# Properties

 Label 4950.2.c.r.199.1 Level $4950$ Weight $2$ Character 4950.199 Analytic conductor $39.526$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4950 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4950.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$39.5259490005$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 66) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 199.1 Root $$-1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 4950.199 Dual form 4950.2.c.r.199.2

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000i q^{2} -1.00000 q^{4} -2.00000i q^{7} +1.00000i q^{8} +O(q^{10})$$ $$q-1.00000i q^{2} -1.00000 q^{4} -2.00000i q^{7} +1.00000i q^{8} +1.00000 q^{11} -4.00000i q^{13} -2.00000 q^{14} +1.00000 q^{16} -6.00000i q^{17} +4.00000 q^{19} -1.00000i q^{22} -6.00000i q^{23} -4.00000 q^{26} +2.00000i q^{28} +6.00000 q^{29} +8.00000 q^{31} -1.00000i q^{32} -6.00000 q^{34} +10.0000i q^{37} -4.00000i q^{38} -6.00000 q^{41} +8.00000i q^{43} -1.00000 q^{44} -6.00000 q^{46} -6.00000i q^{47} +3.00000 q^{49} +4.00000i q^{52} +2.00000 q^{56} -6.00000i q^{58} +8.00000 q^{61} -8.00000i q^{62} -1.00000 q^{64} +4.00000i q^{67} +6.00000i q^{68} -6.00000 q^{71} +2.00000i q^{73} +10.0000 q^{74} -4.00000 q^{76} -2.00000i q^{77} -14.0000 q^{79} +6.00000i q^{82} +12.0000i q^{83} +8.00000 q^{86} +1.00000i q^{88} -6.00000 q^{89} -8.00000 q^{91} +6.00000i q^{92} -6.00000 q^{94} -14.0000i q^{97} -3.00000i q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{4}+O(q^{10})$$ 2 * q - 2 * q^4 $$2 q - 2 q^{4} + 2 q^{11} - 4 q^{14} + 2 q^{16} + 8 q^{19} - 8 q^{26} + 12 q^{29} + 16 q^{31} - 12 q^{34} - 12 q^{41} - 2 q^{44} - 12 q^{46} + 6 q^{49} + 4 q^{56} + 16 q^{61} - 2 q^{64} - 12 q^{71} + 20 q^{74} - 8 q^{76} - 28 q^{79} + 16 q^{86} - 12 q^{89} - 16 q^{91} - 12 q^{94}+O(q^{100})$$ 2 * q - 2 * q^4 + 2 * q^11 - 4 * q^14 + 2 * q^16 + 8 * q^19 - 8 * q^26 + 12 * q^29 + 16 * q^31 - 12 * q^34 - 12 * q^41 - 2 * q^44 - 12 * q^46 + 6 * q^49 + 4 * q^56 + 16 * q^61 - 2 * q^64 - 12 * q^71 + 20 * q^74 - 8 * q^76 - 28 * q^79 + 16 * q^86 - 12 * q^89 - 16 * q^91 - 12 * q^94

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/4950\mathbb{Z}\right)^\times$$.

 $$n$$ $$551$$ $$2377$$ $$4501$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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
$$3$$ 0 0
$$4$$ −1.00000 −0.500000
$$5$$ 0 0
$$6$$ 0 0
$$7$$ − 2.00000i − 0.755929i −0.925820 0.377964i $$-0.876624\pi$$
0.925820 0.377964i $$-0.123376\pi$$
$$8$$ 1.00000i 0.353553i
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 1.00000 0.301511
$$12$$ 0 0
$$13$$ − 4.00000i − 1.10940i −0.832050 0.554700i $$-0.812833\pi$$
0.832050 0.554700i $$-0.187167\pi$$
$$14$$ −2.00000 −0.534522
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ − 6.00000i − 1.45521i −0.685994 0.727607i $$-0.740633\pi$$
0.685994 0.727607i $$-0.259367\pi$$
$$18$$ 0 0
$$19$$ 4.00000 0.917663 0.458831 0.888523i $$-0.348268\pi$$
0.458831 + 0.888523i $$0.348268\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ − 1.00000i − 0.213201i
$$23$$ − 6.00000i − 1.25109i −0.780189 0.625543i $$-0.784877\pi$$
0.780189 0.625543i $$-0.215123\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ −4.00000 −0.784465
$$27$$ 0 0
$$28$$ 2.00000i 0.377964i
$$29$$ 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
$$36$$ 0 0
$$37$$ 10.0000i 1.64399i 0.569495 + 0.821995i $$0.307139\pi$$
−0.569495 + 0.821995i $$0.692861\pi$$
$$38$$ − 4.00000i − 0.648886i
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −6.00000 −0.937043 −0.468521 0.883452i $$-0.655213\pi$$
−0.468521 + 0.883452i $$0.655213\pi$$
$$42$$ 0 0
$$43$$ 8.00000i 1.21999i 0.792406 + 0.609994i $$0.208828\pi$$
−0.792406 + 0.609994i $$0.791172\pi$$
$$44$$ −1.00000 −0.150756
$$45$$ 0 0
$$46$$ −6.00000 −0.884652
$$47$$ − 6.00000i − 0.875190i −0.899172 0.437595i $$-0.855830\pi$$
0.899172 0.437595i $$-0.144170\pi$$
$$48$$ 0 0
$$49$$ 3.00000 0.428571
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 4.00000i 0.554700i
$$53$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 2.00000 0.267261
$$57$$ 0 0
$$58$$ − 6.00000i − 0.787839i
$$59$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$60$$ 0 0
$$61$$ 8.00000 1.02430 0.512148 0.858898i $$-0.328850\pi$$
0.512148 + 0.858898i $$0.328850\pi$$
$$62$$ − 8.00000i − 1.01600i
$$63$$ 0 0
$$64$$ −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
$$71$$ −6.00000 −0.712069 −0.356034 0.934473i $$-0.615871\pi$$
−0.356034 + 0.934473i $$0.615871\pi$$
$$72$$ 0 0
$$73$$ 2.00000i 0.234082i 0.993127 + 0.117041i $$0.0373409\pi$$
−0.993127 + 0.117041i $$0.962659\pi$$
$$74$$ 10.0000 1.16248
$$75$$ 0 0
$$76$$ −4.00000 −0.458831
$$77$$ − 2.00000i − 0.227921i
$$78$$ 0 0
$$79$$ −14.0000 −1.57512 −0.787562 0.616236i $$-0.788657\pi$$
−0.787562 + 0.616236i $$0.788657\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 6.00000i 0.662589i
$$83$$ 12.0000i 1.31717i 0.752506 + 0.658586i $$0.228845\pi$$
−0.752506 + 0.658586i $$0.771155\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 8.00000 0.862662
$$87$$ 0 0
$$88$$ 1.00000i 0.106600i
$$89$$ −6.00000 −0.635999 −0.317999 0.948091i $$-0.603011\pi$$
−0.317999 + 0.948091i $$0.603011\pi$$
$$90$$ 0 0
$$91$$ −8.00000 −0.838628
$$92$$ 6.00000i 0.625543i
$$93$$ 0 0
$$94$$ −6.00000 −0.618853
$$95$$ 0 0
$$96$$ 0 0
$$97$$ − 14.0000i − 1.42148i −0.703452 0.710742i $$-0.748359\pi$$
0.703452 0.710742i $$-0.251641\pi$$
$$98$$ − 3.00000i − 0.303046i
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −6.00000 −0.597022 −0.298511 0.954406i $$-0.596490\pi$$
−0.298511 + 0.954406i $$0.596490\pi$$
$$102$$ 0 0
$$103$$ − 4.00000i − 0.394132i −0.980390 0.197066i $$-0.936859\pi$$
0.980390 0.197066i $$-0.0631413\pi$$
$$104$$ 4.00000 0.392232
$$105$$ 0 0
$$106$$ 0 0
$$107$$ − 12.0000i − 1.16008i −0.814587 0.580042i $$-0.803036\pi$$
0.814587 0.580042i $$-0.196964\pi$$
$$108$$ 0 0
$$109$$ 4.00000 0.383131 0.191565 0.981480i $$-0.438644\pi$$
0.191565 + 0.981480i $$0.438644\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ − 2.00000i − 0.188982i
$$113$$ − 18.0000i − 1.69330i −0.532152 0.846649i $$-0.678617\pi$$
0.532152 0.846649i $$-0.321383\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ −6.00000 −0.557086
$$117$$ 0 0
$$118$$ 0 0
$$119$$ −12.0000 −1.10004
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ − 8.00000i − 0.724286i
$$123$$ 0 0
$$124$$ −8.00000 −0.718421
$$125$$ 0 0
$$126$$ 0 0
$$127$$ − 14.0000i − 1.24230i −0.783692 0.621150i $$-0.786666\pi$$
0.783692 0.621150i $$-0.213334\pi$$
$$128$$ 1.00000i 0.0883883i
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 12.0000 1.04844 0.524222 0.851581i $$-0.324356\pi$$
0.524222 + 0.851581i $$0.324356\pi$$
$$132$$ 0 0
$$133$$ − 8.00000i − 0.693688i
$$134$$ 4.00000 0.345547
$$135$$ 0 0
$$136$$ 6.00000 0.514496
$$137$$ − 18.0000i − 1.53784i −0.639343 0.768922i $$-0.720793\pi$$
0.639343 0.768922i $$-0.279207\pi$$
$$138$$ 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
$$142$$ 6.00000i 0.503509i
$$143$$ − 4.00000i − 0.334497i
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 2.00000 0.165521
$$147$$ 0 0
$$148$$ − 10.0000i − 0.821995i
$$149$$ −6.00000 −0.491539 −0.245770 0.969328i $$-0.579041\pi$$
−0.245770 + 0.969328i $$0.579041\pi$$
$$150$$ 0 0
$$151$$ −10.0000 −0.813788 −0.406894 0.913475i $$-0.633388\pi$$
−0.406894 + 0.913475i $$0.633388\pi$$
$$152$$ 4.00000i 0.324443i
$$153$$ 0 0
$$154$$ −2.00000 −0.161165
$$155$$ 0 0
$$156$$ 0 0
$$157$$ − 2.00000i − 0.159617i −0.996810 0.0798087i $$-0.974569\pi$$
0.996810 0.0798087i $$-0.0254309\pi$$
$$158$$ 14.0000i 1.11378i
$$159$$ 0 0
$$160$$ 0 0
$$161$$ −12.0000 −0.945732
$$162$$ 0 0
$$163$$ − 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
$$167$$ 12.0000i 0.928588i 0.885681 + 0.464294i $$0.153692\pi$$
−0.885681 + 0.464294i $$0.846308\pi$$
$$168$$ 0 0
$$169$$ −3.00000 −0.230769
$$170$$ 0 0
$$171$$ 0 0
$$172$$ − 8.00000i − 0.609994i
$$173$$ − 6.00000i − 0.456172i −0.973641 0.228086i $$-0.926753\pi$$
0.973641 0.228086i $$-0.0732467\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 1.00000 0.0753778
$$177$$ 0 0
$$178$$ 6.00000i 0.449719i
$$179$$ 24.0000 1.79384 0.896922 0.442189i $$-0.145798\pi$$
0.896922 + 0.442189i $$0.145798\pi$$
$$180$$ 0 0
$$181$$ −22.0000 −1.63525 −0.817624 0.575753i $$-0.804709\pi$$
−0.817624 + 0.575753i $$0.804709\pi$$
$$182$$ 8.00000i 0.592999i
$$183$$ 0 0
$$184$$ 6.00000 0.442326
$$185$$ 0 0
$$186$$ 0 0
$$187$$ − 6.00000i − 0.438763i
$$188$$ 6.00000i 0.437595i
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −18.0000 −1.30243 −0.651217 0.758891i $$-0.725741\pi$$
−0.651217 + 0.758891i $$0.725741\pi$$
$$192$$ 0 0
$$193$$ 14.0000i 1.00774i 0.863779 + 0.503871i $$0.168091\pi$$
−0.863779 + 0.503871i $$0.831909\pi$$
$$194$$ −14.0000 −1.00514
$$195$$ 0 0
$$196$$ −3.00000 −0.214286
$$197$$ 6.00000i 0.427482i 0.976890 + 0.213741i $$0.0685649\pi$$
−0.976890 + 0.213741i $$0.931435\pi$$
$$198$$ 0 0
$$199$$ 4.00000 0.283552 0.141776 0.989899i $$-0.454719\pi$$
0.141776 + 0.989899i $$0.454719\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 6.00000i 0.422159i
$$203$$ − 12.0000i − 0.842235i
$$204$$ 0 0
$$205$$ 0 0
$$206$$ −4.00000 −0.278693
$$207$$ 0 0
$$208$$ − 4.00000i − 0.277350i
$$209$$ 4.00000 0.276686
$$210$$ 0 0
$$211$$ 8.00000 0.550743 0.275371 0.961338i $$-0.411199\pi$$
0.275371 + 0.961338i $$0.411199\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ −12.0000 −0.820303
$$215$$ 0 0
$$216$$ 0 0
$$217$$ − 16.0000i − 1.08615i
$$218$$ − 4.00000i − 0.270914i
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −24.0000 −1.61441
$$222$$ 0 0
$$223$$ − 16.0000i − 1.07144i −0.844396 0.535720i $$-0.820040\pi$$
0.844396 0.535720i $$-0.179960\pi$$
$$224$$ −2.00000 −0.133631
$$225$$ 0 0
$$226$$ −18.0000 −1.19734
$$227$$ − 12.0000i − 0.796468i −0.917284 0.398234i $$-0.869623\pi$$
0.917284 0.398234i $$-0.130377\pi$$
$$228$$ 0 0
$$229$$ 22.0000 1.45380 0.726900 0.686743i $$-0.240960\pi$$
0.726900 + 0.686743i $$0.240960\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 6.00000i 0.393919i
$$233$$ 18.0000i 1.17922i 0.807688 + 0.589610i $$0.200718\pi$$
−0.807688 + 0.589610i $$0.799282\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 12.0000i 0.777844i
$$239$$ −12.0000 −0.776215 −0.388108 0.921614i $$-0.626871\pi$$
−0.388108 + 0.921614i $$0.626871\pi$$
$$240$$ 0 0
$$241$$ −10.0000 −0.644157 −0.322078 0.946713i $$-0.604381\pi$$
−0.322078 + 0.946713i $$0.604381\pi$$
$$242$$ − 1.00000i − 0.0642824i
$$243$$ 0 0
$$244$$ −8.00000 −0.512148
$$245$$ 0 0
$$246$$ 0 0
$$247$$ − 16.0000i − 1.01806i
$$248$$ 8.00000i 0.508001i
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$252$$ 0 0
$$253$$ − 6.00000i − 0.377217i
$$254$$ −14.0000 −0.878438
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ − 30.0000i − 1.87135i −0.352865 0.935674i $$-0.614792\pi$$
0.352865 0.935674i $$-0.385208\pi$$
$$258$$ 0 0
$$259$$ 20.0000 1.24274
$$260$$ 0 0
$$261$$ 0 0
$$262$$ − 12.0000i − 0.741362i
$$263$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ −8.00000 −0.490511
$$267$$ 0 0
$$268$$ − 4.00000i − 0.244339i
$$269$$ −24.0000 −1.46331 −0.731653 0.681677i $$-0.761251\pi$$
−0.731653 + 0.681677i $$0.761251\pi$$
$$270$$ 0 0
$$271$$ 2.00000 0.121491 0.0607457 0.998153i $$-0.480652\pi$$
0.0607457 + 0.998153i $$0.480652\pi$$
$$272$$ − 6.00000i − 0.363803i
$$273$$ 0 0
$$274$$ −18.0000 −1.08742
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 16.0000i 0.961347i 0.876900 + 0.480673i $$0.159608\pi$$
−0.876900 + 0.480673i $$0.840392\pi$$
$$278$$ − 4.00000i − 0.239904i
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 6.00000 0.357930 0.178965 0.983855i $$-0.442725\pi$$
0.178965 + 0.983855i $$0.442725\pi$$
$$282$$ 0 0
$$283$$ 8.00000i 0.475551i 0.971320 + 0.237775i $$0.0764182\pi$$
−0.971320 + 0.237775i $$0.923582\pi$$
$$284$$ 6.00000 0.356034
$$285$$ 0 0
$$286$$ −4.00000 −0.236525
$$287$$ 12.0000i 0.708338i
$$288$$ 0 0
$$289$$ −19.0000 −1.11765
$$290$$ 0 0
$$291$$ 0 0
$$292$$ − 2.00000i − 0.117041i
$$293$$ 6.00000i 0.350524i 0.984522 + 0.175262i $$0.0560772\pi$$
−0.984522 + 0.175262i $$0.943923\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ −10.0000 −0.581238
$$297$$ 0 0
$$298$$ 6.00000i 0.347571i
$$299$$ −24.0000 −1.38796
$$300$$ 0 0
$$301$$ 16.0000 0.922225
$$302$$ 10.0000i 0.575435i
$$303$$ 0 0
$$304$$ 4.00000 0.229416
$$305$$ 0 0
$$306$$ 0 0
$$307$$ − 20.0000i − 1.14146i −0.821138 0.570730i $$-0.806660\pi$$
0.821138 0.570730i $$-0.193340\pi$$
$$308$$ 2.00000i 0.113961i
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 18.0000 1.02069 0.510343 0.859971i $$-0.329518\pi$$
0.510343 + 0.859971i $$0.329518\pi$$
$$312$$ 0 0
$$313$$ 26.0000i 1.46961i 0.678280 + 0.734803i $$0.262726\pi$$
−0.678280 + 0.734803i $$0.737274\pi$$
$$314$$ −2.00000 −0.112867
$$315$$ 0 0
$$316$$ 14.0000 0.787562
$$317$$ 12.0000i 0.673987i 0.941507 + 0.336994i $$0.109410\pi$$
−0.941507 + 0.336994i $$0.890590\pi$$
$$318$$ 0 0
$$319$$ 6.00000 0.335936
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 12.0000i 0.668734i
$$323$$ − 24.0000i − 1.33540i
$$324$$ 0 0
$$325$$ 0 0
$$326$$ −4.00000 −0.221540
$$327$$ 0 0
$$328$$ − 6.00000i − 0.331295i
$$329$$ −12.0000 −0.661581
$$330$$ 0 0
$$331$$ −4.00000 −0.219860 −0.109930 0.993939i $$-0.535063\pi$$
−0.109930 + 0.993939i $$0.535063\pi$$
$$332$$ − 12.0000i − 0.658586i
$$333$$ 0 0
$$334$$ 12.0000 0.656611
$$335$$ 0 0
$$336$$ 0 0
$$337$$ − 2.00000i − 0.108947i −0.998515 0.0544735i $$-0.982652\pi$$
0.998515 0.0544735i $$-0.0173480\pi$$
$$338$$ 3.00000i 0.163178i
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 8.00000 0.433224
$$342$$ 0 0
$$343$$ − 20.0000i − 1.07990i
$$344$$ −8.00000 −0.431331
$$345$$ 0 0
$$346$$ −6.00000 −0.322562
$$347$$ 36.0000i 1.93258i 0.257454 + 0.966291i $$0.417117\pi$$
−0.257454 + 0.966291i $$0.582883\pi$$
$$348$$ 0 0
$$349$$ 4.00000 0.214115 0.107058 0.994253i $$-0.465857\pi$$
0.107058 + 0.994253i $$0.465857\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ − 1.00000i − 0.0533002i
$$353$$ 6.00000i 0.319348i 0.987170 + 0.159674i $$0.0510443\pi$$
−0.987170 + 0.159674i $$0.948956\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 6.00000 0.317999
$$357$$ 0 0
$$358$$ − 24.0000i − 1.26844i
$$359$$ −12.0000 −0.633336 −0.316668 0.948536i $$-0.602564\pi$$
−0.316668 + 0.948536i $$0.602564\pi$$
$$360$$ 0 0
$$361$$ −3.00000 −0.157895
$$362$$ 22.0000i 1.15629i
$$363$$ 0 0
$$364$$ 8.00000 0.419314
$$365$$ 0 0
$$366$$ 0 0
$$367$$ − 8.00000i − 0.417597i −0.977959 0.208798i $$-0.933045\pi$$
0.977959 0.208798i $$-0.0669552\pi$$
$$368$$ − 6.00000i − 0.312772i
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 20.0000i 1.03556i 0.855514 + 0.517780i $$0.173242\pi$$
−0.855514 + 0.517780i $$0.826758\pi$$
$$374$$ −6.00000 −0.310253
$$375$$ 0 0
$$376$$ 6.00000 0.309426
$$377$$ − 24.0000i − 1.23606i
$$378$$ 0 0
$$379$$ −20.0000 −1.02733 −0.513665 0.857991i $$-0.671713\pi$$
−0.513665 + 0.857991i $$0.671713\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 18.0000i 0.920960i
$$383$$ − 6.00000i − 0.306586i −0.988181 0.153293i $$-0.951012\pi$$
0.988181 0.153293i $$-0.0489878\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 14.0000 0.712581
$$387$$ 0 0
$$388$$ 14.0000i 0.710742i
$$389$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$390$$ 0 0
$$391$$ −36.0000 −1.82060
$$392$$ 3.00000i 0.151523i
$$393$$ 0 0
$$394$$ 6.00000 0.302276
$$395$$ 0 0
$$396$$ 0 0
$$397$$ − 26.0000i − 1.30490i −0.757831 0.652451i $$-0.773741\pi$$
0.757831 0.652451i $$-0.226259\pi$$
$$398$$ − 4.00000i − 0.200502i
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −30.0000 −1.49813 −0.749064 0.662497i $$-0.769497\pi$$
−0.749064 + 0.662497i $$0.769497\pi$$
$$402$$ 0 0
$$403$$ − 32.0000i − 1.59403i
$$404$$ 6.00000 0.298511
$$405$$ 0 0
$$406$$ −12.0000 −0.595550
$$407$$ 10.0000i 0.495682i
$$408$$ 0 0
$$409$$ 34.0000 1.68119 0.840596 0.541663i $$-0.182205\pi$$
0.840596 + 0.541663i $$0.182205\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 4.00000i 0.197066i
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 0 0
$$416$$ −4.00000 −0.196116
$$417$$ 0 0
$$418$$ − 4.00000i − 0.195646i
$$419$$ 24.0000 1.17248 0.586238 0.810139i $$-0.300608\pi$$
0.586238 + 0.810139i $$0.300608\pi$$
$$420$$ 0 0
$$421$$ −10.0000 −0.487370 −0.243685 0.969854i $$-0.578356\pi$$
−0.243685 + 0.969854i $$0.578356\pi$$
$$422$$ − 8.00000i − 0.389434i
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ − 16.0000i − 0.774294i
$$428$$ 12.0000i 0.580042i
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$432$$ 0 0
$$433$$ 26.0000i 1.24948i 0.780833 + 0.624740i $$0.214795\pi$$
−0.780833 + 0.624740i $$0.785205\pi$$
$$434$$ −16.0000 −0.768025
$$435$$ 0 0
$$436$$ −4.00000 −0.191565
$$437$$ − 24.0000i − 1.14808i
$$438$$ 0 0
$$439$$ 10.0000 0.477274 0.238637 0.971109i $$-0.423299\pi$$
0.238637 + 0.971109i $$0.423299\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 24.0000i 1.14156i
$$443$$ 24.0000i 1.14027i 0.821549 + 0.570137i $$0.193110\pi$$
−0.821549 + 0.570137i $$0.806890\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ −16.0000 −0.757622
$$447$$ 0 0
$$448$$ 2.00000i 0.0944911i
$$449$$ 6.00000 0.283158 0.141579 0.989927i $$-0.454782\pi$$
0.141579 + 0.989927i $$0.454782\pi$$
$$450$$ 0 0
$$451$$ −6.00000 −0.282529
$$452$$ 18.0000i 0.846649i
$$453$$ 0 0
$$454$$ −12.0000 −0.563188
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 10.0000i 0.467780i 0.972263 + 0.233890i $$0.0751456\pi$$
−0.972263 + 0.233890i $$0.924854\pi$$
$$458$$ − 22.0000i − 1.02799i
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −42.0000 −1.95614 −0.978068 0.208288i $$-0.933211\pi$$
−0.978068 + 0.208288i $$0.933211\pi$$
$$462$$ 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
$$467$$ − 12.0000i − 0.555294i −0.960683 0.277647i $$-0.910445\pi$$
0.960683 0.277647i $$-0.0895545\pi$$
$$468$$ 0 0
$$469$$ 8.00000 0.369406
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 8.00000i 0.367840i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 12.0000 0.550019
$$477$$ 0 0
$$478$$ 12.0000i 0.548867i
$$479$$ −24.0000 −1.09659 −0.548294 0.836286i $$-0.684723\pi$$
−0.548294 + 0.836286i $$0.684723\pi$$
$$480$$ 0 0
$$481$$ 40.0000 1.82384
$$482$$ 10.0000i 0.455488i
$$483$$ 0 0
$$484$$ −1.00000 −0.0454545
$$485$$ 0 0
$$486$$ 0 0
$$487$$ − 20.0000i − 0.906287i −0.891438 0.453143i $$-0.850303\pi$$
0.891438 0.453143i $$-0.149697\pi$$
$$488$$ 8.00000i 0.362143i
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −12.0000 −0.541552 −0.270776 0.962642i $$-0.587280\pi$$
−0.270776 + 0.962642i $$0.587280\pi$$
$$492$$ 0 0
$$493$$ − 36.0000i − 1.62136i
$$494$$ −16.0000 −0.719874
$$495$$ 0 0
$$496$$ 8.00000 0.359211
$$497$$ 12.0000i 0.538274i
$$498$$ 0 0
$$499$$ 4.00000 0.179065 0.0895323 0.995984i $$-0.471463\pi$$
0.0895323 + 0.995984i $$0.471463\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ − 12.0000i − 0.535054i −0.963550 0.267527i $$-0.913794\pi$$
0.963550 0.267527i $$-0.0862064\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ −6.00000 −0.266733
$$507$$ 0 0
$$508$$ 14.0000i 0.621150i
$$509$$ 24.0000 1.06378 0.531891 0.846813i $$-0.321482\pi$$
0.531891 + 0.846813i $$0.321482\pi$$
$$510$$ 0 0
$$511$$ 4.00000 0.176950
$$512$$ − 1.00000i − 0.0441942i
$$513$$ 0 0
$$514$$ −30.0000 −1.32324
$$515$$ 0 0
$$516$$ 0 0
$$517$$ − 6.00000i − 0.263880i
$$518$$ − 20.0000i − 0.878750i
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 18.0000 0.788594 0.394297 0.918983i $$-0.370988\pi$$
0.394297 + 0.918983i $$0.370988\pi$$
$$522$$ 0 0
$$523$$ − 16.0000i − 0.699631i −0.936819 0.349816i $$-0.886244\pi$$
0.936819 0.349816i $$-0.113756\pi$$
$$524$$ −12.0000 −0.524222
$$525$$ 0 0
$$526$$ 0 0
$$527$$ − 48.0000i − 2.09091i
$$528$$ 0 0
$$529$$ −13.0000 −0.565217
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 8.00000i 0.346844i
$$533$$ 24.0000i 1.03956i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ −4.00000 −0.172774
$$537$$ 0 0
$$538$$ 24.0000i 1.03471i
$$539$$ 3.00000 0.129219
$$540$$ 0 0
$$541$$ 20.0000 0.859867 0.429934 0.902861i $$-0.358537\pi$$
0.429934 + 0.902861i $$0.358537\pi$$
$$542$$ − 2.00000i − 0.0859074i
$$543$$ 0 0
$$544$$ −6.00000 −0.257248
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 28.0000i 1.19719i 0.801050 + 0.598597i $$0.204275\pi$$
−0.801050 + 0.598597i $$0.795725\pi$$
$$548$$ 18.0000i 0.768922i
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 24.0000 1.02243
$$552$$ 0 0
$$553$$ 28.0000i 1.19068i
$$554$$ 16.0000 0.679775
$$555$$ 0 0
$$556$$ −4.00000 −0.169638
$$557$$ 18.0000i 0.762684i 0.924434 + 0.381342i $$0.124538\pi$$
−0.924434 + 0.381342i $$0.875462\pi$$
$$558$$ 0 0
$$559$$ 32.0000 1.35346
$$560$$ 0 0
$$561$$ 0 0
$$562$$ − 6.00000i − 0.253095i
$$563$$ 12.0000i 0.505740i 0.967500 + 0.252870i $$0.0813744\pi$$
−0.967500 + 0.252870i $$0.918626\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 8.00000 0.336265
$$567$$ 0 0
$$568$$ − 6.00000i − 0.251754i
$$569$$ 18.0000 0.754599 0.377300 0.926091i $$-0.376853\pi$$
0.377300 + 0.926091i $$0.376853\pi$$
$$570$$ 0 0
$$571$$ −28.0000 −1.17176 −0.585882 0.810397i $$-0.699252\pi$$
−0.585882 + 0.810397i $$0.699252\pi$$
$$572$$ 4.00000i 0.167248i
$$573$$ 0 0
$$574$$ 12.0000 0.500870
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 34.0000i 1.41544i 0.706494 + 0.707719i $$0.250276\pi$$
−0.706494 + 0.707719i $$0.749724\pi$$
$$578$$ 19.0000i 0.790296i
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 24.0000 0.995688
$$582$$ 0 0
$$583$$ 0 0
$$584$$ −2.00000 −0.0827606
$$585$$ 0 0
$$586$$ 6.00000 0.247858
$$587$$ 24.0000i 0.990586i 0.868726 + 0.495293i $$0.164939\pi$$
−0.868726 + 0.495293i $$0.835061\pi$$
$$588$$ 0 0
$$589$$ 32.0000 1.31854
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 10.0000i 0.410997i
$$593$$ 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
$$597$$ 0 0
$$598$$ 24.0000i 0.981433i
$$599$$ 30.0000 1.22577 0.612883 0.790173i $$-0.290010\pi$$
0.612883 + 0.790173i $$0.290010\pi$$
$$600$$ 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
$$603$$ 0 0
$$604$$ 10.0000 0.406894
$$605$$ 0 0
$$606$$ 0 0
$$607$$ − 14.0000i − 0.568242i −0.958788 0.284121i $$-0.908298\pi$$
0.958788 0.284121i $$-0.0917018\pi$$
$$608$$ − 4.00000i − 0.162221i
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −24.0000 −0.970936
$$612$$ 0 0
$$613$$ − 16.0000i − 0.646234i −0.946359 0.323117i $$-0.895269\pi$$
0.946359 0.323117i $$-0.104731\pi$$
$$614$$ −20.0000 −0.807134
$$615$$ 0 0
$$616$$ 2.00000 0.0805823
$$617$$ − 30.0000i − 1.20775i −0.797077 0.603877i $$-0.793622\pi$$
0.797077 0.603877i $$-0.206378\pi$$
$$618$$ 0 0
$$619$$ −44.0000 −1.76851 −0.884255 0.467005i $$-0.845333\pi$$
−0.884255 + 0.467005i $$0.845333\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ − 18.0000i − 0.721734i
$$623$$ 12.0000i 0.480770i
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 26.0000 1.03917
$$627$$ 0 0
$$628$$ 2.00000i 0.0798087i
$$629$$ 60.0000 2.39236
$$630$$ 0 0
$$631$$ −16.0000 −0.636950 −0.318475 0.947931i $$-0.603171\pi$$
−0.318475 + 0.947931i $$0.603171\pi$$
$$632$$ − 14.0000i − 0.556890i
$$633$$ 0 0
$$634$$ 12.0000 0.476581
$$635$$ 0 0
$$636$$ 0 0
$$637$$ − 12.0000i − 0.475457i
$$638$$ − 6.00000i − 0.237542i
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −6.00000 −0.236986 −0.118493 0.992955i $$-0.537806\pi$$
−0.118493 + 0.992955i $$0.537806\pi$$
$$642$$ 0 0
$$643$$ − 4.00000i − 0.157745i −0.996885 0.0788723i $$-0.974868\pi$$
0.996885 0.0788723i $$-0.0251319\pi$$
$$644$$ 12.0000 0.472866
$$645$$ 0 0
$$646$$ −24.0000 −0.944267
$$647$$ 6.00000i 0.235884i 0.993020 + 0.117942i $$0.0376297\pi$$
−0.993020 + 0.117942i $$0.962370\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 4.00000i 0.156652i
$$653$$ 36.0000i 1.40879i 0.709809 + 0.704394i $$0.248781\pi$$
−0.709809 + 0.704394i $$0.751219\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ −6.00000 −0.234261
$$657$$ 0 0
$$658$$ 12.0000i 0.467809i
$$659$$ 36.0000 1.40236 0.701180 0.712984i $$-0.252657\pi$$
0.701180 + 0.712984i $$0.252657\pi$$
$$660$$ 0 0
$$661$$ −22.0000 −0.855701 −0.427850 0.903850i $$-0.640729\pi$$
−0.427850 + 0.903850i $$0.640729\pi$$
$$662$$ 4.00000i 0.155464i
$$663$$ 0 0
$$664$$ −12.0000 −0.465690
$$665$$ 0 0
$$666$$ 0 0
$$667$$ − 36.0000i − 1.39393i
$$668$$ − 12.0000i − 0.464294i
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 8.00000 0.308837
$$672$$ 0 0
$$673$$ 14.0000i 0.539660i 0.962908 + 0.269830i $$0.0869676\pi$$
−0.962908 + 0.269830i $$0.913032\pi$$
$$674$$ −2.00000 −0.0770371
$$675$$ 0 0
$$676$$ 3.00000 0.115385
$$677$$ 30.0000i 1.15299i 0.817099 + 0.576497i $$0.195581\pi$$
−0.817099 + 0.576497i $$0.804419\pi$$
$$678$$ 0 0
$$679$$ −28.0000 −1.07454
$$680$$ 0 0
$$681$$ 0 0
$$682$$ − 8.00000i − 0.306336i
$$683$$ 24.0000i 0.918334i 0.888350 + 0.459167i $$0.151852\pi$$
−0.888350 + 0.459167i $$0.848148\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ −20.0000 −0.763604
$$687$$ 0 0
$$688$$ 8.00000i 0.304997i
$$689$$ 0 0
$$690$$ 0 0
$$691$$ 20.0000 0.760836 0.380418 0.924815i $$-0.375780\pi$$
0.380418 + 0.924815i $$0.375780\pi$$
$$692$$ 6.00000i 0.228086i
$$693$$ 0 0
$$694$$ 36.0000 1.36654
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 36.0000i 1.36360i
$$698$$ − 4.00000i − 0.151402i
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 6.00000 0.226617 0.113308 0.993560i $$-0.463855\pi$$
0.113308 + 0.993560i $$0.463855\pi$$
$$702$$ 0 0
$$703$$ 40.0000i 1.50863i
$$704$$ −1.00000 −0.0376889
$$705$$ 0 0
$$706$$ 6.00000 0.225813
$$707$$ 12.0000i 0.451306i
$$708$$ 0 0
$$709$$ −26.0000 −0.976450 −0.488225 0.872718i $$-0.662356\pi$$
−0.488225 + 0.872718i $$0.662356\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ − 6.00000i − 0.224860i
$$713$$ − 48.0000i − 1.79761i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −24.0000 −0.896922
$$717$$ 0 0
$$718$$ 12.0000i 0.447836i
$$719$$ 30.0000 1.11881 0.559406 0.828894i $$-0.311029\pi$$
0.559406 + 0.828894i $$0.311029\pi$$
$$720$$ 0 0
$$721$$ −8.00000 −0.297936
$$722$$ 3.00000i 0.111648i
$$723$$ 0 0
$$724$$ 22.0000 0.817624
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 28.0000i 1.03846i 0.854634 + 0.519231i $$0.173782\pi$$
−0.854634 + 0.519231i $$0.826218\pi$$
$$728$$ − 8.00000i − 0.296500i
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 48.0000 1.77534
$$732$$ 0 0
$$733$$ − 4.00000i − 0.147743i −0.997268 0.0738717i $$-0.976464\pi$$
0.997268 0.0738717i $$-0.0235355\pi$$
$$734$$ −8.00000 −0.295285
$$735$$ 0 0
$$736$$ −6.00000 −0.221163
$$737$$ 4.00000i 0.147342i
$$738$$ 0 0
$$739$$ −8.00000 −0.294285 −0.147142 0.989115i $$-0.547008\pi$$
−0.147142 + 0.989115i $$0.547008\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ − 36.0000i − 1.32071i −0.750953 0.660356i $$-0.770405\pi$$
0.750953 0.660356i $$-0.229595\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 20.0000 0.732252
$$747$$ 0 0
$$748$$ 6.00000i 0.219382i
$$749$$ −24.0000 −0.876941
$$750$$ 0 0
$$751$$ 8.00000 0.291924 0.145962 0.989290i $$-0.453372\pi$$
0.145962 + 0.989290i $$0.453372\pi$$
$$752$$ − 6.00000i − 0.218797i
$$753$$ 0 0
$$754$$ −24.0000 −0.874028
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 34.0000i 1.23575i 0.786276 + 0.617876i $$0.212006\pi$$
−0.786276 + 0.617876i $$0.787994\pi$$
$$758$$ 20.0000i 0.726433i
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −18.0000 −0.652499 −0.326250 0.945284i $$-0.605785\pi$$
−0.326250 + 0.945284i $$0.605785\pi$$
$$762$$ 0 0
$$763$$ − 8.00000i − 0.289619i
$$764$$ 18.0000 0.651217
$$765$$ 0 0
$$766$$ −6.00000 −0.216789
$$767$$ 0 0
$$768$$ 0 0
$$769$$ 34.0000 1.22607 0.613036 0.790055i $$-0.289948\pi$$
0.613036 + 0.790055i $$0.289948\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ − 14.0000i − 0.503871i
$$773$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 14.0000 0.502571
$$777$$ 0 0
$$778$$ 0 0
$$779$$ −24.0000 −0.859889
$$780$$ 0 0
$$781$$ −6.00000 −0.214697
$$782$$ 36.0000i 1.28736i
$$783$$ 0 0
$$784$$ 3.00000 0.107143
$$785$$ 0 0
$$786$$ 0 0
$$787$$ − 32.0000i − 1.14068i −0.821410 0.570338i $$-0.806812\pi$$
0.821410 0.570338i $$-0.193188\pi$$
$$788$$ − 6.00000i − 0.213741i
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −36.0000 −1.28001
$$792$$ 0 0
$$793$$ − 32.0000i − 1.13635i
$$794$$ −26.0000 −0.922705
$$795$$ 0 0
$$796$$ −4.00000 −0.141776
$$797$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$798$$ 0 0
$$799$$ −36.0000 −1.27359
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 30.0000i 1.05934i
$$803$$ 2.00000i 0.0705785i
$$804$$ 0 0
$$805$$ 0 0
$$806$$ −32.0000 −1.12715
$$807$$ 0 0
$$808$$ − 6.00000i − 0.211079i
$$809$$ −42.0000 −1.47664 −0.738321 0.674450i $$-0.764381\pi$$
−0.738321 + 0.674450i $$0.764381\pi$$
$$810$$ 0 0
$$811$$ 8.00000 0.280918 0.140459 0.990086i $$-0.455142\pi$$
0.140459 + 0.990086i $$0.455142\pi$$
$$812$$ 12.0000i 0.421117i
$$813$$ 0 0
$$814$$ 10.0000 0.350500
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 32.0000i 1.11954i
$$818$$ − 34.0000i − 1.18878i
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 18.0000 0.628204 0.314102 0.949389i $$-0.398297\pi$$
0.314102 + 0.949389i $$0.398297\pi$$
$$822$$ 0 0
$$823$$ 8.00000i 0.278862i 0.990232 + 0.139431i $$0.0445274\pi$$
−0.990232 + 0.139431i $$0.955473\pi$$
$$824$$ 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
$$829$$ −14.0000 −0.486240 −0.243120 0.969996i $$-0.578171\pi$$
−0.243120 + 0.969996i $$0.578171\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 4.00000i 0.138675i
$$833$$ − 18.0000i − 0.623663i
$$834$$ 0 0
$$835$$ 0 0
$$836$$ −4.00000 −0.138343
$$837$$ 0 0
$$838$$ − 24.0000i − 0.829066i
$$839$$ 18.0000 0.621429 0.310715 0.950503i $$-0.399432\pi$$
0.310715 + 0.950503i $$0.399432\pi$$
$$840$$ 0 0
$$841$$ 7.00000 0.241379
$$842$$ 10.0000i 0.344623i
$$843$$ 0 0
$$844$$ −8.00000 −0.275371
$$845$$ 0 0
$$846$$ 0 0
$$847$$ − 2.00000i − 0.0687208i
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 60.0000 2.05677
$$852$$ 0 0
$$853$$ 8.00000i 0.273915i 0.990577 + 0.136957i $$0.0437323\pi$$
−0.990577 + 0.136957i $$0.956268\pi$$
$$854$$ −16.0000 −0.547509
$$855$$ 0 0
$$856$$ 12.0000 0.410152
$$857$$ 54.0000i 1.84460i 0.386469 + 0.922302i $$0.373695\pi$$
−0.386469 + 0.922302i $$0.626305\pi$$
$$858$$ 0 0
$$859$$ −20.0000 −0.682391 −0.341196 0.939992i $$-0.610832\pi$$
−0.341196 + 0.939992i $$0.610832\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ − 42.0000i − 1.42970i −0.699280 0.714848i $$-0.746496\pi$$
0.699280 0.714848i $$-0.253504\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 26.0000 0.883516
$$867$$ 0 0
$$868$$ 16.0000i 0.543075i
$$869$$ −14.0000 −0.474917
$$870$$ 0 0
$$871$$ 16.0000 0.542139
$$872$$ 4.00000i 0.135457i
$$873$$ 0 0
$$874$$ −24.0000 −0.811812
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 52.0000i 1.75592i 0.478738 + 0.877958i $$0.341094\pi$$
−0.478738 + 0.877958i $$0.658906\pi$$
$$878$$ − 10.0000i − 0.337484i
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 30.0000 1.01073 0.505363 0.862907i $$-0.331359\pi$$
0.505363 + 0.862907i $$0.331359\pi$$
$$882$$ 0 0
$$883$$ − 28.0000i − 0.942275i −0.882060 0.471138i $$-0.843844\pi$$
0.882060 0.471138i $$-0.156156\pi$$
$$884$$ 24.0000 0.807207
$$885$$ 0 0
$$886$$ 24.0000 0.806296
$$887$$ − 36.0000i − 1.20876i −0.796696 0.604381i $$-0.793421\pi$$
0.796696 0.604381i $$-0.206579\pi$$
$$888$$ 0 0
$$889$$ −28.0000 −0.939090
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 16.0000i 0.535720i
$$893$$ − 24.0000i − 0.803129i
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 2.00000 0.0668153
$$897$$ 0 0
$$898$$ − 6.00000i − 0.200223i
$$899$$ 48.0000 1.60089
$$900$$ 0 0
$$901$$ 0 0
$$902$$ 6.00000i 0.199778i
$$903$$ 0 0
$$904$$ 18.0000 0.598671
$$905$$ 0 0
$$906$$ 0 0
$$907$$ − 20.0000i − 0.664089i −0.943264 0.332045i $$-0.892262\pi$$
0.943264 0.332045i $$-0.107738\pi$$
$$908$$ 12.0000i 0.398234i
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 42.0000 1.39152 0.695761 0.718273i $$-0.255067\pi$$
0.695761 + 0.718273i $$0.255067\pi$$
$$912$$ 0 0
$$913$$ 12.0000i 0.397142i
$$914$$ 10.0000 0.330771
$$915$$ 0 0
$$916$$ −22.0000 −0.726900
$$917$$ − 24.0000i − 0.792550i
$$918$$ 0 0
$$919$$ −2.00000 −0.0659739 −0.0329870 0.999456i $$-0.510502\pi$$
−0.0329870 + 0.999456i $$0.510502\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 42.0000i 1.38320i
$$923$$ 24.0000i 0.789970i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ −4.00000 −0.131448
$$927$$ 0 0
$$928$$ − 6.00000i − 0.196960i
$$929$$ 30.0000 0.984268 0.492134 0.870519i $$-0.336217\pi$$
0.492134 + 0.870519i $$0.336217\pi$$
$$930$$ 0 0
$$931$$ 12.0000 0.393284
$$932$$ − 18.0000i − 0.589610i
$$933$$ 0 0
$$934$$ −12.0000 −0.392652
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 22.0000i 0.718709i 0.933201 + 0.359354i $$0.117003\pi$$
−0.933201 + 0.359354i $$0.882997\pi$$
$$938$$ − 8.00000i − 0.261209i
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 18.0000 0.586783 0.293392 0.955992i $$-0.405216\pi$$
0.293392 + 0.955992i $$0.405216\pi$$
$$942$$ 0 0
$$943$$ 36.0000i 1.17232i
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 8.00000 0.260102
$$947$$ 12.0000i 0.389948i 0.980808 + 0.194974i $$0.0624622\pi$$
−0.980808 + 0.194974i $$0.937538\pi$$
$$948$$ 0 0
$$949$$ 8.00000 0.259691
$$950$$ 0 0
$$951$$ 0 0
$$952$$ − 12.0000i − 0.388922i
$$953$$ 42.0000i 1.36051i 0.732974 + 0.680257i $$0.238132\pi$$
−0.732974 + 0.680257i $$0.761868\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 12.0000 0.388108
$$957$$ 0 0
$$958$$ 24.0000i 0.775405i
$$959$$ −36.0000 −1.16250
$$960$$ 0 0
$$961$$ 33.0000 1.06452
$$962$$ − 40.0000i − 1.28965i
$$963$$ 0 0
$$964$$ 10.0000 0.322078
$$965$$ 0 0
$$966$$ 0 0
$$967$$ − 14.0000i − 0.450210i −0.974335 0.225105i $$-0.927728\pi$$
0.974335 0.225105i $$-0.0722725\pi$$
$$968$$ 1.00000i 0.0321412i
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$972$$ 0 0
$$973$$ − 8.00000i − 0.256468i
$$974$$ −20.0000 −0.640841
$$975$$ 0 0
$$976$$ 8.00000 0.256074
$$977$$ 54.0000i 1.72761i 0.503824 + 0.863807i $$0.331926\pi$$
−0.503824 + 0.863807i $$0.668074\pi$$
$$978$$ 0 0
$$979$$ −6.00000 −0.191761
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 12.0000i 0.382935i
$$983$$ − 30.0000i − 0.956851i −0.878128 0.478426i $$-0.841208\pi$$
0.878128 0.478426i $$-0.158792\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ −36.0000 −1.14647
$$987$$ 0 0
$$988$$ 16.0000i 0.509028i
$$989$$ 48.0000 1.52631
$$990$$ 0 0
$$991$$ 56.0000 1.77890 0.889449 0.457034i $$-0.151088\pi$$
0.889449 + 0.457034i $$0.151088\pi$$
$$992$$ − 8.00000i − 0.254000i
$$993$$ 0 0
$$994$$ 12.0000 0.380617
$$995$$ 0 0
$$996$$ 0 0
$$997$$ − 44.0000i − 1.39349i −0.717317 0.696747i $$-0.754630\pi$$
0.717317 0.696747i $$-0.245370\pi$$
$$998$$ − 4.00000i − 0.126618i
$$999$$ 0 0
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 4950.2.c.r.199.1 2
3.2 odd 2 1650.2.c.d.199.2 2
5.2 odd 4 198.2.a.e.1.1 1
5.3 odd 4 4950.2.a.g.1.1 1
5.4 even 2 inner 4950.2.c.r.199.2 2
15.2 even 4 66.2.a.a.1.1 1
15.8 even 4 1650.2.a.m.1.1 1
15.14 odd 2 1650.2.c.d.199.1 2
20.7 even 4 1584.2.a.h.1.1 1
35.27 even 4 9702.2.a.bu.1.1 1
40.27 even 4 6336.2.a.bf.1.1 1
40.37 odd 4 6336.2.a.bj.1.1 1
45.2 even 12 1782.2.e.s.595.1 2
45.7 odd 12 1782.2.e.f.595.1 2
45.22 odd 12 1782.2.e.f.1189.1 2
45.32 even 12 1782.2.e.s.1189.1 2
55.32 even 4 2178.2.a.b.1.1 1
60.47 odd 4 528.2.a.d.1.1 1
105.62 odd 4 3234.2.a.d.1.1 1
120.77 even 4 2112.2.a.i.1.1 1
120.107 odd 4 2112.2.a.v.1.1 1
165.2 odd 20 726.2.e.b.565.1 4
165.17 odd 20 726.2.e.b.487.1 4
165.32 odd 4 726.2.a.i.1.1 1
165.47 even 20 726.2.e.k.493.1 4
165.62 odd 20 726.2.e.b.511.1 4
165.92 even 20 726.2.e.k.511.1 4
165.107 odd 20 726.2.e.b.493.1 4
165.137 even 20 726.2.e.k.487.1 4
165.152 even 20 726.2.e.k.565.1 4
660.527 even 4 5808.2.a.l.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
66.2.a.a.1.1 1 15.2 even 4
198.2.a.e.1.1 1 5.2 odd 4
528.2.a.d.1.1 1 60.47 odd 4
726.2.a.i.1.1 1 165.32 odd 4
726.2.e.b.487.1 4 165.17 odd 20
726.2.e.b.493.1 4 165.107 odd 20
726.2.e.b.511.1 4 165.62 odd 20
726.2.e.b.565.1 4 165.2 odd 20
726.2.e.k.487.1 4 165.137 even 20
726.2.e.k.493.1 4 165.47 even 20
726.2.e.k.511.1 4 165.92 even 20
726.2.e.k.565.1 4 165.152 even 20
1584.2.a.h.1.1 1 20.7 even 4
1650.2.a.m.1.1 1 15.8 even 4
1650.2.c.d.199.1 2 15.14 odd 2
1650.2.c.d.199.2 2 3.2 odd 2
1782.2.e.f.595.1 2 45.7 odd 12
1782.2.e.f.1189.1 2 45.22 odd 12
1782.2.e.s.595.1 2 45.2 even 12
1782.2.e.s.1189.1 2 45.32 even 12
2112.2.a.i.1.1 1 120.77 even 4
2112.2.a.v.1.1 1 120.107 odd 4
2178.2.a.b.1.1 1 55.32 even 4
3234.2.a.d.1.1 1 105.62 odd 4
4950.2.a.g.1.1 1 5.3 odd 4
4950.2.c.r.199.1 2 1.1 even 1 trivial
4950.2.c.r.199.2 2 5.4 even 2 inner
5808.2.a.l.1.1 1 660.527 even 4
6336.2.a.bf.1.1 1 40.27 even 4
6336.2.a.bj.1.1 1 40.37 odd 4
9702.2.a.bu.1.1 1 35.27 even 4