# Properties

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

## $q$-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 $$0.272814\pi$$
−0.654654 + 0.755929i $$0.727186\pi$$
$$8$$ 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 $$-0.848268\pi$$
0.888523 0.458831i $$-0.151732\pi$$
$$20$$ 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 $$0.255135\pi$$
−0.695608 + 0.718421i $$0.744865\pi$$
$$32$$ − 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 $$-0.947432\pi$$
0.986394 0.164399i $$-0.0525685\pi$$
$$38$$ −4.00000 −0.648886
$$39$$ 0 0
$$40$$ 1.00000 0.158114
$$41$$ − 6.00000i − 0.937043i −0.883452 0.468521i $$-0.844787\pi$$
0.883452 0.468521i $$-0.155213\pi$$
$$42$$ 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 $$-0.921429\pi$$
0.969690 0.244339i $$-0.0785709\pi$$
$$68$$ 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 $$-0.962659\pi$$
0.993127 0.117041i $$-0.0373409\pi$$
$$74$$ −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 $$0.228845\pi$$
−0.752506 + 0.658586i $$0.771155\pi$$
$$84$$ − 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 $$-0.596924\pi$$
0.299813 0.953998i $$-0.403076\pi$$
$$90$$ −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 $$0.0323753\pi$$
−0.994832 + 0.101535i $$0.967625\pi$$
$$98$$ 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 $$-0.841031\pi$$
0.877862 0.478913i $$-0.158969\pi$$
$$110$$ 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 $$-0.917494\pi$$
0.966595 0.256307i $$-0.0825059\pi$$
$$138$$ 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 $$-0.920959\pi$$
0.969328 0.245770i $$-0.0790407\pi$$
$$150$$ 1.00000i 0.0816497i
$$151$$ − 8.00000i − 0.651031i −0.945537 0.325515i $$-0.894462\pi$$
0.945537 0.325515i $$-0.105538\pi$$
$$152$$ 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 $$0.0500701\pi$$
−0.987654 + 0.156652i $$0.949930\pi$$
$$164$$ 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 $$0.290854\pi$$
−0.610784 + 0.791797i $$0.709146\pi$$
$$194$$ 2.00000 0.143592
$$195$$ 0 0
$$196$$ 9.00000 0.642857
$$197$$ − 6.00000i − 0.427482i −0.976890 0.213741i $$-0.931435\pi$$
0.976890 0.213741i $$-0.0685649\pi$$
$$198$$ 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 $$0.233556\pi$$
−0.742677 + 0.669650i $$0.766444\pi$$
$$224$$ 4.00000 0.267261
$$225$$ −1.00000 −0.0666667
$$226$$ 18.0000i 1.19734i
$$227$$ − 12.0000i − 0.796468i −0.917284 0.398234i $$-0.869623\pi$$
0.917284 0.398234i $$-0.130377\pi$$
$$228$$ 4.00000i 0.264906i
$$229$$ 10.0000i 0.660819i 0.943838 + 0.330409i $$0.107187\pi$$
−0.943838 + 0.330409i $$0.892813\pi$$
$$230$$ 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 $$0.282863\pi$$
−0.630468 + 0.776215i $$0.717137\pi$$
$$240$$ − 1.00000i − 0.0645497i
$$241$$ − 2.00000i − 0.128831i −0.997923 0.0644157i $$-0.979482\pi$$
0.997923 0.0644157i $$-0.0205183\pi$$
$$242$$ − 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 $$0.161532\pi$$
−0.873978 + 0.485965i $$0.838468\pi$$
$$272$$ −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 $$-0.819597\pi$$
0.843649 0.536895i $$-0.180403\pi$$
$$282$$ 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 $$0.0560772\pi$$
−0.984522 + 0.175262i $$0.943923\pi$$
$$294$$ 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 $$-0.806660\pi$$
0.821138 0.570730i $$-0.193340\pi$$
$$308$$ 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 $$0.168688\pi$$
−0.862832 + 0.505490i $$0.831312\pi$$
$$318$$ 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 $$-0.720501\pi$$
0.638635 0.769510i $$-0.279499\pi$$
$$332$$ − 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 $$0.0862451\pi$$
−0.963518 + 0.267644i $$0.913755\pi$$
$$350$$ −4.00000 −0.213809
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 6.00000i 0.319348i 0.987170 + 0.159674i $$0.0510443\pi$$
−0.987170 + 0.159674i $$0.948956\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 18.0000i 0.953998i
$$357$$ − 24.0000i − 1.27021i
$$358$$ 24.0000i 1.26844i
$$359$$ 24.0000i 1.26667i 0.773877 + 0.633336i $$0.218315\pi$$
−0.773877 + 0.633336i $$0.781685\pi$$
$$360$$ 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 $$-0.967241\pi$$
0.994709 0.102733i $$-0.0327588\pi$$
$$380$$ 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 $$0.186163\pi$$
−0.833795 + 0.552074i $$0.813837\pi$$
$$398$$ 8.00000i 0.401004i
$$399$$ 16.0000 0.801002
$$400$$ −1.00000 −0.0500000
$$401$$ 6.00000i 0.299626i 0.988714 + 0.149813i $$0.0478671\pi$$
−0.988714 + 0.149813i $$0.952133\pi$$
$$402$$ −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 $$0.222231\pi$$
−0.766027 + 0.642809i $$0.777769\pi$$
$$410$$ 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 $$-0.921644\pi$$
0.969854 0.243685i $$-0.0783563\pi$$
$$422$$ − 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 $$0.0452178\pi$$
−0.989927 + 0.141579i $$0.954782\pi$$
$$450$$ 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 $$0.208074\pi$$
−0.793849 + 0.608114i $$0.791926\pi$$
$$458$$ 10.0000 0.467269
$$459$$ −6.00000 −0.280056
$$460$$ 0 0
$$461$$ − 30.0000i − 1.39724i −0.715493 0.698620i $$-0.753798\pi$$
0.715493 0.698620i $$-0.246202\pi$$
$$462$$ 0 0
$$463$$ 4.00000i 0.185896i 0.995671 + 0.0929479i $$0.0296290\pi$$
−0.995671 + 0.0929479i $$0.970371\pi$$
$$464$$ −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 $$0.184723\pi$$
−0.836286 + 0.548294i $$0.815277\pi$$
$$480$$ −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 $$-0.781247\pi$$
0.773004 0.634401i $$-0.218753\pi$$
$$488$$ − 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 $$-0.971463\pi$$
0.995984 0.0895323i $$-0.0285372\pi$$
$$500$$ − 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 $$-0.957548\pi$$
0.991120 0.132973i $$-0.0424523\pi$$
$$510$$ 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 $$0.0689643\pi$$
−0.976621 + 0.214967i $$0.931036\pi$$
$$542$$ 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 $$-0.875462\pi$$
0.924434 0.381342i $$-0.124538\pi$$
$$558$$ 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 $$0.0132552\pi$$
−0.999133 + 0.0416305i $$0.986745\pi$$
$$578$$ − 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 $$-0.920343\pi$$
0.968850 0.247647i $$-0.0796572\pi$$
$$588$$ 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 $$-0.788762\pi$$
0.787765 0.615976i $$-0.211238\pi$$
$$594$$ 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 $$0.0128599\pi$$
−0.999184 + 0.0403896i $$0.987140\pi$$
$$614$$ −20.0000 −0.807134
$$615$$ −6.00000 −0.241943
$$616$$ 0 0
$$617$$ 30.0000i 1.20775i 0.797077 + 0.603877i $$0.206378\pi$$
−0.797077 + 0.603877i $$0.793622\pi$$
$$618$$ − 4.00000i − 0.160904i
$$619$$ − 44.0000i − 1.76851i −0.467005 0.884255i $$-0.654667\pi$$
0.467005 0.884255i $$-0.345333\pi$$
$$620$$ −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 $$-0.780196\pi$$
0.770905 0.636950i $$-0.219804\pi$$
$$632$$ 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 $$-0.974868\pi$$
0.996885 0.0788723i $$-0.0251319\pi$$
$$644$$ 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 $$-0.912226\pi$$
0.962221 0.272268i $$-0.0877739\pi$$
$$662$$ −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 $$0.0737364\pi$$
−0.973289 + 0.229584i $$0.926264\pi$$
$$684$$ 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 $$0.315646\pi$$
−0.547326 + 0.836919i $$0.684354\pi$$
$$692$$ 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 $$-0.747082\pi$$
0.700594 0.713560i $$-0.252918\pi$$
$$710$$ 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 $$-0.866821\pi$$
0.913742 0.406294i $$-0.133179\pi$$
$$734$$ 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 $$0.405687\pi$$
−0.291977 + 0.956425i $$0.594313\pi$$
$$740$$ 2.00000 0.0735215
$$741$$ 0 0
$$742$$ −24.0000 −0.881068
$$743$$ − 24.0000i − 0.880475i −0.897881 0.440237i $$-0.854894\pi$$
0.897881 0.440237i $$-0.145106\pi$$
$$744$$ −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 $$-0.894215\pi$$
0.945284 0.326250i $$-0.105785\pi$$
$$762$$ 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 $$0.0114810\pi$$
−0.999350 + 0.0360609i $$0.988519\pi$$
$$770$$ 0 0
$$771$$ 18.0000 0.648254
$$772$$ − 22.0000i − 0.791797i
$$773$$ 42.0000i 1.51064i 0.655359 + 0.755318i $$0.272517\pi$$
−0.655359 + 0.755318i $$0.727483\pi$$
$$774$$ − 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 $$0.0227123\pi$$
−0.997455 + 0.0712923i $$0.977288\pi$$
$$788$$ 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 $$-0.977627\pi$$
0.997531 0.0702295i $$-0.0223732\pi$$
$$812$$ 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 $$0.101703\pi$$
−0.949389 + 0.314102i $$0.898297\pi$$
$$822$$ −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 $$-0.933096\pi$$
0.977992 0.208640i $$-0.0669038\pi$$
$$828$$ 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 $$-0.864031\pi$$
0.910147 0.414286i $$-0.135969\pi$$
$$840$$ 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 $$0.288628\pi$$
−0.616308 + 0.787505i $$0.711372\pi$$
$$854$$ −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 $$0.133943\pi$$
−0.912765 + 0.408485i $$0.866057\pi$$
$$864$$ − 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 $$0.0107506\pi$$
−0.999430 + 0.0337676i $$0.989249\pi$$
$$878$$ 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 $$-0.968619\pi$$
0.995144 0.0984268i $$-0.0313810\pi$$
$$930$$ − 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 $$0.0947840\pi$$
−0.955992 + 0.293392i $$0.905216\pi$$
$$942$$ − 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 $$-0.801125\pi$$
0.811090 0.584921i $$-0.198875\pi$$
$$948$$ −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 $$-0.979514\pi$$
0.997930 0.0643157i $$-0.0204865\pi$$
$$968$$ 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 $$-0.765500\pi$$
0.740688 0.671850i $$-0.234500\pi$$
$$978$$ 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 $$0.125020\pi$$
−0.923856 + 0.382741i $$0.874980\pi$$
$$984$$ 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.1351.1 2
13.5 odd 4 30.2.a.a.1.1 1
13.8 odd 4 5070.2.a.w.1.1 1
13.12 even 2 inner 5070.2.b.k.1351.2 2
39.5 even 4 90.2.a.c.1.1 1
52.31 even 4 240.2.a.b.1.1 1
65.18 even 4 150.2.c.a.49.2 2
65.44 odd 4 150.2.a.b.1.1 1
65.57 even 4 150.2.c.a.49.1 2
91.5 even 12 1470.2.i.q.361.1 2
91.18 odd 12 1470.2.i.o.961.1 2
91.31 even 12 1470.2.i.q.961.1 2
91.44 odd 12 1470.2.i.o.361.1 2
91.83 even 4 1470.2.a.d.1.1 1
104.5 odd 4 960.2.a.e.1.1 1
104.83 even 4 960.2.a.p.1.1 1
117.5 even 12 810.2.e.b.541.1 2
117.31 odd 12 810.2.e.l.541.1 2
117.70 odd 12 810.2.e.l.271.1 2
117.83 even 12 810.2.e.b.271.1 2
143.109 even 4 3630.2.a.w.1.1 1
156.83 odd 4 720.2.a.j.1.1 1
195.44 even 4 450.2.a.d.1.1 1
195.83 odd 4 450.2.c.b.199.1 2
195.122 odd 4 450.2.c.b.199.2 2
208.5 odd 4 3840.2.k.y.1921.1 2
208.83 even 4 3840.2.k.f.1921.1 2
208.109 odd 4 3840.2.k.y.1921.2 2
208.187 even 4 3840.2.k.f.1921.2 2
221.135 odd 4 8670.2.a.g.1.1 1
260.83 odd 4 1200.2.f.e.49.1 2
260.187 odd 4 1200.2.f.e.49.2 2
260.239 even 4 1200.2.a.k.1.1 1
273.83 odd 4 4410.2.a.z.1.1 1
312.5 even 4 2880.2.a.a.1.1 1
312.83 odd 4 2880.2.a.q.1.1 1
455.174 even 4 7350.2.a.ct.1.1 1
520.83 odd 4 4800.2.f.w.3649.2 2
520.109 odd 4 4800.2.a.cq.1.1 1
520.187 odd 4 4800.2.f.w.3649.1 2
520.213 even 4 4800.2.f.p.3649.1 2
520.317 even 4 4800.2.f.p.3649.2 2
520.499 even 4 4800.2.a.d.1.1 1
780.83 even 4 3600.2.f.i.2449.1 2
780.239 odd 4 3600.2.a.f.1.1 1
780.707 even 4 3600.2.f.i.2449.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
30.2.a.a.1.1 1 13.5 odd 4
90.2.a.c.1.1 1 39.5 even 4
150.2.a.b.1.1 1 65.44 odd 4
150.2.c.a.49.1 2 65.57 even 4
150.2.c.a.49.2 2 65.18 even 4
240.2.a.b.1.1 1 52.31 even 4
450.2.a.d.1.1 1 195.44 even 4
450.2.c.b.199.1 2 195.83 odd 4
450.2.c.b.199.2 2 195.122 odd 4
720.2.a.j.1.1 1 156.83 odd 4
810.2.e.b.271.1 2 117.83 even 12
810.2.e.b.541.1 2 117.5 even 12
810.2.e.l.271.1 2 117.70 odd 12
810.2.e.l.541.1 2 117.31 odd 12
960.2.a.e.1.1 1 104.5 odd 4
960.2.a.p.1.1 1 104.83 even 4
1200.2.a.k.1.1 1 260.239 even 4
1200.2.f.e.49.1 2 260.83 odd 4
1200.2.f.e.49.2 2 260.187 odd 4
1470.2.a.d.1.1 1 91.83 even 4
1470.2.i.o.361.1 2 91.44 odd 12
1470.2.i.o.961.1 2 91.18 odd 12
1470.2.i.q.361.1 2 91.5 even 12
1470.2.i.q.961.1 2 91.31 even 12
2880.2.a.a.1.1 1 312.5 even 4
2880.2.a.q.1.1 1 312.83 odd 4
3600.2.a.f.1.1 1 780.239 odd 4
3600.2.f.i.2449.1 2 780.83 even 4
3600.2.f.i.2449.2 2 780.707 even 4
3630.2.a.w.1.1 1 143.109 even 4
3840.2.k.f.1921.1 2 208.83 even 4
3840.2.k.f.1921.2 2 208.187 even 4
3840.2.k.y.1921.1 2 208.5 odd 4
3840.2.k.y.1921.2 2 208.109 odd 4
4410.2.a.z.1.1 1 273.83 odd 4
4800.2.a.d.1.1 1 520.499 even 4
4800.2.a.cq.1.1 1 520.109 odd 4
4800.2.f.p.3649.1 2 520.213 even 4
4800.2.f.p.3649.2 2 520.317 even 4
4800.2.f.w.3649.1 2 520.187 odd 4
4800.2.f.w.3649.2 2 520.83 odd 4
5070.2.a.w.1.1 1 13.8 odd 4
5070.2.b.k.1351.1 2 1.1 even 1 trivial
5070.2.b.k.1351.2 2 13.12 even 2 inner
7350.2.a.ct.1.1 1 455.174 even 4
8670.2.a.g.1.1 1 221.135 odd 4