ −2.00000 −0.147043
$$186$$ 8.00000 0.586588
$$187$$ 0 0
$$188$$ 0 0
$$189$$ − 4.00000i − 0.290957i
$$190$$ − 4.00000i − 0.290191i
$$191$$ −24.0000 −1.73658 −0.868290 0.496058i $$-0.834780\pi$$
−0.868290 + 0.496058i $$0.834780\pi$$
$$192$$ −1.00000 −0.0721688
$$193$$ − 22.0000i − 1.58359i −0.610784 0.791797i $$-0.709146\pi$$
0.610784 0.791797i $$-0.290854\pi$$
$$194$$ 2.00000 0.143592
$$195$$ 0 0
$$196$$ 9.00000 0.642857
$$197$$ 6.00000i 0.427482i 0.976890 + 0.213741i $$0.0685649\pi$$
−0.976890 + 0.213741i $$0.931435\pi$$
$$198$$ 0 0
$$199$$ −8.00000 −0.567105 −0.283552 0.958957i $$-0.591513\pi$$
−0.283552 + 0.958957i $$0.591513\pi$$
$$200$$ 1.00000i 0.0707107i
$$201$$ 4.00000i 0.282138i
$$202$$ − 18.0000i − 1.26648i
$$203$$ 24.0000i 1.68447i
$$204$$ 6.00000 0.420084
$$205$$ −6.00000 −0.419058
$$206$$ 4.00000i 0.278693i
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 0 0
$$210$$ 4.00000i 0.276026i
$$211$$ 20.0000 1.37686 0.688428 0.725304i $$-0.258301\pi$$
0.688428 + 0.725304i $$0.258301\pi$$
$$212$$ 6.00000 0.412082
$$213$$ 0 0
$$214$$ − 12.0000i − 0.820303i
$$215$$ 4.00000i 0.272798i
$$216$$ − 1.00000i − 0.0680414i
$$217$$ −32.0000 −2.17230
$$218$$ −10.0000 −0.677285
$$219$$ 2.00000i 0.135147i
$$220$$ 0 0
$$221$$ 0 0
$$222$$ −2.00000 −0.134231
$$223$$ − 20.0000i − 1.33930i −0.742677 0.669650i $$-0.766444\pi$$
0.742677 0.669650i $$-0.233556\pi$$
$$224$$ 4.00000 0.267261
$$225$$ −1.00000 −0.0666667
$$226$$ − 18.0000i − 1.19734i
$$227$$ 12.0000i 0.796468i 0.917284 + 0.398234i $$0.130377\pi$$
−0.917284 + 0.398234i $$0.869623\pi$$
$$228$$ − 4.00000i − 0.264906i
$$229$$ − 10.0000i − 0.660819i −0.943838 0.330409i $$-0.892813\pi$$
0.943838 0.330409i $$-0.107187\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 6.00000i 0.393919i
$$233$$ 18.0000 1.17922 0.589610 0.807688i $$-0.299282\pi$$
0.589610 + 0.807688i $$0.299282\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 8.00000 0.519656
$$238$$ −24.0000 −1.55569
$$239$$ − 24.0000i − 1.55243i −0.630468 0.776215i $$-0.717137\pi$$
0.630468 0.776215i $$-0.282863\pi$$
$$240$$ 1.00000i 0.0645497i
$$241$$ 2.00000i 0.128831i 0.997923 + 0.0644157i $$0.0205183\pi$$
−0.997923 + 0.0644157i $$0.979482\pi$$
$$242$$ 11.0000i 0.707107i
$$243$$ 1.00000 0.0641500
$$244$$ 10.0000 0.640184
$$245$$ − 9.00000i − 0.574989i
$$246$$ −6.00000 −0.382546
$$247$$ 0 0
$$248$$ −8.00000 −0.508001
$$249$$ − 12.0000i − 0.760469i
$$250$$ 1.00000 0.0632456
$$251$$ 24.0000 1.51487 0.757433 0.652913i $$-0.226453\pi$$
0.757433 + 0.652913i $$0.226453\pi$$
$$252$$ 4.00000i 0.251976i
$$253$$ 0 0
$$254$$ − 20.0000i − 1.25491i
$$255$$ − 6.00000i − 0.375735i
$$256$$ 1.00000 0.0625000
$$257$$ 18.0000 1.12281 0.561405 0.827541i $$-0.310261\pi$$
0.561405 + 0.827541i $$0.310261\pi$$
$$258$$ 4.00000i 0.249029i
$$259$$ 8.00000 0.497096
$$260$$ 0 0
$$261$$ −6.00000 −0.371391
$$262$$ 0 0
$$263$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$264$$ 0 0
$$265$$ − 6.00000i − 0.368577i
$$266$$ 16.0000i 0.981023i
$$267$$ 18.0000i 1.10158i
$$268$$ − 4.00000i − 0.244339i
$$269$$ −6.00000 −0.365826 −0.182913 0.983129i $$-0.558553\pi$$
−0.182913 + 0.983129i $$0.558553\pi$$
$$270$$ −1.00000 −0.0608581
$$271$$ − 16.0000i − 0.971931i −0.873978 0.485965i $$-0.838468\pi$$
0.873978 0.485965i $$-0.161532\pi$$
$$272$$ −6.00000 −0.363803
$$273$$ 0 0
$$274$$ −6.00000 −0.362473
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −2.00000 −0.120168 −0.0600842 0.998193i $$-0.519137\pi$$
−0.0600842 + 0.998193i $$0.519137\pi$$
$$278$$ − 4.00000i − 0.239904i
$$279$$ − 8.00000i − 0.478947i
$$280$$ − 4.00000i − 0.239046i
$$281$$ 18.0000i 1.07379i 0.843649 + 0.536895i $$0.180403\pi$$
−0.843649 + 0.536895i $$0.819597\pi$$
$$282$$ 0 0
$$283$$ 28.0000 1.66443 0.832214 0.554455i $$-0.187073\pi$$
0.832214 + 0.554455i $$0.187073\pi$$
$$284$$ 0 0
$$285$$ −4.00000 −0.236940
$$286$$ 0 0
$$287$$ 24.0000 1.41668
$$288$$ 1.00000i 0.0589256i
$$289$$ 19.0000 1.11765
$$290$$ 6.00000 0.352332
$$291$$ − 2.00000i − 0.117242i
$$292$$ − 2.00000i − 0.117041i
$$293$$ − 6.00000i − 0.350524i −0.984522 0.175262i $$-0.943923\pi$$
0.984522 0.175262i $$-0.0560772\pi$$
$$294$$ − 9.00000i − 0.524891i
$$295$$ 0 0
$$296$$ 2.00000 0.116248
$$297$$ 0 0
$$298$$ −6.00000 −0.347571
$$299$$ 0 0
$$300$$ 1.00000 0.0577350
$$301$$ − 16.0000i − 0.922225i
$$302$$ −8.00000 −0.460348
$$303$$ −18.0000 −1.03407
$$304$$ 4.00000i 0.229416i
$$305$$ − 10.0000i − 0.572598i
$$306$$ − 6.00000i − 0.342997i
$$307$$ 20.0000i 1.14146i 0.821138 + 0.570730i $$0.193340\pi$$
−0.821138 + 0.570730i $$0.806660\pi$$
$$308$$ 0 0
$$309$$ 4.00000 0.227552
$$310$$ 8.00000i 0.454369i
$$311$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$312$$ 0 0
$$313$$ 2.00000 0.113047 0.0565233 0.998401i $$-0.481998\pi$$
0.0565233 + 0.998401i $$0.481998\pi$$
$$314$$ 2.00000i 0.112867i
$$315$$ 4.00000 0.225374
$$316$$ −8.00000 −0.450035
$$317$$ − 18.0000i − 1.01098i −0.862832 0.505490i $$-0.831312\pi$$
0.862832 0.505490i $$-0.168688\pi$$
$$318$$ − 6.00000i − 0.336463i
$$319$$ 0 0
$$320$$ − 1.00000i − 0.0559017i
$$321$$ −12.0000 −0.669775
$$322$$ 0 0
$$323$$ − 24.0000i − 1.33540i
$$324$$ −1.00000 −0.0555556
$$325$$ 0 0
$$326$$ 4.00000 0.221540
$$327$$ 10.0000i 0.553001i
$$328$$ 6.00000 0.331295
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 28.0000i 1.53902i 0.638635 + 0.769510i $$0.279499\pi$$
−0.638635 + 0.769510i $$0.720501\pi$$
$$332$$ 12.0000i 0.658586i
$$333$$ 2.00000i 0.109599i
$$334$$ 0 0
$$335$$ −4.00000 −0.218543
$$336$$ − 4.00000i − 0.218218i
$$337$$ −26.0000 −1.41631 −0.708155 0.706057i $$-0.750472\pi$$
−0.708155 + 0.706057i $$0.750472\pi$$
$$338$$ 0 0
$$339$$ −18.0000 −0.977626
$$340$$ 6.00000i 0.325396i
$$341$$ 0 0
$$342$$ −4.00000 −0.216295
$$343$$ 8.00000i 0.431959i
$$344$$ − 4.00000i − 0.215666i
$$345$$ 0 0
$$346$$ − 18.0000i − 0.967686i
$$347$$ 12.0000 0.644194 0.322097 0.946707i $$-0.395612\pi$$
0.322097 + 0.946707i $$0.395612\pi$$
$$348$$ 6.00000 0.321634
$$349$$ − 10.0000i − 0.535288i −0.963518 0.267644i $$-0.913755\pi$$
0.963518 0.267644i $$-0.0862451\pi$$
$$350$$ −4.00000 −0.213809
$$351$$ 0 0
$$352$$ 0 0
$$353$$ − 6.00000i − 0.319348i −0.987170 0.159674i $$-0.948956\pi$$
0.987170 0.159674i $$-0.0510443\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ − 18.0000i − 0.953998i
$$357$$ 24.0000i 1.27021i
$$358$$ − 24.0000i − 1.26844i
$$359$$ − 24.0000i − 1.26667i −0.773877 0.633336i $$-0.781685\pi$$
0.773877 0.633336i $$-0.218315\pi$$
$$360$$ 1.00000 0.0527046
$$361$$ 3.00000 0.157895
$$362$$ − 14.0000i − 0.735824i
$$363$$ 11.0000 0.577350
$$364$$ 0 0
$$365$$ −2.00000 −0.104685
$$366$$ − 10.0000i − 0.522708i
$$367$$ −28.0000 −1.46159 −0.730794 0.682598i $$-0.760850\pi$$
−0.730794 + 0.682598i $$0.760850\pi$$
$$368$$ 0 0
$$369$$ 6.00000i 0.312348i
$$370$$ − 2.00000i − 0.103975i
$$371$$ 24.0000i 1.24602i
$$372$$ 8.00000i 0.414781i
$$373$$ 26.0000 1.34623 0.673114 0.739538i $$-0.264956\pi$$
0.673114 + 0.739538i $$0.264956\pi$$
$$374$$ 0 0
$$375$$ − 1.00000i − 0.0516398i
$$376$$ 0 0
$$377$$ 0 0
$$378$$ 4.00000 0.205738
$$379$$ 4.00000i 0.205466i 0.994709 + 0.102733i $$0.0327588\pi$$
−0.994709 + 0.102733i $$0.967241\pi$$
$$380$$ 4.00000 0.205196
$$381$$ −20.0000 −1.02463
$$382$$ − 24.0000i − 1.22795i
$$383$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$384$$ − 1.00000i − 0.0510310i
$$385$$ 0 0
$$386$$ 22.0000 1.11977
$$387$$ 4.00000 0.203331
$$388$$ 2.00000i 0.101535i
$$389$$ 6.00000 0.304212 0.152106 0.988364i $$-0.451394\pi$$
0.152106 + 0.988364i $$0.451394\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 9.00000i 0.454569i
$$393$$ 0 0
$$394$$ −6.00000 −0.302276
$$395$$ 8.00000i 0.402524i
$$396$$ 0 0
$$397$$ − 22.0000i − 1.10415i −0.833795 0.552074i $$-0.813837\pi$$
0.833795 0.552074i $$-0.186163\pi$$
$$398$$ − 8.00000i − 0.401004i
$$399$$ 16.0000 0.801002
$$400$$ −1.00000 −0.0500000
$$401$$ − 6.00000i − 0.299626i −0.988714 0.149813i $$-0.952133\pi$$
0.988714 0.149813i $$-0.0478671\pi$$
$$402$$ −4.00000 −0.199502
$$403$$ 0 0
$$404$$ 18.0000 0.895533
$$405$$ 1.00000i 0.0496904i
$$406$$ −24.0000 −1.19110
$$407$$ 0 0
$$408$$ 6.00000i 0.297044i
$$409$$ − 26.0000i − 1.28562i −0.766027 0.642809i $$-0.777769\pi$$
0.766027 0.642809i $$-0.222231\pi$$
$$410$$ − 6.00000i − 0.296319i
$$411$$ 6.00000i 0.295958i
$$412$$ −4.00000 −0.197066
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 12.0000 0.589057
$$416$$ 0 0
$$417$$ −4.00000 −0.195881
$$418$$ 0 0
$$419$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$420$$ −4.00000 −0.195180
$$421$$ 10.0000i 0.487370i 0.969854 + 0.243685i $$0.0783563\pi$$
−0.969854 + 0.243685i $$0.921644\pi$$
$$422$$ 20.0000i 0.973585i
$$423$$ 0 0
$$424$$ 6.00000i 0.291386i
$$425$$ 6.00000 0.291043
$$426$$ 0 0
$$427$$ 40.0000i 1.93574i
$$428$$ 12.0000 0.580042
$$429$$ 0 0
$$430$$ −4.00000 −0.192897
$$431$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$432$$ 1.00000 0.0481125
$$433$$ −26.0000 −1.24948 −0.624740 0.780833i $$-0.714795\pi$$
−0.624740 + 0.780833i $$0.714795\pi$$
$$434$$ − 32.0000i − 1.53605i
$$435$$ − 6.00000i − 0.287678i
$$436$$ − 10.0000i − 0.478913i
$$437$$ 0 0
$$438$$ −2.00000 −0.0955637
$$439$$ −8.00000 −0.381819 −0.190910 0.981608i $$-0.561144\pi$$
−0.190910 + 0.981608i $$0.561144\pi$$
$$440$$ 0 0
$$441$$ −9.00000 −0.428571
$$442$$ 0 0
$$443$$ 12.0000 0.570137 0.285069 0.958507i $$-0.407984\pi$$
0.285069 + 0.958507i $$0.407984\pi$$
$$444$$ − 2.00000i − 0.0949158i
$$445$$ −18.0000 −0.853282
$$446$$ 20.0000 0.947027
$$447$$ 6.00000i 0.283790i
$$448$$ 4.00000i 0.188982i
$$449$$ − 6.00000i − 0.283158i −0.989927 0.141579i $$-0.954782\pi$$
0.989927 0.141579i $$-0.0452178\pi$$
$$450$$ − 1.00000i − 0.0471405i
$$451$$ 0 0
$$452$$ 18.0000 0.846649
$$453$$ 8.00000i 0.375873i
$$454$$ −12.0000 −0.563188
$$455$$ 0 0
$$456$$ 4.00000 0.187317
$$457$$ − 26.0000i − 1.21623i −0.793849 0.608114i $$-0.791926\pi$$
0.793849 0.608114i $$-0.208074\pi$$
$$458$$ 10.0000 0.467269
$$459$$ −6.00000 −0.280056
$$460$$ 0 0
$$461$$ 30.0000i 1.39724i 0.715493 + 0.698620i $$0.246202\pi$$
−0.715493 + 0.698620i $$0.753798\pi$$
$$462$$ 0 0
$$463$$ − 4.00000i − 0.185896i −0.995671 0.0929479i $$-0.970371\pi$$
0.995671 0.0929479i $$-0.0296290\pi$$
$$464$$ −6.00000 −0.278543
$$465$$ 8.00000 0.370991
$$466$$ 18.0000i 0.833834i
$$467$$ 36.0000 1.66588 0.832941 0.553362i $$-0.186655\pi$$
0.832941 + 0.553362i $$0.186655\pi$$
$$468$$ 0 0
$$469$$ 16.0000 0.738811
$$470$$ 0 0
$$471$$ 2.00000 0.0921551
$$472$$ 0 0
$$473$$ 0 0
$$474$$ 8.00000i 0.367452i
$$475$$ − 4.00000i − 0.183533i
$$476$$ − 24.0000i − 1.10004i
$$477$$ −6.00000 −0.274721
$$478$$ 24.0000 1.09773
$$479$$ − 24.0000i − 1.09659i −0.836286 0.548294i $$-0.815277\pi$$
0.836286 0.548294i $$-0.184723\pi$$
$$480$$ −1.00000 −0.0456435
$$481$$ 0 0
$$482$$ −2.00000 −0.0910975
$$483$$ 0 0
$$484$$ −11.0000 −0.500000
$$485$$ 2.00000 0.0908153
$$486$$ 1.00000i 0.0453609i
$$487$$ 28.0000i 1.26880i 0.773004 + 0.634401i $$0.218753\pi$$
−0.773004 + 0.634401i $$0.781247\pi$$
$$488$$ 10.0000i 0.452679i
$$489$$ − 4.00000i − 0.180886i
$$490$$ 9.00000 0.406579
$$491$$ −24.0000 −1.08310 −0.541552 0.840667i $$-0.682163\pi$$
−0.541552 + 0.840667i $$0.682163\pi$$
$$492$$ − 6.00000i − 0.270501i
$$493$$ 36.0000 1.62136
$$494$$ 0 0
$$495$$ 0 0
$$496$$ − 8.00000i − 0.359211i
$$497$$ 0 0
$$498$$ 12.0000 0.537733
$$499$$ 4.00000i 0.179065i 0.995984 + 0.0895323i $$0.0285372\pi$$
−0.995984 + 0.0895323i $$0.971463\pi$$
$$500$$ 1.00000i 0.0447214i
$$501$$ 0 0
$$502$$ 24.0000i 1.07117i
$$503$$ 24.0000 1.07011 0.535054 0.844818i $$-0.320291\pi$$
0.535054 + 0.844818i $$0.320291\pi$$
$$504$$ −4.00000 −0.178174
$$505$$ − 18.0000i − 0.800989i
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 20.0000 0.887357
$$509$$ 6.00000i 0.265945i 0.991120 + 0.132973i $$0.0424523\pi$$
−0.991120 + 0.132973i $$0.957548\pi$$
$$510$$ 6.00000 0.265684
$$511$$ 8.00000 0.353899
$$512$$ 1.00000i 0.0441942i
$$513$$ 4.00000i 0.176604i
$$514$$ 18.0000i 0.793946i
$$515$$ 4.00000i 0.176261i
$$516$$ −4.00000 −0.176090
$$517$$ 0 0
$$518$$ 8.00000i 0.351500i
$$519$$ −18.0000 −0.790112
$$520$$ 0 0
$$521$$ 18.0000 0.788594 0.394297 0.918983i $$-0.370988\pi$$
0.394297 + 0.918983i $$0.370988\pi$$
$$522$$ − 6.00000i − 0.262613i
$$523$$ 20.0000 0.874539 0.437269 0.899331i $$-0.355946\pi$$
0.437269 + 0.899331i $$0.355946\pi$$
$$524$$ 0 0
$$525$$ 4.00000i 0.174574i
$$526$$ 0 0
$$527$$ 48.0000i 2.09091i
$$528$$ 0 0
$$529$$ −23.0000 −1.00000
$$530$$ 6.00000 0.260623
$$531$$ 0 0
$$532$$ −16.0000 −0.693688
$$533$$ 0 0
$$534$$ −18.0000 −0.778936
$$535$$ − 12.0000i − 0.518805i
$$536$$ 4.00000 0.172774
$$537$$ −24.0000 −1.03568
$$538$$ − 6.00000i − 0.258678i
$$539$$ 0 0
$$540$$ − 1.00000i − 0.0430331i
$$541$$ − 10.0000i − 0.429934i −0.976621 0.214967i $$-0.931036\pi$$
0.976621 0.214967i $$-0.0689643\pi$$
$$542$$ 16.0000 0.687259
$$543$$ −14.0000 −0.600798
$$544$$ − 6.00000i − 0.257248i
$$545$$ −10.0000 −0.428353
$$546$$ 0 0
$$547$$ −28.0000 −1.19719 −0.598597 0.801050i $$-0.704275\pi$$
−0.598597 + 0.801050i $$0.704275\pi$$
$$548$$ − 6.00000i − 0.256307i
$$549$$ −10.0000 −0.426790
$$550$$ 0 0
$$551$$ − 24.0000i − 1.02243i
$$552$$ 0 0
$$553$$ − 32.0000i − 1.36078i
$$554$$ − 2.00000i − 0.0849719i
$$555$$ −2.00000 −0.0848953
$$556$$ 4.00000 0.169638
$$557$$ 18.0000i 0.762684i 0.924434 + 0.381342i $$0.124538\pi$$
−0.924434 + 0.381342i $$0.875462\pi$$
$$558$$ 8.00000 0.338667
$$559$$ 0 0
$$560$$ 4.00000 0.169031
$$561$$ 0 0
$$562$$ −18.0000 −0.759284
$$563$$ 12.0000 0.505740 0.252870 0.967500i $$-0.418626\pi$$
0.252870 + 0.967500i $$0.418626\pi$$
$$564$$ 0 0
$$565$$ − 18.0000i − 0.757266i
$$566$$ 28.0000i 1.17693i
$$567$$ − 4.00000i − 0.167984i
$$568$$ 0 0
$$569$$ −18.0000 −0.754599 −0.377300 0.926091i $$-0.623147\pi$$
−0.377300 + 0.926091i $$0.623147\pi$$
$$570$$ − 4.00000i − 0.167542i
$$571$$ −20.0000 −0.836974 −0.418487 0.908223i $$-0.637439\pi$$
−0.418487 + 0.908223i $$0.637439\pi$$
$$572$$ 0 0
$$573$$ −24.0000 −1.00261
$$574$$ 24.0000i 1.00174i
$$575$$ 0 0
$$576$$ −1.00000 −0.0416667
$$577$$ − 2.00000i − 0.0832611i −0.999133 0.0416305i $$-0.986745\pi$$
0.999133 0.0416305i $$-0.0132552\pi$$
$$578$$ 19.0000i 0.790296i
$$579$$ − 22.0000i − 0.914289i
$$580$$ 6.00000i 0.249136i
$$581$$ −48.0000 −1.99138
$$582$$ 2.00000 0.0829027
$$583$$ 0 0
$$584$$ 2.00000 0.0827606
$$585$$ 0 0
$$586$$ 6.00000 0.247858
$$587$$ 12.0000i 0.495293i 0.968850 + 0.247647i $$0.0796572\pi$$
−0.968850 + 0.247647i $$0.920343\pi$$
$$588$$ 9.00000 0.371154
$$589$$ 32.0000 1.31854
$$590$$ 0 0
$$591$$ 6.00000i 0.246807i
$$592$$ 2.00000i 0.0821995i
$$593$$ 30.0000i 1.23195i 0.787765 + 0.615976i $$0.211238\pi$$
−0.787765 + 0.615976i $$0.788762\pi$$
$$594$$ 0 0
$$595$$ −24.0000 −0.983904
$$596$$ − 6.00000i − 0.245770i
$$597$$ −8.00000 −0.327418
$$598$$ 0 0
$$599$$ −24.0000 −0.980613 −0.490307 0.871550i $$-0.663115\pi$$
−0.490307 + 0.871550i $$0.663115\pi$$
$$600$$ 1.00000i 0.0408248i
$$601$$ −22.0000 −0.897399 −0.448699 0.893683i $$-0.648113\pi$$
−0.448699 + 0.893683i $$0.648113\pi$$
$$602$$ 16.0000 0.652111
$$603$$ 4.00000i 0.162893i
$$604$$ − 8.00000i − 0.325515i
$$605$$ 11.0000i 0.447214i
$$606$$ − 18.0000i − 0.731200i
$$607$$ −4.00000 −0.162355 −0.0811775 0.996700i $$-0.525868\pi$$
−0.0811775 + 0.996700i $$0.525868\pi$$
$$608$$ −4.00000 −0.162221
$$609$$ 24.0000i 0.972529i
$$610$$ 10.0000 0.404888
$$611$$ 0 0
$$612$$ 6.00000 0.242536
$$613$$ − 2.00000i − 0.0807792i −0.999184 0.0403896i $$-0.987140\pi$$
0.999184 0.0403896i $$-0.0128599\pi$$
$$614$$ −20.0000 −0.807134
$$615$$ −6.00000 −0.241943
$$616$$ 0 0
$$617$$ − 30.0000i − 1.20775i −0.797077 0.603877i $$-0.793622\pi$$
0.797077 0.603877i $$-0.206378\pi$$
$$618$$ 4.00000i 0.160904i
$$619$$ 44.0000i 1.76851i 0.467005 + 0.884255i $$0.345333\pi$$
−0.467005 + 0.884255i $$0.654667\pi$$
$$620$$ −8.00000 −0.321288
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 72.0000 2.88462
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 2.00000i 0.0799361i
$$627$$ 0 0
$$628$$ −2.00000 −0.0798087
$$629$$ − 12.0000i − 0.478471i
$$630$$ 4.00000i 0.159364i
$$631$$ 32.0000i 1.27390i 0.770905 + 0.636950i $$0.219804\pi$$
−0.770905 + 0.636950i $$0.780196\pi$$
$$632$$ − 8.00000i − 0.318223i
$$633$$ 20.0000 0.794929
$$634$$ 18.0000 0.714871
$$635$$ − 20.0000i − 0.793676i
$$636$$ 6.00000 0.237915
$$637$$ 0 0
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 1.00000 0.0395285
$$641$$ 30.0000 1.18493 0.592464 0.805597i $$-0.298155\pi$$
0.592464 + 0.805597i $$0.298155\pi$$
$$642$$ − 12.0000i − 0.473602i
$$643$$ 4.00000i 0.157745i 0.996885 + 0.0788723i $$0.0251319\pi$$
−0.996885 + 0.0788723i $$0.974868\pi$$
$$644$$ 0 0
$$645$$ 4.00000i 0.157500i
$$646$$ 24.0000 0.944267
$$647$$ −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
$$651$$ −32.0000 −1.25418
$$652$$ 4.00000i 0.156652i
$$653$$ 18.0000 0.704394 0.352197 0.935926i $$-0.385435\pi$$
0.352197 + 0.935926i $$0.385435\pi$$
$$654$$ −10.0000 −0.391031
$$655$$ 0 0
$$656$$ 6.00000i 0.234261i
$$657$$ 2.00000i 0.0780274i
$$658$$ 0 0
$$659$$ 48.0000 1.86981 0.934907 0.354892i $$-0.115482\pi$$
0.934907 + 0.354892i $$0.115482\pi$$
$$660$$ 0 0
$$661$$ 14.0000i 0.544537i 0.962221 + 0.272268i $$0.0877739\pi$$
−0.962221 + 0.272268i $$0.912226\pi$$
$$662$$ −28.0000 −1.08825
$$663$$ 0 0
$$664$$ −12.0000 −0.465690
$$665$$ 16.0000i 0.620453i
$$666$$ −2.00000 −0.0774984
$$667$$ 0 0
$$668$$ 0 0
$$669$$ − 20.0000i − 0.773245i
$$670$$ − 4.00000i − 0.154533i
$$671$$ 0 0
$$672$$ 4.00000 0.154303
$$673$$ −26.0000 −1.00223 −0.501113 0.865382i $$-0.667076\pi$$
−0.501113 + 0.865382i $$0.667076\pi$$
$$674$$ − 26.0000i − 1.00148i
$$675$$ −1.00000 −0.0384900
$$676$$ 0 0
$$677$$ −6.00000 −0.230599 −0.115299 0.993331i $$-0.536783\pi$$
−0.115299 + 0.993331i $$0.536783\pi$$
$$678$$ − 18.0000i − 0.691286i
$$679$$ −8.00000 −0.307012
$$680$$ −6.00000 −0.230089
$$681$$ 12.0000i 0.459841i
$$682$$ 0 0
$$683$$ − 12.0000i − 0.459167i −0.973289 0.229584i $$-0.926264\pi$$
0.973289 0.229584i $$-0.0737364\pi$$
$$684$$ − 4.00000i − 0.152944i
$$685$$ −6.00000 −0.229248
$$686$$ −8.00000 −0.305441
$$687$$ − 10.0000i − 0.381524i
$$688$$ 4.00000 0.152499
$$689$$ 0 0
$$690$$ 0 0
$$691$$ − 44.0000i − 1.67384i −0.547326 0.836919i $$-0.684354\pi$$
0.547326 0.836919i $$-0.315646\pi$$
$$692$$ 18.0000 0.684257
$$693$$ 0 0
$$694$$ 12.0000i 0.455514i
$$695$$ − 4.00000i − 0.151729i
$$696$$ 6.00000i 0.227429i
$$697$$ − 36.0000i − 1.36360i
$$698$$ 10.0000 0.378506
$$699$$ 18.0000 0.680823
$$700$$ − 4.00000i − 0.151186i
$$701$$ 6.00000 0.226617 0.113308 0.993560i $$-0.463855\pi$$
0.113308 + 0.993560i $$0.463855\pi$$
$$702$$ 0 0
$$703$$ −8.00000 −0.301726
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 6.00000 0.225813
$$707$$ 72.0000i 2.70784i
$$708$$ 0 0
$$709$$ 38.0000i 1.42712i 0.700594 + 0.713560i $$0.252918\pi$$
−0.700594 + 0.713560i $$0.747082\pi$$
$$710$$ 0 0
$$711$$ 8.00000 0.300023
$$712$$ 18.0000 0.674579
$$713$$ 0 0
$$714$$ −24.0000 −0.898177
$$715$$ 0 0
$$716$$ 24.0000 0.896922
$$717$$ − 24.0000i − 0.896296i
$$718$$ 24.0000 0.895672
$$719$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$720$$ 1.00000i 0.0372678i
$$721$$ − 16.0000i − 0.595871i
$$722$$ 3.00000i 0.111648i
$$723$$ 2.00000i 0.0743808i
$$724$$ 14.0000 0.520306
$$725$$ 6.00000 0.222834
$$726$$ 11.0000i 0.408248i
$$727$$ 28.0000 1.03846 0.519231 0.854634i $$-0.326218\pi$$
0.519231 + 0.854634i $$0.326218\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ − 2.00000i − 0.0740233i
$$731$$ −24.0000 −0.887672
$$732$$ 10.0000 0.369611
$$733$$ 22.0000i 0.812589i 0.913742 + 0.406294i $$0.133179\pi$$
−0.913742 + 0.406294i $$0.866821\pi$$
$$734$$ − 28.0000i − 1.03350i
$$735$$ − 9.00000i − 0.331970i
$$736$$ 0 0
$$737$$ 0 0
$$738$$ −6.00000 −0.220863
$$739$$ − 52.0000i − 1.91285i −0.291977 0.956425i $$-0.594313\pi$$
0.291977 0.956425i $$-0.405687\pi$$
$$740$$ 2.00000 0.0735215
$$741$$ 0 0
$$742$$ −24.0000 −0.881068
$$743$$ 24.0000i 0.880475i 0.897881 + 0.440237i $$0.145106\pi$$
−0.897881 + 0.440237i $$0.854894\pi$$
$$744$$ −8.00000 −0.293294
$$745$$ −6.00000 −0.219823
$$746$$ 26.0000i 0.951928i
$$747$$ − 12.0000i − 0.439057i
$$748$$ 0 0
$$749$$ 48.0000i 1.75388i
$$750$$ 1.00000 0.0365148
$$751$$ 40.0000 1.45962 0.729810 0.683650i $$-0.239608\pi$$
0.729810 + 0.683650i $$0.239608\pi$$
$$752$$ 0 0
$$753$$ 24.0000 0.874609
$$754$$ 0 0
$$755$$ −8.00000 −0.291150
$$756$$ 4.00000i 0.145479i
$$757$$ 2.00000 0.0726912 0.0363456 0.999339i $$-0.488428\pi$$
0.0363456 + 0.999339i $$0.488428\pi$$
$$758$$ −4.00000 −0.145287
$$759$$ 0 0
$$760$$ 4.00000i 0.145095i
$$761$$ 18.0000i 0.652499i 0.945284 + 0.326250i $$0.105785\pi$$
−0.945284 + 0.326250i $$0.894215\pi$$
$$762$$ − 20.0000i − 0.724524i
$$763$$ 40.0000 1.44810
$$764$$ 24.0000 0.868290
$$765$$ − 6.00000i − 0.216930i
$$766$$ 0 0
$$767$$ 0 0
$$768$$ 1.00000 0.0360844
$$769$$ − 2.00000i − 0.0721218i −0.999350 0.0360609i $$-0.988519\pi$$
0.999350 0.0360609i $$-0.0114810\pi$$
$$770$$ 0 0
$$771$$ 18.0000 0.648254
$$772$$ 22.0000i 0.791797i
$$773$$ − 42.0000i − 1.51064i −0.655359 0.755318i $$-0.727483\pi$$
0.655359 0.755318i $$-0.272517\pi$$
$$774$$ 4.00000i 0.143777i
$$775$$ 8.00000i 0.287368i
$$776$$ −2.00000 −0.0717958
$$777$$ 8.00000 0.286998
$$778$$ 6.00000i 0.215110i
$$779$$ −24.0000 −0.859889
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ −6.00000 −0.214423
$$784$$ −9.00000 −0.321429
$$785$$ 2.00000i 0.0713831i
$$786$$ 0 0
$$787$$ − 4.00000i − 0.142585i −0.997455 0.0712923i $$-0.977288\pi$$
0.997455 0.0712923i $$-0.0227123\pi$$
$$788$$ − 6.00000i − 0.213741i
$$789$$ 0 0
$$790$$ −8.00000 −0.284627
$$791$$ 72.0000i 2.56003i
$$792$$ 0 0
$$793$$ 0 0
$$794$$ 22.0000 0.780751
$$795$$ − 6.00000i − 0.212798i
$$796$$ 8.00000 0.283552
$$797$$ 30.0000 1.06265 0.531327 0.847167i $$-0.321693\pi$$
0.531327 + 0.847167i $$0.321693\pi$$
$$798$$ 16.0000i 0.566394i
$$799$$ 0 0
$$800$$ − 1.00000i − 0.0353553i
$$801$$ 18.0000i 0.635999i
$$802$$ 6.00000 0.211867
$$803$$ 0 0
$$804$$ − 4.00000i − 0.141069i
$$805$$ 0 0
$$806$$ 0 0
$$807$$ −6.00000 −0.211210
$$808$$ 18.0000i 0.633238i
$$809$$ −54.0000 −1.89854 −0.949269 0.314464i $$-0.898175\pi$$
−0.949269 + 0.314464i $$0.898175\pi$$
$$810$$ −1.00000 −0.0351364
$$811$$ 4.00000i 0.140459i 0.997531 + 0.0702295i $$0.0223732\pi$$
−0.997531 + 0.0702295i $$0.977627\pi$$
$$812$$ − 24.0000i − 0.842235i
$$813$$ − 16.0000i − 0.561144i
$$814$$ 0 0
$$815$$ 4.00000 0.140114
$$816$$ −6.00000 −0.210042
$$817$$ 16.0000i 0.559769i
$$818$$ 26.0000 0.909069
$$819$$ 0 0
$$820$$ 6.00000 0.209529
$$821$$ − 18.0000i − 0.628204i −0.949389 0.314102i $$-0.898297\pi$$
0.949389 0.314102i $$-0.101703\pi$$
$$822$$ −6.00000 −0.209274
$$823$$ −20.0000 −0.697156 −0.348578 0.937280i $$-0.613335\pi$$
−0.348578 + 0.937280i $$0.613335\pi$$
$$824$$ − 4.00000i − 0.139347i
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 12.0000i 0.417281i 0.977992 + 0.208640i $$0.0669038\pi$$
−0.977992 + 0.208640i $$0.933096\pi$$
$$828$$ 0 0
$$829$$ −38.0000 −1.31979 −0.659897 0.751356i $$-0.729400\pi$$
−0.659897 + 0.751356i $$0.729400\pi$$
$$830$$ 12.0000i 0.416526i
$$831$$ −2.00000 −0.0693792
$$832$$ 0 0
$$833$$ 54.0000 1.87099
$$834$$ − 4.00000i − 0.138509i
$$835$$ 0 0
$$836$$ 0 0
$$837$$ − 8.00000i − 0.276520i
$$838$$ 0 0
$$839$$ 24.0000i 0.828572i 0.910147 + 0.414286i $$0.135969\pi$$
−0.910147 + 0.414286i $$0.864031\pi$$
$$840$$ − 4.00000i − 0.138013i
$$841$$ 7.00000 0.241379
$$842$$ −10.0000 −0.344623
$$843$$ 18.0000i 0.619953i
$$844$$ −20.0000 −0.688428
$$845$$ 0 0
$$846$$ 0 0
$$847$$ − 44.0000i − 1.51186i
$$848$$ −6.00000 −0.206041
$$849$$ 28.0000 0.960958
$$850$$ 6.00000i 0.205798i
$$851$$ 0 0
$$852$$ 0 0
$$853$$ − 46.0000i − 1.57501i −0.616308 0.787505i $$-0.711372\pi$$
0.616308 0.787505i $$-0.288628\pi$$
$$854$$ −40.0000 −1.36877
$$855$$ −4.00000 −0.136797
$$856$$ 12.0000i 0.410152i
$$857$$ 18.0000 0.614868 0.307434 0.951569i $$-0.400530\pi$$
0.307434 + 0.951569i $$0.400530\pi$$
$$858$$ 0 0
$$859$$ −4.00000 −0.136478 −0.0682391 0.997669i $$-0.521738\pi$$
−0.0682391 + 0.997669i $$0.521738\pi$$
$$860$$ − 4.00000i − 0.136399i
$$861$$ 24.0000 0.817918
$$862$$ 0 0
$$863$$ − 24.0000i − 0.816970i −0.912765 0.408485i $$-0.866057\pi$$
0.912765 0.408485i $$-0.133943\pi$$
$$864$$ 1.00000i 0.0340207i
$$865$$ − 18.0000i − 0.612018i
$$866$$ − 26.0000i − 0.883516i
$$867$$ 19.0000 0.645274
$$868$$ 32.0000 1.08615
$$869$$ 0 0
$$870$$ 6.00000 0.203419
$$871$$ 0 0
$$872$$ 10.0000 0.338643
$$873$$ − 2.00000i − 0.0676897i
$$874$$ 0 0
$$875$$ −4.00000 −0.135225
$$876$$ − 2.00000i − 0.0675737i
$$877$$ − 2.00000i − 0.0675352i −0.999430 0.0337676i $$-0.989249\pi$$
0.999430 0.0337676i $$-0.0107506\pi$$
$$878$$ − 8.00000i − 0.269987i
$$879$$ − 6.00000i − 0.202375i
$$880$$ 0 0
$$881$$ 54.0000 1.81931 0.909653 0.415369i $$-0.136347\pi$$
0.909653 + 0.415369i $$0.136347\pi$$
$$882$$ − 9.00000i − 0.303046i
$$883$$ 4.00000 0.134611 0.0673054 0.997732i $$-0.478560\pi$$
0.0673054 + 0.997732i $$0.478560\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 12.0000i 0.403148i
$$887$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$888$$ 2.00000 0.0671156
$$889$$ 80.0000i 2.68311i
$$890$$ − 18.0000i − 0.603361i
$$891$$ 0 0
$$892$$ 20.0000i 0.669650i
$$893$$ 0 0
$$894$$ −6.00000 −0.200670
$$895$$ − 24.0000i − 0.802232i
$$896$$ −4.00000 −0.133631
$$897$$ 0 0
$$898$$ 6.00000 0.200223
$$899$$ 48.0000i 1.60089i
$$900$$ 1.00000 0.0333333
$$901$$ 36.0000 1.19933
$$902$$ 0 0
$$903$$ − 16.0000i − 0.532447i
$$904$$ 18.0000i 0.598671i
$$905$$ − 14.0000i − 0.465376i
$$906$$ −8.00000 −0.265782
$$907$$ −44.0000 −1.46100 −0.730498 0.682915i $$-0.760712\pi$$
−0.730498 + 0.682915i $$0.760712\pi$$
$$908$$ − 12.0000i − 0.398234i
$$909$$ −18.0000 −0.597022
$$910$$ 0 0
$$911$$ −48.0000 −1.59031 −0.795155 0.606406i $$-0.792611\pi$$
−0.795155 + 0.606406i $$0.792611\pi$$
$$912$$ 4.00000i 0.132453i
$$913$$ 0 0
$$914$$ 26.0000 0.860004
$$915$$ − 10.0000i − 0.330590i
$$916$$ 10.0000i 0.330409i
$$917$$ 0 0
$$918$$ − 6.00000i − 0.198030i
$$919$$ −16.0000 −0.527791 −0.263896 0.964551i $$-0.585007\pi$$
−0.263896 + 0.964551i $$0.585007\pi$$
$$920$$ 0 0
$$921$$ 20.0000i 0.659022i
$$922$$ −30.0000 −0.987997
$$923$$ 0 0
$$924$$ 0 0
$$925$$ − 2.00000i − 0.0657596i
$$926$$ 4.00000 0.131448
$$927$$ 4.00000 0.131377
$$928$$ − 6.00000i − 0.196960i
$$929$$ 6.00000i 0.196854i 0.995144 + 0.0984268i $$0.0313810\pi$$
−0.995144 + 0.0984268i $$0.968619\pi$$
$$930$$ 8.00000i 0.262330i
$$931$$ − 36.0000i − 1.17985i
$$932$$ −18.0000 −0.589610
$$933$$ 0 0
$$934$$ 36.0000i 1.17796i
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 26.0000 0.849383 0.424691 0.905338i $$-0.360383\pi$$
0.424691 + 0.905338i $$0.360383\pi$$
$$938$$ 16.0000i 0.522419i
$$939$$ 2.00000 0.0652675
$$940$$ 0 0
$$941$$ − 18.0000i − 0.586783i −0.955992 0.293392i $$-0.905216\pi$$
0.955992 0.293392i $$-0.0947840\pi$$
$$942$$ 2.00000i 0.0651635i
$$943$$ 0 0
$$944$$ 0 0
$$945$$ 4.00000 0.130120
$$946$$ 0 0
$$947$$ 36.0000i 1.16984i 0.811090 + 0.584921i $$0.198875\pi$$
−0.811090 + 0.584921i $$0.801125\pi$$
$$948$$ −8.00000 −0.259828
$$949$$ 0 0
$$950$$ 4.00000 0.129777
$$951$$ − 18.0000i − 0.583690i
$$952$$ 24.0000 0.777844
$$953$$ −6.00000 −0.194359 −0.0971795 0.995267i $$-0.530982\pi$$
−0.0971795 + 0.995267i $$0.530982\pi$$
$$954$$ − 6.00000i − 0.194257i
$$955$$ − 24.0000i − 0.776622i
$$956$$ 24.0000i 0.776215i
$$957$$ 0 0
$$958$$ 24.0000 0.775405
$$959$$ 24.0000 0.775000
$$960$$ − 1.00000i − 0.0322749i
$$961$$ −33.0000 −1.06452
$$962$$ 0 0
$$963$$ −12.0000 −0.386695
$$964$$ − 2.00000i − 0.0644157i
$$965$$ 22.0000 0.708205
$$966$$ 0 0
$$967$$ 4.00000i 0.128631i 0.997930 + 0.0643157i $$0.0204865\pi$$
−0.997930 + 0.0643157i $$0.979514\pi$$
$$968$$ − 11.0000i − 0.353553i
$$969$$ − 24.0000i − 0.770991i
$$970$$ 2.00000i 0.0642161i
$$971$$ 24.0000 0.770197 0.385098 0.922876i $$-0.374168\pi$$
0.385098 + 0.922876i $$0.374168\pi$$
$$972$$ −1.00000 −0.0320750
$$973$$ 16.0000i 0.512936i
$$974$$ −28.0000 −0.897178
$$975$$ 0 0
$$976$$ −10.0000 −0.320092
$$977$$ 42.0000i 1.34370i 0.740688 + 0.671850i $$0.234500\pi$$
−0.740688 + 0.671850i $$0.765500\pi$$
$$978$$ 4.00000 0.127906
$$979$$ 0 0
$$980$$ 9.00000i 0.287494i
$$981$$ 10.0000i 0.319275i
$$982$$ − 24.0000i − 0.765871i
$$983$$ − 24.0000i − 0.765481i −0.923856 0.382741i $$-0.874980\pi$$
0.923856 0.382741i $$-0.125020\pi$$
$$984$$ 6.00000 0.191273
$$985$$ −6.00000 −0.191176
$$986$$ 36.0000i 1.14647i
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 0 0
$$990$$ 0 0
$$991$$ −16.0000 −0.508257 −0.254128 0.967170i $$-0.581789\pi$$
−0.254128 + 0.967170i $$0.581789\pi$$
$$992$$ 8.00000 0.254000
$$993$$ 28.0000i 0.888553i
$$994$$ 0 0
$$995$$ − 8.00000i − 0.253617i
$$996$$ 12.0000i 0.380235i
$$997$$ 26.0000 0.823428 0.411714 0.911313i $$-0.364930\pi$$
0.411714 + 0.911313i $$0.364930\pi$$
$$998$$ −4.00000 −0.126618
$$999$$ 2.00000i 0.0632772i
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 5070.2.b.k.1351.2 2
13.5 odd 4 5070.2.a.w.1.1 1
13.8 odd 4 30.2.a.a.1.1 1
13.12 even 2 inner 5070.2.b.k.1351.1 2
39.8 even 4 90.2.a.c.1.1 1
52.47 even 4 240.2.a.b.1.1 1
65.8 even 4 150.2.c.a.49.2 2
65.34 odd 4 150.2.a.b.1.1 1
65.47 even 4 150.2.c.a.49.1 2
91.34 even 4 1470.2.a.d.1.1 1
91.47 even 12 1470.2.i.q.361.1 2
91.60 odd 12 1470.2.i.o.961.1 2
91.73 even 12 1470.2.i.q.961.1 2
91.86 odd 12 1470.2.i.o.361.1 2
104.21 odd 4 960.2.a.e.1.1 1
104.99 even 4 960.2.a.p.1.1 1
117.34 odd 12 810.2.e.l.271.1 2
117.47 even 12 810.2.e.b.271.1 2
117.86 even 12 810.2.e.b.541.1 2
117.112 odd 12 810.2.e.l.541.1 2
143.21 even 4 3630.2.a.w.1.1 1
156.47 odd 4 720.2.a.j.1.1 1
195.8 odd 4 450.2.c.b.199.1 2
195.47 odd 4 450.2.c.b.199.2 2
195.164 even 4 450.2.a.d.1.1 1
208.21 odd 4 3840.2.k.y.1921.1 2
208.99 even 4 3840.2.k.f.1921.1 2
208.125 odd 4 3840.2.k.y.1921.2 2
208.203 even 4 3840.2.k.f.1921.2 2
221.203 odd 4 8670.2.a.g.1.1 1
260.47 odd 4 1200.2.f.e.49.2 2
260.99 even 4 1200.2.a.k.1.1 1
260.203 odd 4 1200.2.f.e.49.1 2
273.125 odd 4 4410.2.a.z.1.1 1
312.125 even 4 2880.2.a.a.1.1 1
312.203 odd 4 2880.2.a.q.1.1 1
455.34 even 4 7350.2.a.ct.1.1 1
520.99 even 4 4800.2.a.d.1.1 1
520.203 odd 4 4800.2.f.w.3649.2 2
520.229 odd 4 4800.2.a.cq.1.1 1
520.307 odd 4 4800.2.f.w.3649.1 2
520.333 even 4 4800.2.f.p.3649.1 2
520.437 even 4 4800.2.f.p.3649.2 2
780.47 even 4 3600.2.f.i.2449.2 2
780.203 even 4 3600.2.f.i.2449.1 2
780.359 odd 4 3600.2.a.f.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
30.2.a.a.1.1 1 13.8 odd 4
90.2.a.c.1.1 1 39.8 even 4
150.2.a.b.1.1 1 65.34 odd 4
150.2.c.a.49.1 2 65.47 even 4
150.2.c.a.49.2 2 65.8 even 4
240.2.a.b.1.1 1 52.47 even 4
450.2.a.d.1.1 1 195.164 even 4
450.2.c.b.199.1 2 195.8 odd 4
450.2.c.b.199.2 2 195.47 odd 4
720.2.a.j.1.1 1 156.47 odd 4
810.2.e.b.271.1 2 117.47 even 12
810.2.e.b.541.1 2 117.86 even 12
810.2.e.l.271.1 2 117.34 odd 12
810.2.e.l.541.1 2 117.112 odd 12
960.2.a.e.1.1 1 104.21 odd 4
960.2.a.p.1.1 1 104.99 even 4
1200.2.a.k.1.1 1 260.99 even 4
1200.2.f.e.49.1 2 260.203 odd 4
1200.2.f.e.49.2 2 260.47 odd 4
1470.2.a.d.1.1 1 91.34 even 4
1470.2.i.o.361.1 2 91.86 odd 12
1470.2.i.o.961.1 2 91.60 odd 12
1470.2.i.q.361.1 2 91.47 even 12
1470.2.i.q.961.1 2 91.73 even 12
2880.2.a.a.1.1 1 312.125 even 4
2880.2.a.q.1.1 1 312.203 odd 4
3600.2.a.f.1.1 1 780.359 odd 4
3600.2.f.i.2449.1 2 780.203 even 4
3600.2.f.i.2449.2 2 780.47 even 4
3630.2.a.w.1.1 1 143.21 even 4
3840.2.k.f.1921.1 2 208.99 even 4
3840.2.k.f.1921.2 2 208.203 even 4
3840.2.k.y.1921.1 2 208.21 odd 4
3840.2.k.y.1921.2 2 208.125 odd 4
4410.2.a.z.1.1 1 273.125 odd 4
4800.2.a.d.1.1 1 520.99 even 4
4800.2.a.cq.1.1 1 520.229 odd 4
4800.2.f.p.3649.1 2 520.333 even 4
4800.2.f.p.3649.2 2 520.437 even 4
4800.2.f.w.3649.1 2 520.307 odd 4
4800.2.f.w.3649.2 2 520.203 odd 4
5070.2.a.w.1.1 1 13.5 odd 4
5070.2.b.k.1351.1 2 13.12 even 2 inner
5070.2.b.k.1351.2 2 1.1 even 1 trivial
7350.2.a.ct.1.1 1 455.34 even 4
8670.2.a.g.1.1 1 221.203 odd 4