# Properties

 Label 150.2.c.a.49.2 Level $150$ Weight $2$ Character 150.49 Analytic conductor $1.198$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Learn more about

## Newspace parameters

 Level: $$N$$ $$=$$ $$150 = 2 \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 150.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.19775603032$$ Analytic rank: $$0$$ 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 49.2 Root $$1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 150.49 Dual form 150.2.c.a.49.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +4.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$$ $$q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +4.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} -1.00000i q^{12} +2.00000i q^{13} -4.00000 q^{14} +1.00000 q^{16} -6.00000i q^{17} -1.00000i q^{18} +4.00000 q^{19} -4.00000 q^{21} +1.00000 q^{24} -2.00000 q^{26} -1.00000i q^{27} -4.00000i q^{28} +6.00000 q^{29} +8.00000 q^{31} +1.00000i q^{32} +6.00000 q^{34} +1.00000 q^{36} -2.00000i q^{37} +4.00000i q^{38} -2.00000 q^{39} -6.00000 q^{41} -4.00000i q^{42} -4.00000i q^{43} +1.00000i q^{48} -9.00000 q^{49} +6.00000 q^{51} -2.00000i q^{52} -6.00000i q^{53} +1.00000 q^{54} +4.00000 q^{56} +4.00000i q^{57} +6.00000i q^{58} -10.0000 q^{61} +8.00000i q^{62} -4.00000i q^{63} -1.00000 q^{64} +4.00000i q^{67} +6.00000i q^{68} +1.00000i q^{72} +2.00000i q^{73} +2.00000 q^{74} -4.00000 q^{76} -2.00000i q^{78} -8.00000 q^{79} +1.00000 q^{81} -6.00000i q^{82} +12.0000i q^{83} +4.00000 q^{84} +4.00000 q^{86} +6.00000i q^{87} -18.0000 q^{89} -8.00000 q^{91} +8.00000i q^{93} -1.00000 q^{96} -2.00000i q^{97} -9.00000i q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{4} - 2q^{6} - 2q^{9} + O(q^{10})$$ $$2q - 2q^{4} - 2q^{6} - 2q^{9} - 8q^{14} + 2q^{16} + 8q^{19} - 8q^{21} + 2q^{24} - 4q^{26} + 12q^{29} + 16q^{31} + 12q^{34} + 2q^{36} - 4q^{39} - 12q^{41} - 18q^{49} + 12q^{51} + 2q^{54} + 8q^{56} - 20q^{61} - 2q^{64} + 4q^{74} - 8q^{76} - 16q^{79} + 2q^{81} + 8q^{84} + 8q^{86} - 36q^{89} - 16q^{91} - 2q^{96} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/150\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$127$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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
$$3$$ 1.00000i 0.577350i
$$4$$ −1.00000 −0.500000
$$5$$ 0 0
$$6$$ −1.00000 −0.408248
$$7$$ 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
$$10$$ 0 0
$$11$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$12$$ − 1.00000i − 0.288675i
$$13$$ 2.00000i 0.554700i 0.960769 + 0.277350i $$0.0894562\pi$$
−0.960769 + 0.277350i $$0.910544\pi$$
$$14$$ −4.00000 −1.06904
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ − 6.00000i − 1.45521i −0.685994 0.727607i $$-0.740633\pi$$
0.685994 0.727607i $$-0.259367\pi$$
$$18$$ − 1.00000i − 0.235702i
$$19$$ 4.00000 0.917663 0.458831 0.888523i $$-0.348268\pi$$
0.458831 + 0.888523i $$0.348268\pi$$
$$20$$ 0 0
$$21$$ −4.00000 −0.872872
$$22$$ 0 0
$$23$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$24$$ 1.00000 0.204124
$$25$$ 0 0
$$26$$ −2.00000 −0.392232
$$27$$ − 1.00000i − 0.192450i
$$28$$ − 4.00000i − 0.755929i
$$29$$ 6.00000 1.11417 0.557086 0.830455i $$-0.311919\pi$$
0.557086 + 0.830455i $$0.311919\pi$$
$$30$$ 0 0
$$31$$ 8.00000 1.43684 0.718421 0.695608i $$-0.244865\pi$$
0.718421 + 0.695608i $$0.244865\pi$$
$$32$$ 1.00000i 0.176777i
$$33$$ 0 0
$$34$$ 6.00000 1.02899
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ − 2.00000i − 0.328798i −0.986394 0.164399i $$-0.947432\pi$$
0.986394 0.164399i $$-0.0525685\pi$$
$$38$$ 4.00000i 0.648886i
$$39$$ −2.00000 −0.320256
$$40$$ 0 0
$$41$$ −6.00000 −0.937043 −0.468521 0.883452i $$-0.655213\pi$$
−0.468521 + 0.883452i $$0.655213\pi$$
$$42$$ − 4.00000i − 0.617213i
$$43$$ − 4.00000i − 0.609994i −0.952353 0.304997i $$-0.901344\pi$$
0.952353 0.304997i $$-0.0986555\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$48$$ 1.00000i 0.144338i
$$49$$ −9.00000 −1.28571
$$50$$ 0 0
$$51$$ 6.00000 0.840168
$$52$$ − 2.00000i − 0.277350i
$$53$$ − 6.00000i − 0.824163i −0.911147 0.412082i $$-0.864802\pi$$
0.911147 0.412082i $$-0.135198\pi$$
$$54$$ 1.00000 0.136083
$$55$$ 0 0
$$56$$ 4.00000 0.534522
$$57$$ 4.00000i 0.529813i
$$58$$ 6.00000i 0.787839i
$$59$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$60$$ 0 0
$$61$$ −10.0000 −1.28037 −0.640184 0.768221i $$-0.721142\pi$$
−0.640184 + 0.768221i $$0.721142\pi$$
$$62$$ 8.00000i 1.01600i
$$63$$ − 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 $$0.921429\pi$$
$$68$$ 6.00000i 0.727607i
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$72$$ 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
$$75$$ 0 0
$$76$$ −4.00000 −0.458831
$$77$$ 0 0
$$78$$ − 2.00000i − 0.226455i
$$79$$ −8.00000 −0.900070 −0.450035 0.893011i $$-0.648589\pi$$
−0.450035 + 0.893011i $$0.648589\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ − 6.00000i − 0.662589i
$$83$$ 12.0000i 1.31717i 0.752506 + 0.658586i $$0.228845\pi$$
−0.752506 + 0.658586i $$0.771155\pi$$
$$84$$ 4.00000 0.436436
$$85$$ 0 0
$$86$$ 4.00000 0.431331
$$87$$ 6.00000i 0.643268i
$$88$$ 0 0
$$89$$ −18.0000 −1.90800 −0.953998 0.299813i $$-0.903076\pi$$
−0.953998 + 0.299813i $$0.903076\pi$$
$$90$$ 0 0
$$91$$ −8.00000 −0.838628
$$92$$ 0 0
$$93$$ 8.00000i 0.829561i
$$94$$ 0 0
$$95$$ 0 0
$$96$$ −1.00000 −0.102062
$$97$$ − 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
$$100$$ 0 0
$$101$$ 18.0000 1.79107 0.895533 0.444994i $$-0.146794\pi$$
0.895533 + 0.444994i $$0.146794\pi$$
$$102$$ 6.00000i 0.594089i
$$103$$ − 4.00000i − 0.394132i −0.980390 0.197066i $$-0.936859\pi$$
0.980390 0.197066i $$-0.0631413\pi$$
$$104$$ 2.00000 0.196116
$$105$$ 0 0
$$106$$ 6.00000 0.582772
$$107$$ 12.0000i 1.16008i 0.814587 + 0.580042i $$0.196964\pi$$
−0.814587 + 0.580042i $$0.803036\pi$$
$$108$$ 1.00000i 0.0962250i
$$109$$ 10.0000 0.957826 0.478913 0.877862i $$-0.341031\pi$$
0.478913 + 0.877862i $$0.341031\pi$$
$$110$$ 0 0
$$111$$ 2.00000 0.189832
$$112$$ 4.00000i 0.377964i
$$113$$ − 18.0000i − 1.69330i −0.532152 0.846649i $$-0.678617\pi$$
0.532152 0.846649i $$-0.321383\pi$$
$$114$$ −4.00000 −0.374634
$$115$$ 0 0
$$116$$ −6.00000 −0.557086
$$117$$ − 2.00000i − 0.184900i
$$118$$ 0 0
$$119$$ 24.0000 2.20008
$$120$$ 0 0
$$121$$ −11.0000 −1.00000
$$122$$ − 10.0000i − 0.905357i
$$123$$ − 6.00000i − 0.541002i
$$124$$ −8.00000 −0.718421
$$125$$ 0 0
$$126$$ 4.00000 0.356348
$$127$$ − 20.0000i − 1.77471i −0.461084 0.887357i $$-0.652539\pi$$
0.461084 0.887357i $$-0.347461\pi$$
$$128$$ − 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
$$133$$ 16.0000i 1.38738i
$$134$$ −4.00000 −0.345547
$$135$$ 0 0
$$136$$ −6.00000 −0.514496
$$137$$ − 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 $$-0.445740\pi$$
0.169638 + 0.985506i $$0.445740\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 0 0
$$144$$ −1.00000 −0.0833333
$$145$$ 0 0
$$146$$ −2.00000 −0.165521
$$147$$ − 9.00000i − 0.742307i
$$148$$ 2.00000i 0.164399i
$$149$$ 6.00000 0.491539 0.245770 0.969328i $$-0.420959\pi$$
0.245770 + 0.969328i $$0.420959\pi$$
$$150$$ 0 0
$$151$$ 8.00000 0.651031 0.325515 0.945537i $$-0.394462\pi$$
0.325515 + 0.945537i $$0.394462\pi$$
$$152$$ − 4.00000i − 0.324443i
$$153$$ 6.00000i 0.485071i
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 2.00000 0.160128
$$157$$ − 2.00000i − 0.159617i −0.996810 0.0798087i $$-0.974569\pi$$
0.996810 0.0798087i $$-0.0254309\pi$$
$$158$$ − 8.00000i − 0.636446i
$$159$$ 6.00000 0.475831
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 1.00000i 0.0785674i
$$163$$ − 4.00000i − 0.313304i −0.987654 0.156652i $$-0.949930\pi$$
0.987654 0.156652i $$-0.0500701\pi$$
$$164$$ 6.00000 0.468521
$$165$$ 0 0
$$166$$ −12.0000 −0.931381
$$167$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$168$$ 4.00000i 0.308607i
$$169$$ 9.00000 0.692308
$$170$$ 0 0
$$171$$ −4.00000 −0.305888
$$172$$ 4.00000i 0.304997i
$$173$$ 18.0000i 1.36851i 0.729241 + 0.684257i $$0.239873\pi$$
−0.729241 + 0.684257i $$0.760127\pi$$
$$174$$ −6.00000 −0.454859
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 0 0
$$178$$ − 18.0000i − 1.34916i
$$179$$ −24.0000 −1.79384 −0.896922 0.442189i $$-0.854202\pi$$
−0.896922 + 0.442189i $$0.854202\pi$$
$$180$$ 0 0
$$181$$ 14.0000 1.04061 0.520306 0.853980i $$-0.325818\pi$$
0.520306 + 0.853980i $$0.325818\pi$$
$$182$$ − 8.00000i − 0.592999i
$$183$$ − 10.0000i − 0.739221i
$$184$$ 0 0
$$185$$ 0 0
$$186$$ −8.00000 −0.586588
$$187$$ 0 0
$$188$$ 0 0
$$189$$ 4.00000 0.290957
$$190$$ 0 0
$$191$$ −24.0000 −1.73658 −0.868290 0.496058i $$-0.834780\pi$$
−0.868290 + 0.496058i $$0.834780\pi$$
$$192$$ − 1.00000i − 0.0721688i
$$193$$ − 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 $$0.591513\pi$$
$$200$$ 0 0
$$201$$ −4.00000 −0.282138
$$202$$ 18.0000i 1.26648i
$$203$$ 24.0000i 1.68447i
$$204$$ −6.00000 −0.420084
$$205$$ 0 0
$$206$$ 4.00000 0.278693
$$207$$ 0 0
$$208$$ 2.00000i 0.138675i
$$209$$ 0 0
$$210$$ 0 0
$$211$$ 20.0000 1.37686 0.688428 0.725304i $$-0.258301\pi$$
0.688428 + 0.725304i $$0.258301\pi$$
$$212$$ 6.00000i 0.412082i
$$213$$ 0 0
$$214$$ −12.0000 −0.820303
$$215$$ 0 0
$$216$$ −1.00000 −0.0680414
$$217$$ 32.0000i 2.17230i
$$218$$ 10.0000i 0.677285i
$$219$$ −2.00000 −0.135147
$$220$$ 0 0
$$221$$ 12.0000 0.807207
$$222$$ 2.00000i 0.134231i
$$223$$ 20.0000i 1.33930i 0.742677 + 0.669650i $$0.233556\pi$$
−0.742677 + 0.669650i $$0.766444\pi$$
$$224$$ −4.00000 −0.267261
$$225$$ 0 0
$$226$$ 18.0000 1.19734
$$227$$ 12.0000i 0.796468i 0.917284 + 0.398234i $$0.130377\pi$$
−0.917284 + 0.398234i $$0.869623\pi$$
$$228$$ − 4.00000i − 0.264906i
$$229$$ 10.0000 0.660819 0.330409 0.943838i $$-0.392813\pi$$
0.330409 + 0.943838i $$0.392813\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ − 6.00000i − 0.393919i
$$233$$ − 18.0000i − 1.17922i −0.807688 0.589610i $$-0.799282\pi$$
0.807688 0.589610i $$-0.200718\pi$$
$$234$$ 2.00000 0.130744
$$235$$ 0 0
$$236$$ 0 0
$$237$$ − 8.00000i − 0.519656i
$$238$$ 24.0000i 1.55569i
$$239$$ −24.0000 −1.55243 −0.776215 0.630468i $$-0.782863\pi$$
−0.776215 + 0.630468i $$0.782863\pi$$
$$240$$ 0 0
$$241$$ 2.00000 0.128831 0.0644157 0.997923i $$-0.479482\pi$$
0.0644157 + 0.997923i $$0.479482\pi$$
$$242$$ − 11.0000i − 0.707107i
$$243$$ 1.00000i 0.0641500i
$$244$$ 10.0000 0.640184
$$245$$ 0 0
$$246$$ 6.00000 0.382546
$$247$$ 8.00000i 0.509028i
$$248$$ − 8.00000i − 0.508001i
$$249$$ −12.0000 −0.760469
$$250$$ 0 0
$$251$$ −24.0000 −1.51487 −0.757433 0.652913i $$-0.773547\pi$$
−0.757433 + 0.652913i $$0.773547\pi$$
$$252$$ 4.00000i 0.251976i
$$253$$ 0 0
$$254$$ 20.0000 1.25491
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 18.0000i 1.12281i 0.827541 + 0.561405i $$0.189739\pi$$
−0.827541 + 0.561405i $$0.810261\pi$$
$$258$$ 4.00000i 0.249029i
$$259$$ 8.00000 0.497096
$$260$$ 0 0
$$261$$ −6.00000 −0.371391
$$262$$ 0 0
$$263$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ −16.0000 −0.981023
$$267$$ − 18.0000i − 1.10158i
$$268$$ − 4.00000i − 0.244339i
$$269$$ 6.00000 0.365826 0.182913 0.983129i $$-0.441447\pi$$
0.182913 + 0.983129i $$0.441447\pi$$
$$270$$ 0 0
$$271$$ −16.0000 −0.971931 −0.485965 0.873978i $$-0.661532\pi$$
−0.485965 + 0.873978i $$0.661532\pi$$
$$272$$ − 6.00000i − 0.363803i
$$273$$ − 8.00000i − 0.484182i
$$274$$ 6.00000 0.362473
$$275$$ 0 0
$$276$$ 0 0
$$277$$ − 2.00000i − 0.120168i −0.998193 0.0600842i $$-0.980863\pi$$
0.998193 0.0600842i $$-0.0191369\pi$$
$$278$$ 4.00000i 0.239904i
$$279$$ −8.00000 −0.478947
$$280$$ 0 0
$$281$$ 18.0000 1.07379 0.536895 0.843649i $$-0.319597\pi$$
0.536895 + 0.843649i $$0.319597\pi$$
$$282$$ 0 0
$$283$$ − 28.0000i − 1.66443i −0.554455 0.832214i $$-0.687073\pi$$
0.554455 0.832214i $$-0.312927\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ − 24.0000i − 1.41668i
$$288$$ − 1.00000i − 0.0589256i
$$289$$ −19.0000 −1.11765
$$290$$ 0 0
$$291$$ 2.00000 0.117242
$$292$$ − 2.00000i − 0.117041i
$$293$$ − 6.00000i − 0.350524i −0.984522 0.175262i $$-0.943923\pi$$
0.984522 0.175262i $$-0.0560772\pi$$
$$294$$ 9.00000 0.524891
$$295$$ 0 0
$$296$$ −2.00000 −0.116248
$$297$$ 0 0
$$298$$ 6.00000i 0.347571i
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 16.0000 0.922225
$$302$$ 8.00000i 0.460348i
$$303$$ 18.0000i 1.03407i
$$304$$ 4.00000 0.229416
$$305$$ 0 0
$$306$$ −6.00000 −0.342997
$$307$$ − 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
$$310$$ 0 0
$$311$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$312$$ 2.00000i 0.113228i
$$313$$ 2.00000i 0.113047i 0.998401 + 0.0565233i $$0.0180015\pi$$
−0.998401 + 0.0565233i $$0.981998\pi$$
$$314$$ 2.00000 0.112867
$$315$$ 0 0
$$316$$ 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
$$320$$ 0 0
$$321$$ −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
$$328$$ 6.00000i 0.331295i
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −28.0000 −1.53902 −0.769510 0.638635i $$-0.779499\pi$$
−0.769510 + 0.638635i $$0.779499\pi$$
$$332$$ − 12.0000i − 0.658586i
$$333$$ 2.00000i 0.109599i
$$334$$ 0 0
$$335$$ 0 0
$$336$$ −4.00000 −0.218218
$$337$$ − 26.0000i − 1.41631i −0.706057 0.708155i $$-0.749528\pi$$
0.706057 0.708155i $$-0.250472\pi$$
$$338$$ 9.00000i 0.489535i
$$339$$ 18.0000 0.977626
$$340$$ 0 0
$$341$$ 0 0
$$342$$ − 4.00000i − 0.216295i
$$343$$ − 8.00000i − 0.431959i
$$344$$ −4.00000 −0.215666
$$345$$ 0 0
$$346$$ −18.0000 −0.967686
$$347$$ − 12.0000i − 0.644194i −0.946707 0.322097i $$-0.895612\pi$$
0.946707 0.322097i $$-0.104388\pi$$
$$348$$ − 6.00000i − 0.321634i
$$349$$ 10.0000 0.535288 0.267644 0.963518i $$-0.413755\pi$$
0.267644 + 0.963518i $$0.413755\pi$$
$$350$$ 0 0
$$351$$ 2.00000 0.106752
$$352$$ 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
$$356$$ 18.0000 0.953998
$$357$$ 24.0000i 1.27021i
$$358$$ − 24.0000i − 1.26844i
$$359$$ 24.0000 1.26667 0.633336 0.773877i $$-0.281685\pi$$
0.633336 + 0.773877i $$0.281685\pi$$
$$360$$ 0 0
$$361$$ −3.00000 −0.157895
$$362$$ 14.0000i 0.735824i
$$363$$ − 11.0000i − 0.577350i
$$364$$ 8.00000 0.419314
$$365$$ 0 0
$$366$$ 10.0000 0.522708
$$367$$ 28.0000i 1.46159i 0.682598 + 0.730794i $$0.260850\pi$$
−0.682598 + 0.730794i $$0.739150\pi$$
$$368$$ 0 0
$$369$$ 6.00000 0.312348
$$370$$ 0 0
$$371$$ 24.0000 1.24602
$$372$$ − 8.00000i − 0.414781i
$$373$$ 26.0000i 1.34623i 0.739538 + 0.673114i $$0.235044\pi$$
−0.739538 + 0.673114i $$0.764956\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 12.0000i 0.618031i
$$378$$ 4.00000i 0.205738i
$$379$$ 4.00000 0.205466 0.102733 0.994709i $$-0.467241\pi$$
0.102733 + 0.994709i $$0.467241\pi$$
$$380$$ 0 0
$$381$$ 20.0000 1.02463
$$382$$ − 24.0000i − 1.22795i
$$383$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$384$$ 1.00000 0.0510310
$$385$$ 0 0
$$386$$ 22.0000 1.11977
$$387$$ 4.00000i 0.203331i
$$388$$ 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
$$395$$ 0 0
$$396$$ 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
$$400$$ 0 0
$$401$$ −6.00000 −0.299626 −0.149813 0.988714i $$-0.547867\pi$$
−0.149813 + 0.988714i $$0.547867\pi$$
$$402$$ − 4.00000i − 0.199502i
$$403$$ 16.0000i 0.797017i
$$404$$ −18.0000 −0.895533
$$405$$ 0 0
$$406$$ −24.0000 −1.19110
$$407$$ 0 0
$$408$$ − 6.00000i − 0.297044i
$$409$$ −26.0000 −1.28562 −0.642809 0.766027i $$-0.722231\pi$$
−0.642809 + 0.766027i $$0.722231\pi$$
$$410$$ 0 0
$$411$$ 6.00000 0.295958
$$412$$ 4.00000i 0.197066i
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 0 0
$$416$$ −2.00000 −0.0980581
$$417$$ 4.00000i 0.195881i
$$418$$ 0 0
$$419$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$420$$ 0 0
$$421$$ −10.0000 −0.487370 −0.243685 0.969854i $$-0.578356\pi$$
−0.243685 + 0.969854i $$0.578356\pi$$
$$422$$ 20.0000i 0.973585i
$$423$$ 0 0
$$424$$ −6.00000 −0.291386
$$425$$ 0 0
$$426$$ 0 0
$$427$$ − 40.0000i − 1.93574i
$$428$$ − 12.0000i − 0.580042i
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$432$$ − 1.00000i − 0.0481125i
$$433$$ 26.0000i 1.24948i 0.780833 + 0.624740i $$0.214795\pi$$
−0.780833 + 0.624740i $$0.785205\pi$$
$$434$$ −32.0000 −1.53605
$$435$$ 0 0
$$436$$ −10.0000 −0.478913
$$437$$ 0 0
$$438$$ − 2.00000i − 0.0955637i
$$439$$ −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
$$442$$ 12.0000i 0.570782i
$$443$$ 12.0000i 0.570137i 0.958507 + 0.285069i $$0.0920164\pi$$
−0.958507 + 0.285069i $$0.907984\pi$$
$$444$$ −2.00000 −0.0949158
$$445$$ 0 0
$$446$$ −20.0000 −0.947027
$$447$$ 6.00000i 0.283790i
$$448$$ − 4.00000i − 0.188982i
$$449$$ 6.00000 0.283158 0.141579 0.989927i $$-0.454782\pi$$
0.141579 + 0.989927i $$0.454782\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 18.0000i 0.846649i
$$453$$ 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 $$-0.208074\pi$$
$$458$$ 10.0000i 0.467269i
$$459$$ −6.00000 −0.280056
$$460$$ 0 0
$$461$$ −30.0000 −1.39724 −0.698620 0.715493i $$-0.746202\pi$$
−0.698620 + 0.715493i $$0.746202\pi$$
$$462$$ 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
$$465$$ 0 0
$$466$$ 18.0000 0.833834
$$467$$ 36.0000i 1.66588i 0.553362 + 0.832941i $$0.313345\pi$$
−0.553362 + 0.832941i $$0.686655\pi$$
$$468$$ 2.00000i 0.0924500i
$$469$$ −16.0000 −0.738811
$$470$$ 0 0
$$471$$ 2.00000 0.0921551
$$472$$ 0 0
$$473$$ 0 0
$$474$$ 8.00000 0.367452
$$475$$ 0 0
$$476$$ −24.0000 −1.10004
$$477$$ 6.00000i 0.274721i
$$478$$ − 24.0000i − 1.09773i
$$479$$ 24.0000 1.09659 0.548294 0.836286i $$-0.315277\pi$$
0.548294 + 0.836286i $$0.315277\pi$$
$$480$$ 0 0
$$481$$ 4.00000 0.182384
$$482$$ 2.00000i 0.0910975i
$$483$$ 0 0
$$484$$ 11.0000 0.500000
$$485$$ 0 0
$$486$$ −1.00000 −0.0453609
$$487$$ 28.0000i 1.26880i 0.773004 + 0.634401i $$0.218753\pi$$
−0.773004 + 0.634401i $$0.781247\pi$$
$$488$$ 10.0000i 0.452679i
$$489$$ 4.00000 0.180886
$$490$$ 0 0
$$491$$ 24.0000 1.08310 0.541552 0.840667i $$-0.317837\pi$$
0.541552 + 0.840667i $$0.317837\pi$$
$$492$$ 6.00000i 0.270501i
$$493$$ − 36.0000i − 1.62136i
$$494$$ −8.00000 −0.359937
$$495$$ 0 0
$$496$$ 8.00000 0.359211
$$497$$ 0 0
$$498$$ − 12.0000i − 0.537733i
$$499$$ 4.00000 0.179065 0.0895323 0.995984i $$-0.471463\pi$$
0.0895323 + 0.995984i $$0.471463\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ − 24.0000i − 1.07117i
$$503$$ 24.0000i 1.07011i 0.844818 + 0.535054i $$0.179709\pi$$
−0.844818 + 0.535054i $$0.820291\pi$$
$$504$$ −4.00000 −0.178174
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 9.00000i 0.399704i
$$508$$ 20.0000i 0.887357i
$$509$$ 6.00000 0.265945 0.132973 0.991120i $$-0.457548\pi$$
0.132973 + 0.991120i $$0.457548\pi$$
$$510$$ 0 0
$$511$$ −8.00000 −0.353899
$$512$$ 1.00000i 0.0441942i
$$513$$ − 4.00000i − 0.176604i
$$514$$ −18.0000 −0.793946
$$515$$ 0 0
$$516$$ −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
$$523$$ 20.0000i 0.874539i 0.899331 + 0.437269i $$0.144054\pi$$
−0.899331 + 0.437269i $$0.855946\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ − 48.0000i − 2.09091i
$$528$$ 0 0
$$529$$ 23.0000 1.00000
$$530$$ 0 0
$$531$$ 0 0
$$532$$ − 16.0000i − 0.693688i
$$533$$ − 12.0000i − 0.519778i
$$534$$ 18.0000 0.778936
$$535$$ 0 0
$$536$$ 4.00000 0.172774
$$537$$ − 24.0000i − 1.03568i
$$538$$ 6.00000i 0.258678i
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −10.0000 −0.429934 −0.214967 0.976621i $$-0.568964\pi$$
−0.214967 + 0.976621i $$0.568964\pi$$
$$542$$ − 16.0000i − 0.687259i
$$543$$ 14.0000i 0.600798i
$$544$$ 6.00000 0.257248
$$545$$ 0 0
$$546$$ 8.00000 0.342368
$$547$$ 28.0000i 1.19719i 0.801050 + 0.598597i $$0.204275\pi$$
−0.801050 + 0.598597i $$0.795725\pi$$
$$548$$ 6.00000i 0.256307i
$$549$$ 10.0000 0.426790
$$550$$ 0 0
$$551$$ 24.0000 1.02243
$$552$$ 0 0
$$553$$ − 32.0000i − 1.36078i
$$554$$ 2.00000 0.0849719
$$555$$ 0 0
$$556$$ −4.00000 −0.169638
$$557$$ − 18.0000i − 0.762684i −0.924434 0.381342i $$-0.875462\pi$$
0.924434 0.381342i $$-0.124538\pi$$
$$558$$ − 8.00000i − 0.338667i
$$559$$ 8.00000 0.338364
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 18.0000i 0.759284i
$$563$$ − 12.0000i − 0.505740i −0.967500 0.252870i $$-0.918626\pi$$
0.967500 0.252870i $$-0.0813744\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 28.0000 1.17693
$$567$$ 4.00000i 0.167984i
$$568$$ 0 0
$$569$$ −18.0000 −0.754599 −0.377300 0.926091i $$-0.623147\pi$$
−0.377300 + 0.926091i $$0.623147\pi$$
$$570$$ 0 0
$$571$$ 20.0000 0.836974 0.418487 0.908223i $$-0.362561\pi$$
0.418487 + 0.908223i $$0.362561\pi$$
$$572$$ 0 0
$$573$$ − 24.0000i − 1.00261i
$$574$$ 24.0000 1.00174
$$575$$ 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
$$579$$ 22.0000 0.914289
$$580$$ 0 0
$$581$$ −48.0000 −1.99138
$$582$$ 2.00000i 0.0829027i
$$583$$ 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 $$0.920343\pi$$
$$588$$ 9.00000i 0.371154i
$$589$$ 32.0000 1.31854
$$590$$ 0 0
$$591$$ −6.00000 −0.246807
$$592$$ − 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
$$595$$ 0 0
$$596$$ −6.00000 −0.245770
$$597$$ − 8.00000i − 0.327418i
$$598$$ 0 0
$$599$$ 24.0000 0.980613 0.490307 0.871550i $$-0.336885\pi$$
0.490307 + 0.871550i $$0.336885\pi$$
$$600$$ 0 0
$$601$$ −22.0000 −0.897399 −0.448699 0.893683i $$-0.648113\pi$$
−0.448699 + 0.893683i $$0.648113\pi$$
$$602$$ 16.0000i 0.652111i
$$603$$ − 4.00000i − 0.162893i
$$604$$ −8.00000 −0.325515
$$605$$ 0 0
$$606$$ −18.0000 −0.731200
$$607$$ 4.00000i 0.162355i 0.996700 + 0.0811775i $$0.0258681\pi$$
−0.996700 + 0.0811775i $$0.974132\pi$$
$$608$$ 4.00000i 0.162221i
$$609$$ −24.0000 −0.972529
$$610$$ 0 0
$$611$$ 0 0
$$612$$ − 6.00000i − 0.242536i
$$613$$ 2.00000i 0.0807792i 0.999184 + 0.0403896i $$0.0128599\pi$$
−0.999184 + 0.0403896i $$0.987140\pi$$
$$614$$ 20.0000 0.807134
$$615$$ 0 0
$$616$$ 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
$$619$$ −44.0000 −1.76851 −0.884255 0.467005i $$-0.845333\pi$$
−0.884255 + 0.467005i $$0.845333\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ − 72.0000i − 2.88462i
$$624$$ −2.00000 −0.0800641
$$625$$ 0 0
$$626$$ −2.00000 −0.0799361
$$627$$ 0 0
$$628$$ 2.00000i 0.0798087i
$$629$$ −12.0000 −0.478471
$$630$$ 0 0
$$631$$ 32.0000 1.27390 0.636950 0.770905i $$-0.280196\pi$$
0.636950 + 0.770905i $$0.280196\pi$$
$$632$$ 8.00000i 0.318223i
$$633$$ 20.0000i 0.794929i
$$634$$ 18.0000 0.714871
$$635$$ 0 0
$$636$$ −6.00000 −0.237915
$$637$$ − 18.0000i − 0.713186i
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −30.0000 −1.18493 −0.592464 0.805597i $$-0.701845\pi$$
−0.592464 + 0.805597i $$0.701845\pi$$
$$642$$ − 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
$$645$$ 0 0
$$646$$ 24.0000 0.944267
$$647$$ − 24.0000i − 0.943537i −0.881722 0.471769i $$-0.843616\pi$$
0.881722 0.471769i $$-0.156384\pi$$
$$648$$ − 1.00000i − 0.0392837i
$$649$$ 0 0
$$650$$ 0 0
$$651$$ −32.0000 −1.25418
$$652$$ 4.00000i 0.156652i
$$653$$ 18.0000i 0.704394i 0.935926 + 0.352197i $$0.114565\pi$$
−0.935926 + 0.352197i $$0.885435\pi$$
$$654$$ −10.0000 −0.391031
$$655$$ 0 0
$$656$$ −6.00000 −0.234261
$$657$$ − 2.00000i − 0.0780274i
$$658$$ 0 0
$$659$$ −48.0000 −1.86981 −0.934907 0.354892i $$-0.884518\pi$$
−0.934907 + 0.354892i $$0.884518\pi$$
$$660$$ 0 0
$$661$$ 14.0000 0.544537 0.272268 0.962221i $$-0.412226\pi$$
0.272268 + 0.962221i $$0.412226\pi$$
$$662$$ − 28.0000i − 1.08825i
$$663$$ 12.0000i 0.466041i
$$664$$ 12.0000 0.465690
$$665$$ 0 0
$$666$$ −2.00000 −0.0774984
$$667$$ 0 0
$$668$$ 0 0
$$669$$ −20.0000 −0.773245
$$670$$ 0 0
$$671$$ 0 0
$$672$$ − 4.00000i − 0.154303i
$$673$$ 26.0000i 1.00223i 0.865382 + 0.501113i $$0.167076\pi$$
−0.865382 + 0.501113i $$0.832924\pi$$
$$674$$ 26.0000 1.00148
$$675$$ 0 0
$$676$$ −9.00000 −0.346154
$$677$$ 6.00000i 0.230599i 0.993331 + 0.115299i $$0.0367827\pi$$
−0.993331 + 0.115299i $$0.963217\pi$$
$$678$$ 18.0000i 0.691286i
$$679$$ 8.00000 0.307012
$$680$$ 0 0
$$681$$ −12.0000 −0.459841
$$682$$ 0 0
$$683$$ − 12.0000i − 0.459167i −0.973289 0.229584i $$-0.926264\pi$$
0.973289 0.229584i $$-0.0737364\pi$$
$$684$$ 4.00000 0.152944
$$685$$ 0 0
$$686$$ 8.00000 0.305441
$$687$$ 10.0000i 0.381524i
$$688$$ − 4.00000i − 0.152499i
$$689$$ 12.0000 0.457164
$$690$$ 0 0
$$691$$ 44.0000 1.67384 0.836919 0.547326i $$-0.184354\pi$$
0.836919 + 0.547326i $$0.184354\pi$$
$$692$$ − 18.0000i − 0.684257i
$$693$$ 0 0
$$694$$ 12.0000 0.455514
$$695$$ 0 0
$$696$$ 6.00000 0.227429
$$697$$ 36.0000i 1.36360i
$$698$$ 10.0000i 0.378506i
$$699$$ 18.0000 0.680823
$$700$$ 0 0
$$701$$ −6.00000 −0.226617 −0.113308 0.993560i $$-0.536145\pi$$
−0.113308 + 0.993560i $$0.536145\pi$$
$$702$$ 2.00000i 0.0754851i
$$703$$ − 8.00000i − 0.301726i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ −6.00000 −0.225813
$$707$$ 72.0000i 2.70784i
$$708$$ 0 0
$$709$$ −38.0000 −1.42712 −0.713560 0.700594i $$-0.752918\pi$$
−0.713560 + 0.700594i $$0.752918\pi$$
$$710$$ 0 0
$$711$$ 8.00000 0.300023
$$712$$ 18.0000i 0.674579i
$$713$$ 0 0
$$714$$ −24.0000 −0.898177
$$715$$ 0 0
$$716$$ 24.0000 0.896922
$$717$$ − 24.0000i − 0.896296i
$$718$$ 24.0000i 0.895672i
$$719$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$720$$ 0 0
$$721$$ 16.0000 0.595871
$$722$$ − 3.00000i − 0.111648i
$$723$$ 2.00000i 0.0743808i
$$724$$ −14.0000 −0.520306
$$725$$ 0 0
$$726$$ 11.0000 0.408248
$$727$$ 28.0000i 1.03846i 0.854634 + 0.519231i $$0.173782\pi$$
−0.854634 + 0.519231i $$0.826218\pi$$
$$728$$ 8.00000i 0.296500i
$$729$$ −1.00000 −0.0370370
$$730$$ 0 0
$$731$$ −24.0000 −0.887672
$$732$$ 10.0000i 0.369611i
$$733$$ − 22.0000i − 0.812589i −0.913742 0.406294i $$-0.866821\pi$$
0.913742 0.406294i $$-0.133179\pi$$
$$734$$ −28.0000 −1.03350
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 6.00000i 0.220863i
$$739$$ 52.0000 1.91285 0.956425 0.291977i $$-0.0943129\pi$$
0.956425 + 0.291977i $$0.0943129\pi$$
$$740$$ 0 0
$$741$$ −8.00000 −0.293887
$$742$$ 24.0000i 0.881068i
$$743$$ − 24.0000i − 0.880475i −0.897881 0.440237i $$-0.854894\pi$$
0.897881 0.440237i $$-0.145106\pi$$
$$744$$ 8.00000 0.293294
$$745$$ 0 0
$$746$$ −26.0000 −0.951928
$$747$$ − 12.0000i − 0.439057i
$$748$$ 0 0
$$749$$ −48.0000 −1.75388
$$750$$ 0 0
$$751$$ −40.0000 −1.45962 −0.729810 0.683650i $$-0.760392\pi$$
−0.729810 + 0.683650i $$0.760392\pi$$
$$752$$ 0 0
$$753$$ − 24.0000i − 0.874609i
$$754$$ −12.0000 −0.437014
$$755$$ 0 0
$$756$$ −4.00000 −0.145479
$$757$$ − 2.00000i − 0.0726912i −0.999339 0.0363456i $$-0.988428\pi$$
0.999339 0.0363456i $$-0.0115717\pi$$
$$758$$ 4.00000i 0.145287i
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 18.0000 0.652499 0.326250 0.945284i $$-0.394215\pi$$
0.326250 + 0.945284i $$0.394215\pi$$
$$762$$ 20.0000i 0.724524i
$$763$$ 40.0000i 1.44810i
$$764$$ 24.0000 0.868290
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 0 0
$$768$$ 1.00000i 0.0360844i
$$769$$ −2.00000 −0.0721218 −0.0360609 0.999350i $$-0.511481\pi$$
−0.0360609 + 0.999350i $$0.511481\pi$$
$$770$$ 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 $$0.727483\pi$$
$$774$$ −4.00000 −0.143777
$$775$$ 0 0
$$776$$ −2.00000 −0.0717958
$$777$$ 8.00000i 0.286998i
$$778$$ 6.00000i 0.215110i
$$779$$ −24.0000 −0.859889
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ − 6.00000i − 0.214423i
$$784$$ −9.00000 −0.321429
$$785$$ 0 0
$$786$$ 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
$$790$$ 0 0
$$791$$ 72.0000 2.56003
$$792$$ 0 0
$$793$$ − 20.0000i − 0.710221i
$$794$$ −22.0000 −0.780751
$$795$$ 0 0
$$796$$ 8.00000 0.283552
$$797$$ 30.0000i 1.06265i 0.847167 + 0.531327i $$0.178307\pi$$
−0.847167 + 0.531327i $$0.821693\pi$$
$$798$$ − 16.0000i − 0.566394i
$$799$$ 0 0
$$800$$ 0 0
$$801$$ 18.0000 0.635999
$$802$$ − 6.00000i − 0.211867i
$$803$$ 0 0
$$804$$ 4.00000 0.141069
$$805$$ 0 0
$$806$$ −16.0000 −0.563576
$$807$$ 6.00000i 0.211210i
$$808$$ − 18.0000i − 0.633238i
$$809$$ 54.0000 1.89854 0.949269 0.314464i $$-0.101825\pi$$
0.949269 + 0.314464i $$0.101825\pi$$
$$810$$ 0 0
$$811$$ −4.00000 −0.140459 −0.0702295 0.997531i $$-0.522373\pi$$
−0.0702295 + 0.997531i $$0.522373\pi$$
$$812$$ − 24.0000i − 0.842235i
$$813$$ − 16.0000i − 0.561144i
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 6.00000 0.210042
$$817$$ − 16.0000i − 0.559769i
$$818$$ − 26.0000i − 0.909069i
$$819$$ 8.00000 0.279543
$$820$$ 0 0
$$821$$ 18.0000 0.628204 0.314102 0.949389i $$-0.398297\pi$$
0.314102 + 0.949389i $$0.398297\pi$$
$$822$$ 6.00000i 0.209274i
$$823$$ 20.0000i 0.697156i 0.937280 + 0.348578i $$0.113335\pi$$
−0.937280 + 0.348578i $$0.886665\pi$$
$$824$$ −4.00000 −0.139347
$$825$$ 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 $$0.729400\pi$$
$$830$$ 0 0
$$831$$ 2.00000 0.0693792
$$832$$ − 2.00000i − 0.0693375i
$$833$$ 54.0000i 1.87099i
$$834$$ −4.00000 −0.138509
$$835$$ 0 0
$$836$$ 0 0
$$837$$ − 8.00000i − 0.276520i
$$838$$ 0 0
$$839$$ −24.0000 −0.828572 −0.414286 0.910147i $$-0.635969\pi$$
−0.414286 + 0.910147i $$0.635969\pi$$
$$840$$ 0 0
$$841$$ 7.00000 0.241379
$$842$$ − 10.0000i − 0.344623i
$$843$$ 18.0000i 0.619953i
$$844$$ −20.0000 −0.688428
$$845$$ 0 0
$$846$$ 0 0
$$847$$ − 44.0000i − 1.51186i
$$848$$ − 6.00000i − 0.206041i
$$849$$ 28.0000 0.960958
$$850$$ 0 0
$$851$$ 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
$$855$$ 0 0
$$856$$ 12.0000 0.410152
$$857$$ 18.0000i 0.614868i 0.951569 + 0.307434i $$0.0994704\pi$$
−0.951569 + 0.307434i $$0.900530\pi$$
$$858$$ 0 0
$$859$$ 4.00000 0.136478 0.0682391 0.997669i $$-0.478262\pi$$
0.0682391 + 0.997669i $$0.478262\pi$$
$$860$$ 0 0
$$861$$ 24.0000 0.817918
$$862$$ 0 0
$$863$$ 24.0000i 0.816970i 0.912765 + 0.408485i $$0.133943\pi$$
−0.912765 + 0.408485i $$0.866057\pi$$
$$864$$ 1.00000 0.0340207
$$865$$ 0 0
$$866$$ −26.0000 −0.883516
$$867$$ − 19.0000i − 0.645274i
$$868$$ − 32.0000i − 1.08615i
$$869$$ 0 0
$$870$$ 0 0
$$871$$ −8.00000 −0.271070
$$872$$ − 10.0000i − 0.338643i
$$873$$ 2.00000i 0.0676897i
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 2.00000 0.0675737
$$877$$ − 2.00000i − 0.0675352i −0.999430 0.0337676i $$-0.989249\pi$$
0.999430 0.0337676i $$-0.0107506\pi$$
$$878$$ − 8.00000i − 0.269987i
$$879$$ 6.00000 0.202375
$$880$$ 0 0
$$881$$ −54.0000 −1.81931 −0.909653 0.415369i $$-0.863653\pi$$
−0.909653 + 0.415369i $$0.863653\pi$$
$$882$$ 9.00000i 0.303046i
$$883$$ − 4.00000i − 0.134611i −0.997732 0.0673054i $$-0.978560\pi$$
0.997732 0.0673054i $$-0.0214402\pi$$
$$884$$ −12.0000 −0.403604
$$885$$ 0 0
$$886$$ −12.0000 −0.403148
$$887$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$888$$ − 2.00000i − 0.0671156i
$$889$$ 80.0000 2.68311
$$890$$ 0 0
$$891$$ 0 0
$$892$$ − 20.0000i − 0.669650i
$$893$$ 0 0
$$894$$ −6.00000 −0.200670
$$895$$ 0 0
$$896$$ 4.00000 0.133631
$$897$$ 0 0
$$898$$ 6.00000i 0.200223i
$$899$$ 48.0000 1.60089
$$900$$ 0 0
$$901$$ −36.0000 −1.19933
$$902$$ 0 0
$$903$$ 16.0000i 0.532447i
$$904$$ −18.0000 −0.598671
$$905$$ 0 0
$$906$$ −8.00000 −0.265782
$$907$$ − 44.0000i − 1.46100i −0.682915 0.730498i $$-0.739288\pi$$
0.682915 0.730498i $$-0.260712\pi$$
$$908$$ − 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
$$915$$ 0 0
$$916$$ −10.0000 −0.330409
$$917$$ 0 0
$$918$$ − 6.00000i − 0.198030i
$$919$$ 16.0000 0.527791 0.263896 0.964551i $$-0.414993\pi$$
0.263896 + 0.964551i $$0.414993\pi$$
$$920$$ 0 0
$$921$$ 20.0000 0.659022
$$922$$ − 30.0000i − 0.987997i
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 4.00000 0.131448
$$927$$ 4.00000i 0.131377i
$$928$$ 6.00000i 0.196960i
$$929$$ 6.00000 0.196854 0.0984268 0.995144i $$-0.468619\pi$$
0.0984268 + 0.995144i $$0.468619\pi$$
$$930$$ 0 0
$$931$$ −36.0000 −1.17985
$$932$$ 18.0000i 0.589610i
$$933$$ 0 0
$$934$$ −36.0000 −1.17796
$$935$$ 0 0
$$936$$ −2.00000 −0.0653720
$$937$$ − 26.0000i − 0.849383i −0.905338 0.424691i $$-0.860383\pi$$
0.905338 0.424691i $$-0.139617\pi$$
$$938$$ − 16.0000i − 0.522419i
$$939$$ −2.00000 −0.0652675
$$940$$ 0 0
$$941$$ 18.0000 0.586783 0.293392 0.955992i $$-0.405216\pi$$
0.293392 + 0.955992i $$0.405216\pi$$
$$942$$ 2.00000i 0.0651635i
$$943$$ 0 0
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ − 36.0000i − 1.16984i −0.811090 0.584921i $$-0.801125\pi$$
0.811090 0.584921i $$-0.198875\pi$$
$$948$$ 8.00000i 0.259828i
$$949$$ −4.00000 −0.129845
$$950$$ 0 0
$$951$$ 18.0000 0.583690
$$952$$ − 24.0000i − 0.777844i
$$953$$ 6.00000i 0.194359i 0.995267 + 0.0971795i $$0.0309821\pi$$
−0.995267 + 0.0971795i $$0.969018\pi$$
$$954$$ −6.00000 −0.194257
$$955$$ 0 0
$$956$$ 24.0000 0.776215
$$957$$ 0 0
$$958$$ 24.0000i 0.775405i
$$959$$ 24.0000 0.775000
$$960$$ 0 0
$$961$$ 33.0000 1.06452
$$962$$ 4.00000i 0.128965i
$$963$$ − 12.0000i − 0.386695i
$$964$$ −2.00000 −0.0644157
$$965$$ 0 0
$$966$$ 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
$$969$$ 24.0000 0.770991
$$970$$ 0 0
$$971$$ 24.0000 0.770197 0.385098 0.922876i $$-0.374168\pi$$
0.385098 + 0.922876i $$0.374168\pi$$
$$972$$ − 1.00000i − 0.0320750i
$$973$$ 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 $$0.765500\pi$$
$$978$$ 4.00000i 0.127906i
$$979$$ 0 0
$$980$$ 0 0
$$981$$ −10.0000 −0.319275
$$982$$ 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
$$985$$ 0 0
$$986$$ 36.0000 1.14647
$$987$$ 0 0
$$988$$ − 8.00000i − 0.254514i
$$989$$ 0 0
$$990$$ 0 0
$$991$$ −16.0000 −0.508257 −0.254128 0.967170i $$-0.581789\pi$$
−0.254128 + 0.967170i $$0.581789\pi$$
$$992$$ 8.00000i 0.254000i
$$993$$ − 28.0000i − 0.888553i
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 12.0000 0.380235
$$997$$ − 26.0000i − 0.823428i −0.911313 0.411714i $$-0.864930\pi$$
0.911313 0.411714i $$-0.135070\pi$$
$$998$$ 4.00000i 0.126618i
$$999$$ −2.00000 −0.0632772
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 150.2.c.a.49.2 2
3.2 odd 2 450.2.c.b.199.1 2
4.3 odd 2 1200.2.f.e.49.1 2
5.2 odd 4 30.2.a.a.1.1 1
5.3 odd 4 150.2.a.b.1.1 1
5.4 even 2 inner 150.2.c.a.49.1 2
8.3 odd 2 4800.2.f.w.3649.2 2
8.5 even 2 4800.2.f.p.3649.1 2
12.11 even 2 3600.2.f.i.2449.1 2
15.2 even 4 90.2.a.c.1.1 1
15.8 even 4 450.2.a.d.1.1 1
15.14 odd 2 450.2.c.b.199.2 2
20.3 even 4 1200.2.a.k.1.1 1
20.7 even 4 240.2.a.b.1.1 1
20.19 odd 2 1200.2.f.e.49.2 2
35.2 odd 12 1470.2.i.o.361.1 2
35.12 even 12 1470.2.i.q.361.1 2
35.13 even 4 7350.2.a.ct.1.1 1
35.17 even 12 1470.2.i.q.961.1 2
35.27 even 4 1470.2.a.d.1.1 1
35.32 odd 12 1470.2.i.o.961.1 2
40.3 even 4 4800.2.a.d.1.1 1
40.13 odd 4 4800.2.a.cq.1.1 1
40.19 odd 2 4800.2.f.w.3649.1 2
40.27 even 4 960.2.a.p.1.1 1
40.29 even 2 4800.2.f.p.3649.2 2
40.37 odd 4 960.2.a.e.1.1 1
45.2 even 12 810.2.e.b.271.1 2
45.7 odd 12 810.2.e.l.271.1 2
45.22 odd 12 810.2.e.l.541.1 2
45.32 even 12 810.2.e.b.541.1 2
55.32 even 4 3630.2.a.w.1.1 1
60.23 odd 4 3600.2.a.f.1.1 1
60.47 odd 4 720.2.a.j.1.1 1
60.59 even 2 3600.2.f.i.2449.2 2
65.12 odd 4 5070.2.a.w.1.1 1
65.47 even 4 5070.2.b.k.1351.1 2
65.57 even 4 5070.2.b.k.1351.2 2
80.27 even 4 3840.2.k.f.1921.2 2
80.37 odd 4 3840.2.k.y.1921.1 2
80.67 even 4 3840.2.k.f.1921.1 2
80.77 odd 4 3840.2.k.y.1921.2 2
85.67 odd 4 8670.2.a.g.1.1 1
105.62 odd 4 4410.2.a.z.1.1 1
120.77 even 4 2880.2.a.a.1.1 1
120.107 odd 4 2880.2.a.q.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
30.2.a.a.1.1 1 5.2 odd 4
90.2.a.c.1.1 1 15.2 even 4
150.2.a.b.1.1 1 5.3 odd 4
150.2.c.a.49.1 2 5.4 even 2 inner
150.2.c.a.49.2 2 1.1 even 1 trivial
240.2.a.b.1.1 1 20.7 even 4
450.2.a.d.1.1 1 15.8 even 4
450.2.c.b.199.1 2 3.2 odd 2
450.2.c.b.199.2 2 15.14 odd 2
720.2.a.j.1.1 1 60.47 odd 4
810.2.e.b.271.1 2 45.2 even 12
810.2.e.b.541.1 2 45.32 even 12
810.2.e.l.271.1 2 45.7 odd 12
810.2.e.l.541.1 2 45.22 odd 12
960.2.a.e.1.1 1 40.37 odd 4
960.2.a.p.1.1 1 40.27 even 4
1200.2.a.k.1.1 1 20.3 even 4
1200.2.f.e.49.1 2 4.3 odd 2
1200.2.f.e.49.2 2 20.19 odd 2
1470.2.a.d.1.1 1 35.27 even 4
1470.2.i.o.361.1 2 35.2 odd 12
1470.2.i.o.961.1 2 35.32 odd 12
1470.2.i.q.361.1 2 35.12 even 12
1470.2.i.q.961.1 2 35.17 even 12
2880.2.a.a.1.1 1 120.77 even 4
2880.2.a.q.1.1 1 120.107 odd 4
3600.2.a.f.1.1 1 60.23 odd 4
3600.2.f.i.2449.1 2 12.11 even 2
3600.2.f.i.2449.2 2 60.59 even 2
3630.2.a.w.1.1 1 55.32 even 4
3840.2.k.f.1921.1 2 80.67 even 4
3840.2.k.f.1921.2 2 80.27 even 4
3840.2.k.y.1921.1 2 80.37 odd 4
3840.2.k.y.1921.2 2 80.77 odd 4
4410.2.a.z.1.1 1 105.62 odd 4
4800.2.a.d.1.1 1 40.3 even 4
4800.2.a.cq.1.1 1 40.13 odd 4
4800.2.f.p.3649.1 2 8.5 even 2
4800.2.f.p.3649.2 2 40.29 even 2
4800.2.f.w.3649.1 2 40.19 odd 2
4800.2.f.w.3649.2 2 8.3 odd 2
5070.2.a.w.1.1 1 65.12 odd 4
5070.2.b.k.1351.1 2 65.47 even 4
5070.2.b.k.1351.2 2 65.57 even 4
7350.2.a.ct.1.1 1 35.13 even 4
8670.2.a.g.1.1 1 85.67 odd 4