# Properties

 Label 3850.2.c.m.1849.1 Level $3850$ Weight $2$ Character 3850.1849 Analytic conductor $30.742$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3850 = 2 \cdot 5^{2} \cdot 7 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3850.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$30.7424047782$$ 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 770) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 1849.1 Root $$-1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 3850.1849 Dual form 3850.2.c.m.1849.2

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000i q^{2} -1.00000 q^{4} +1.00000i q^{7} +1.00000i q^{8} +3.00000 q^{9} +O(q^{10})$$ $$q-1.00000i q^{2} -1.00000 q^{4} +1.00000i q^{7} +1.00000i q^{8} +3.00000 q^{9} -1.00000 q^{11} +6.00000i q^{13} +1.00000 q^{14} +1.00000 q^{16} -2.00000i q^{17} -3.00000i q^{18} +4.00000 q^{19} +1.00000i q^{22} +4.00000i q^{23} +6.00000 q^{26} -1.00000i q^{28} -6.00000 q^{29} -1.00000i q^{32} -2.00000 q^{34} -3.00000 q^{36} -2.00000i q^{37} -4.00000i q^{38} -6.00000 q^{41} +4.00000i q^{43} +1.00000 q^{44} +4.00000 q^{46} +4.00000i q^{47} -1.00000 q^{49} -6.00000i q^{52} +2.00000i q^{53} -1.00000 q^{56} +6.00000i q^{58} -12.0000 q^{59} -2.00000 q^{61} +3.00000i q^{63} -1.00000 q^{64} -8.00000i q^{67} +2.00000i q^{68} -8.00000 q^{71} +3.00000i q^{72} +10.0000i q^{73} -2.00000 q^{74} -4.00000 q^{76} -1.00000i q^{77} +8.00000 q^{79} +9.00000 q^{81} +6.00000i q^{82} +12.0000i q^{83} +4.00000 q^{86} -1.00000i q^{88} -10.0000 q^{89} -6.00000 q^{91} -4.00000i q^{92} +4.00000 q^{94} -6.00000i q^{97} +1.00000i q^{98} -3.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{4} + 6 q^{9}+O(q^{10})$$ 2 * q - 2 * q^4 + 6 * q^9 $$2 q - 2 q^{4} + 6 q^{9} - 2 q^{11} + 2 q^{14} + 2 q^{16} + 8 q^{19} + 12 q^{26} - 12 q^{29} - 4 q^{34} - 6 q^{36} - 12 q^{41} + 2 q^{44} + 8 q^{46} - 2 q^{49} - 2 q^{56} - 24 q^{59} - 4 q^{61} - 2 q^{64} - 16 q^{71} - 4 q^{74} - 8 q^{76} + 16 q^{79} + 18 q^{81} + 8 q^{86} - 20 q^{89} - 12 q^{91} + 8 q^{94} - 6 q^{99}+O(q^{100})$$ 2 * q - 2 * q^4 + 6 * q^9 - 2 * q^11 + 2 * q^14 + 2 * q^16 + 8 * q^19 + 12 * q^26 - 12 * q^29 - 4 * q^34 - 6 * q^36 - 12 * q^41 + 2 * q^44 + 8 * q^46 - 2 * q^49 - 2 * q^56 - 24 * q^59 - 4 * q^61 - 2 * q^64 - 16 * q^71 - 4 * q^74 - 8 * q^76 + 16 * q^79 + 18 * q^81 + 8 * q^86 - 20 * q^89 - 12 * q^91 + 8 * q^94 - 6 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3850\mathbb{Z}\right)^\times$$.

 $$n$$ $$1751$$ $$2201$$ $$2927$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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 1.00000 $$0$$
−1.00000 $$\pi$$
$$4$$ −1.00000 −0.500000
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 1.00000i 0.377964i
$$8$$ 1.00000i 0.353553i
$$9$$ 3.00000 1.00000
$$10$$ 0 0
$$11$$ −1.00000 −0.301511
$$12$$ 0 0
$$13$$ 6.00000i 1.66410i 0.554700 + 0.832050i $$0.312833\pi$$
−0.554700 + 0.832050i $$0.687167\pi$$
$$14$$ 1.00000 0.267261
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ − 2.00000i − 0.485071i −0.970143 0.242536i $$-0.922021\pi$$
0.970143 0.242536i $$-0.0779791\pi$$
$$18$$ − 3.00000i − 0.707107i
$$19$$ 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
$$23$$ 4.00000i 0.834058i 0.908893 + 0.417029i $$0.136929\pi$$
−0.908893 + 0.417029i $$0.863071\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 6.00000 1.17670
$$27$$ 0 0
$$28$$ − 1.00000i − 0.188982i
$$29$$ −6.00000 −1.11417 −0.557086 0.830455i $$-0.688081\pi$$
−0.557086 + 0.830455i $$0.688081\pi$$
$$30$$ 0 0
$$31$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$32$$ − 1.00000i − 0.176777i
$$33$$ 0 0
$$34$$ −2.00000 −0.342997
$$35$$ 0 0
$$36$$ −3.00000 −0.500000
$$37$$ − 2.00000i − 0.328798i −0.986394 0.164399i $$-0.947432\pi$$
0.986394 0.164399i $$-0.0525685\pi$$
$$38$$ − 4.00000i − 0.648886i
$$39$$ 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
$$43$$ 4.00000i 0.609994i 0.952353 + 0.304997i $$0.0986555\pi$$
−0.952353 + 0.304997i $$0.901344\pi$$
$$44$$ 1.00000 0.150756
$$45$$ 0 0
$$46$$ 4.00000 0.589768
$$47$$ 4.00000i 0.583460i 0.956501 + 0.291730i $$0.0942309\pi$$
−0.956501 + 0.291730i $$0.905769\pi$$
$$48$$ 0 0
$$49$$ −1.00000 −0.142857
$$50$$ 0 0
$$51$$ 0 0
$$52$$ − 6.00000i − 0.832050i
$$53$$ 2.00000i 0.274721i 0.990521 + 0.137361i $$0.0438619\pi$$
−0.990521 + 0.137361i $$0.956138\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ −1.00000 −0.133631
$$57$$ 0 0
$$58$$ 6.00000i 0.787839i
$$59$$ −12.0000 −1.56227 −0.781133 0.624364i $$-0.785358\pi$$
−0.781133 + 0.624364i $$0.785358\pi$$
$$60$$ 0 0
$$61$$ −2.00000 −0.256074 −0.128037 0.991769i $$-0.540868\pi$$
−0.128037 + 0.991769i $$0.540868\pi$$
$$62$$ 0 0
$$63$$ 3.00000i 0.377964i
$$64$$ −1.00000 −0.125000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ − 8.00000i − 0.977356i −0.872464 0.488678i $$-0.837479\pi$$
0.872464 0.488678i $$-0.162521\pi$$
$$68$$ 2.00000i 0.242536i
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −8.00000 −0.949425 −0.474713 0.880141i $$-0.657448\pi$$
−0.474713 + 0.880141i $$0.657448\pi$$
$$72$$ 3.00000i 0.353553i
$$73$$ 10.0000i 1.17041i 0.810885 + 0.585206i $$0.198986\pi$$
−0.810885 + 0.585206i $$0.801014\pi$$
$$74$$ −2.00000 −0.232495
$$75$$ 0 0
$$76$$ −4.00000 −0.458831
$$77$$ − 1.00000i − 0.113961i
$$78$$ 0 0
$$79$$ 8.00000 0.900070 0.450035 0.893011i $$-0.351411\pi$$
0.450035 + 0.893011i $$0.351411\pi$$
$$80$$ 0 0
$$81$$ 9.00000 1.00000
$$82$$ 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
$$86$$ 4.00000 0.431331
$$87$$ 0 0
$$88$$ − 1.00000i − 0.106600i
$$89$$ −10.0000 −1.06000 −0.529999 0.847998i $$-0.677808\pi$$
−0.529999 + 0.847998i $$0.677808\pi$$
$$90$$ 0 0
$$91$$ −6.00000 −0.628971
$$92$$ − 4.00000i − 0.417029i
$$93$$ 0 0
$$94$$ 4.00000 0.412568
$$95$$ 0 0
$$96$$ 0 0
$$97$$ − 6.00000i − 0.609208i −0.952479 0.304604i $$-0.901476\pi$$
0.952479 0.304604i $$-0.0985241\pi$$
$$98$$ 1.00000i 0.101015i
$$99$$ −3.00000 −0.301511
$$100$$ 0 0
$$101$$ 14.0000 1.39305 0.696526 0.717532i $$-0.254728\pi$$
0.696526 + 0.717532i $$0.254728\pi$$
$$102$$ 0 0
$$103$$ 4.00000i 0.394132i 0.980390 + 0.197066i $$0.0631413\pi$$
−0.980390 + 0.197066i $$0.936859\pi$$
$$104$$ −6.00000 −0.588348
$$105$$ 0 0
$$106$$ 2.00000 0.194257
$$107$$ 4.00000i 0.386695i 0.981130 + 0.193347i $$0.0619344\pi$$
−0.981130 + 0.193347i $$0.938066\pi$$
$$108$$ 0 0
$$109$$ −14.0000 −1.34096 −0.670478 0.741929i $$-0.733911\pi$$
−0.670478 + 0.741929i $$0.733911\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 1.00000i 0.0944911i
$$113$$ 6.00000i 0.564433i 0.959351 + 0.282216i $$0.0910696\pi$$
−0.959351 + 0.282216i $$0.908930\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 6.00000 0.557086
$$117$$ 18.0000i 1.66410i
$$118$$ 12.0000i 1.10469i
$$119$$ 2.00000 0.183340
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ 2.00000i 0.181071i
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 3.00000 0.267261
$$127$$ − 8.00000i − 0.709885i −0.934888 0.354943i $$-0.884500\pi$$
0.934888 0.354943i $$-0.115500\pi$$
$$128$$ 1.00000i 0.0883883i
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −12.0000 −1.04844 −0.524222 0.851581i $$-0.675644\pi$$
−0.524222 + 0.851581i $$0.675644\pi$$
$$132$$ 0 0
$$133$$ 4.00000i 0.346844i
$$134$$ −8.00000 −0.691095
$$135$$ 0 0
$$136$$ 2.00000 0.171499
$$137$$ 18.0000i 1.53784i 0.639343 + 0.768922i $$0.279207\pi$$
−0.639343 + 0.768922i $$0.720793\pi$$
$$138$$ 0 0
$$139$$ 20.0000 1.69638 0.848189 0.529694i $$-0.177693\pi$$
0.848189 + 0.529694i $$0.177693\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 8.00000i 0.671345i
$$143$$ − 6.00000i − 0.501745i
$$144$$ 3.00000 0.250000
$$145$$ 0 0
$$146$$ 10.0000 0.827606
$$147$$ 0 0
$$148$$ 2.00000i 0.164399i
$$149$$ −6.00000 −0.491539 −0.245770 0.969328i $$-0.579041\pi$$
−0.245770 + 0.969328i $$0.579041\pi$$
$$150$$ 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
$$154$$ −1.00000 −0.0805823
$$155$$ 0 0
$$156$$ 0 0
$$157$$ − 2.00000i − 0.159617i −0.996810 0.0798087i $$-0.974569\pi$$
0.996810 0.0798087i $$-0.0254309\pi$$
$$158$$ − 8.00000i − 0.636446i
$$159$$ 0 0
$$160$$ 0 0
$$161$$ −4.00000 −0.315244
$$162$$ − 9.00000i − 0.707107i
$$163$$ − 8.00000i − 0.626608i −0.949653 0.313304i $$-0.898564\pi$$
0.949653 0.313304i $$-0.101436\pi$$
$$164$$ 6.00000 0.468521
$$165$$ 0 0
$$166$$ 12.0000 0.931381
$$167$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$168$$ 0 0
$$169$$ −23.0000 −1.76923
$$170$$ 0 0
$$171$$ 12.0000 0.917663
$$172$$ − 4.00000i − 0.304997i
$$173$$ 14.0000i 1.06440i 0.846619 + 0.532200i $$0.178635\pi$$
−0.846619 + 0.532200i $$0.821365\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ −1.00000 −0.0753778
$$177$$ 0 0
$$178$$ 10.0000i 0.749532i
$$179$$ −20.0000 −1.49487 −0.747435 0.664335i $$-0.768715\pi$$
−0.747435 + 0.664335i $$0.768715\pi$$
$$180$$ 0 0
$$181$$ 6.00000 0.445976 0.222988 0.974821i $$-0.428419\pi$$
0.222988 + 0.974821i $$0.428419\pi$$
$$182$$ 6.00000i 0.444750i
$$183$$ 0 0
$$184$$ −4.00000 −0.294884
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 2.00000i 0.146254i
$$188$$ − 4.00000i − 0.291730i
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 8.00000 0.578860 0.289430 0.957199i $$-0.406534\pi$$
0.289430 + 0.957199i $$0.406534\pi$$
$$192$$ 0 0
$$193$$ − 6.00000i − 0.431889i −0.976406 0.215945i $$-0.930717\pi$$
0.976406 0.215945i $$-0.0692831\pi$$
$$194$$ −6.00000 −0.430775
$$195$$ 0 0
$$196$$ 1.00000 0.0714286
$$197$$ 18.0000i 1.28245i 0.767354 + 0.641223i $$0.221573\pi$$
−0.767354 + 0.641223i $$0.778427\pi$$
$$198$$ 3.00000i 0.213201i
$$199$$ 24.0000 1.70131 0.850657 0.525720i $$-0.176204\pi$$
0.850657 + 0.525720i $$0.176204\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ − 14.0000i − 0.985037i
$$203$$ − 6.00000i − 0.421117i
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 4.00000 0.278693
$$207$$ 12.0000i 0.834058i
$$208$$ 6.00000i 0.416025i
$$209$$ −4.00000 −0.276686
$$210$$ 0 0
$$211$$ 4.00000 0.275371 0.137686 0.990476i $$-0.456034\pi$$
0.137686 + 0.990476i $$0.456034\pi$$
$$212$$ − 2.00000i − 0.137361i
$$213$$ 0 0
$$214$$ 4.00000 0.273434
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 14.0000i 0.948200i
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 12.0000 0.807207
$$222$$ 0 0
$$223$$ 4.00000i 0.267860i 0.990991 + 0.133930i $$0.0427597\pi$$
−0.990991 + 0.133930i $$0.957240\pi$$
$$224$$ 1.00000 0.0668153
$$225$$ 0 0
$$226$$ 6.00000 0.399114
$$227$$ 4.00000i 0.265489i 0.991150 + 0.132745i $$0.0423790\pi$$
−0.991150 + 0.132745i $$0.957621\pi$$
$$228$$ 0 0
$$229$$ −6.00000 −0.396491 −0.198246 0.980152i $$-0.563524\pi$$
−0.198246 + 0.980152i $$0.563524\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ − 6.00000i − 0.393919i
$$233$$ 10.0000i 0.655122i 0.944830 + 0.327561i $$0.106227\pi$$
−0.944830 + 0.327561i $$0.893773\pi$$
$$234$$ 18.0000 1.17670
$$235$$ 0 0
$$236$$ 12.0000 0.781133
$$237$$ 0 0
$$238$$ − 2.00000i − 0.129641i
$$239$$ 24.0000 1.55243 0.776215 0.630468i $$-0.217137\pi$$
0.776215 + 0.630468i $$0.217137\pi$$
$$240$$ 0 0
$$241$$ 2.00000 0.128831 0.0644157 0.997923i $$-0.479482\pi$$
0.0644157 + 0.997923i $$0.479482\pi$$
$$242$$ − 1.00000i − 0.0642824i
$$243$$ 0 0
$$244$$ 2.00000 0.128037
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 24.0000i 1.52708i
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 20.0000 1.26239 0.631194 0.775625i $$-0.282565\pi$$
0.631194 + 0.775625i $$0.282565\pi$$
$$252$$ − 3.00000i − 0.188982i
$$253$$ − 4.00000i − 0.251478i
$$254$$ −8.00000 −0.501965
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 18.0000i 1.12281i 0.827541 + 0.561405i $$0.189739\pi$$
−0.827541 + 0.561405i $$0.810261\pi$$
$$258$$ 0 0
$$259$$ 2.00000 0.124274
$$260$$ 0 0
$$261$$ −18.0000 −1.11417
$$262$$ 12.0000i 0.741362i
$$263$$ − 16.0000i − 0.986602i −0.869859 0.493301i $$-0.835790\pi$$
0.869859 0.493301i $$-0.164210\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 4.00000 0.245256
$$267$$ 0 0
$$268$$ 8.00000i 0.488678i
$$269$$ 2.00000 0.121942 0.0609711 0.998140i $$-0.480580\pi$$
0.0609711 + 0.998140i $$0.480580\pi$$
$$270$$ 0 0
$$271$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$272$$ − 2.00000i − 0.121268i
$$273$$ 0 0
$$274$$ 18.0000 1.08742
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 26.0000i 1.56219i 0.624413 + 0.781094i $$0.285338\pi$$
−0.624413 + 0.781094i $$0.714662\pi$$
$$278$$ − 20.0000i − 1.19952i
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −30.0000 −1.78965 −0.894825 0.446417i $$-0.852700\pi$$
−0.894825 + 0.446417i $$0.852700\pi$$
$$282$$ 0 0
$$283$$ 20.0000i 1.18888i 0.804141 + 0.594438i $$0.202626\pi$$
−0.804141 + 0.594438i $$0.797374\pi$$
$$284$$ 8.00000 0.474713
$$285$$ 0 0
$$286$$ −6.00000 −0.354787
$$287$$ − 6.00000i − 0.354169i
$$288$$ − 3.00000i − 0.176777i
$$289$$ 13.0000 0.764706
$$290$$ 0 0
$$291$$ 0 0
$$292$$ − 10.0000i − 0.585206i
$$293$$ 14.0000i 0.817889i 0.912559 + 0.408944i $$0.134103\pi$$
−0.912559 + 0.408944i $$0.865897\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 2.00000 0.116248
$$297$$ 0 0
$$298$$ 6.00000i 0.347571i
$$299$$ −24.0000 −1.38796
$$300$$ 0 0
$$301$$ −4.00000 −0.230556
$$302$$ − 8.00000i − 0.460348i
$$303$$ 0 0
$$304$$ 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 $$-0.193340\pi$$
$$308$$ 1.00000i 0.0569803i
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$312$$ 0 0
$$313$$ − 10.0000i − 0.565233i −0.959233 0.282617i $$-0.908798\pi$$
0.959233 0.282617i $$-0.0912024\pi$$
$$314$$ −2.00000 −0.112867
$$315$$ 0 0
$$316$$ −8.00000 −0.450035
$$317$$ 14.0000i 0.786318i 0.919470 + 0.393159i $$0.128618\pi$$
−0.919470 + 0.393159i $$0.871382\pi$$
$$318$$ 0 0
$$319$$ 6.00000 0.335936
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 4.00000i 0.222911i
$$323$$ − 8.00000i − 0.445132i
$$324$$ −9.00000 −0.500000
$$325$$ 0 0
$$326$$ −8.00000 −0.443079
$$327$$ 0 0
$$328$$ − 6.00000i − 0.331295i
$$329$$ −4.00000 −0.220527
$$330$$ 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
$$333$$ − 6.00000i − 0.328798i
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 22.0000i 1.19842i 0.800593 + 0.599208i $$0.204518\pi$$
−0.800593 + 0.599208i $$0.795482\pi$$
$$338$$ 23.0000i 1.25104i
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 0 0
$$342$$ − 12.0000i − 0.648886i
$$343$$ − 1.00000i − 0.0539949i
$$344$$ −4.00000 −0.215666
$$345$$ 0 0
$$346$$ 14.0000 0.752645
$$347$$ − 28.0000i − 1.50312i −0.659665 0.751559i $$-0.729302\pi$$
0.659665 0.751559i $$-0.270698\pi$$
$$348$$ 0 0
$$349$$ 34.0000 1.81998 0.909989 0.414632i $$-0.136090\pi$$
0.909989 + 0.414632i $$0.136090\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 1.00000i 0.0533002i
$$353$$ 14.0000i 0.745145i 0.928003 + 0.372572i $$0.121524\pi$$
−0.928003 + 0.372572i $$0.878476\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 10.0000 0.529999
$$357$$ 0 0
$$358$$ 20.0000i 1.05703i
$$359$$ 16.0000 0.844448 0.422224 0.906492i $$-0.361250\pi$$
0.422224 + 0.906492i $$0.361250\pi$$
$$360$$ 0 0
$$361$$ −3.00000 −0.157895
$$362$$ − 6.00000i − 0.315353i
$$363$$ 0 0
$$364$$ 6.00000 0.314485
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 20.0000i 1.04399i 0.852948 + 0.521996i $$0.174812\pi$$
−0.852948 + 0.521996i $$0.825188\pi$$
$$368$$ 4.00000i 0.208514i
$$369$$ −18.0000 −0.937043
$$370$$ 0 0
$$371$$ −2.00000 −0.103835
$$372$$ 0 0
$$373$$ − 34.0000i − 1.76045i −0.474554 0.880227i $$-0.657390\pi$$
0.474554 0.880227i $$-0.342610\pi$$
$$374$$ 2.00000 0.103418
$$375$$ 0 0
$$376$$ −4.00000 −0.206284
$$377$$ − 36.0000i − 1.85409i
$$378$$ 0 0
$$379$$ 12.0000 0.616399 0.308199 0.951322i $$-0.400274\pi$$
0.308199 + 0.951322i $$0.400274\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ − 8.00000i − 0.409316i
$$383$$ 28.0000i 1.43073i 0.698749 + 0.715367i $$0.253740\pi$$
−0.698749 + 0.715367i $$0.746260\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −6.00000 −0.305392
$$387$$ 12.0000i 0.609994i
$$388$$ 6.00000i 0.304604i
$$389$$ −6.00000 −0.304212 −0.152106 0.988364i $$-0.548606\pi$$
−0.152106 + 0.988364i $$0.548606\pi$$
$$390$$ 0 0
$$391$$ 8.00000 0.404577
$$392$$ − 1.00000i − 0.0505076i
$$393$$ 0 0
$$394$$ 18.0000 0.906827
$$395$$ 0 0
$$396$$ 3.00000 0.150756
$$397$$ − 34.0000i − 1.70641i −0.521575 0.853206i $$-0.674655\pi$$
0.521575 0.853206i $$-0.325345\pi$$
$$398$$ − 24.0000i − 1.20301i
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 34.0000 1.69788 0.848939 0.528490i $$-0.177242\pi$$
0.848939 + 0.528490i $$0.177242\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ −14.0000 −0.696526
$$405$$ 0 0
$$406$$ −6.00000 −0.297775
$$407$$ 2.00000i 0.0991363i
$$408$$ 0 0
$$409$$ −34.0000 −1.68119 −0.840596 0.541663i $$-0.817795\pi$$
−0.840596 + 0.541663i $$0.817795\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ − 4.00000i − 0.197066i
$$413$$ − 12.0000i − 0.590481i
$$414$$ 12.0000 0.589768
$$415$$ 0 0
$$416$$ 6.00000 0.294174
$$417$$ 0 0
$$418$$ 4.00000i 0.195646i
$$419$$ −12.0000 −0.586238 −0.293119 0.956076i $$-0.594693\pi$$
−0.293119 + 0.956076i $$0.594693\pi$$
$$420$$ 0 0
$$421$$ −26.0000 −1.26716 −0.633581 0.773676i $$-0.718416\pi$$
−0.633581 + 0.773676i $$0.718416\pi$$
$$422$$ − 4.00000i − 0.194717i
$$423$$ 12.0000i 0.583460i
$$424$$ −2.00000 −0.0971286
$$425$$ 0 0
$$426$$ 0 0
$$427$$ − 2.00000i − 0.0967868i
$$428$$ − 4.00000i − 0.193347i
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 8.00000 0.385346 0.192673 0.981263i $$-0.438284\pi$$
0.192673 + 0.981263i $$0.438284\pi$$
$$432$$ 0 0
$$433$$ 6.00000i 0.288342i 0.989553 + 0.144171i $$0.0460515\pi$$
−0.989553 + 0.144171i $$0.953949\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 14.0000 0.670478
$$437$$ 16.0000i 0.765384i
$$438$$ 0 0
$$439$$ 24.0000 1.14546 0.572729 0.819745i $$-0.305885\pi$$
0.572729 + 0.819745i $$0.305885\pi$$
$$440$$ 0 0
$$441$$ −3.00000 −0.142857
$$442$$ − 12.0000i − 0.570782i
$$443$$ − 16.0000i − 0.760183i −0.924949 0.380091i $$-0.875893\pi$$
0.924949 0.380091i $$-0.124107\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 4.00000 0.189405
$$447$$ 0 0
$$448$$ − 1.00000i − 0.0472456i
$$449$$ −2.00000 −0.0943858 −0.0471929 0.998886i $$-0.515028\pi$$
−0.0471929 + 0.998886i $$0.515028\pi$$
$$450$$ 0 0
$$451$$ 6.00000 0.282529
$$452$$ − 6.00000i − 0.282216i
$$453$$ 0 0
$$454$$ 4.00000 0.187729
$$455$$ 0 0
$$456$$ 0 0
$$457$$ − 26.0000i − 1.21623i −0.793849 0.608114i $$-0.791926\pi$$
0.793849 0.608114i $$-0.208074\pi$$
$$458$$ 6.00000i 0.280362i
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 6.00000 0.279448 0.139724 0.990190i $$-0.455378\pi$$
0.139724 + 0.990190i $$0.455378\pi$$
$$462$$ 0 0
$$463$$ − 20.0000i − 0.929479i −0.885448 0.464739i $$-0.846148\pi$$
0.885448 0.464739i $$-0.153852\pi$$
$$464$$ −6.00000 −0.278543
$$465$$ 0 0
$$466$$ 10.0000 0.463241
$$467$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$468$$ − 18.0000i − 0.832050i
$$469$$ 8.00000 0.369406
$$470$$ 0 0
$$471$$ 0 0
$$472$$ − 12.0000i − 0.552345i
$$473$$ − 4.00000i − 0.183920i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ −2.00000 −0.0916698
$$477$$ 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
$$481$$ 12.0000 0.547153
$$482$$ − 2.00000i − 0.0910975i
$$483$$ 0 0
$$484$$ −1.00000 −0.0454545
$$485$$ 0 0
$$486$$ 0 0
$$487$$ − 28.0000i − 1.26880i −0.773004 0.634401i $$-0.781247\pi$$
0.773004 0.634401i $$-0.218753\pi$$
$$488$$ − 2.00000i − 0.0905357i
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 20.0000 0.902587 0.451294 0.892375i $$-0.350963\pi$$
0.451294 + 0.892375i $$0.350963\pi$$
$$492$$ 0 0
$$493$$ 12.0000i 0.540453i
$$494$$ 24.0000 1.07981
$$495$$ 0 0
$$496$$ 0 0
$$497$$ − 8.00000i − 0.358849i
$$498$$ 0 0
$$499$$ −28.0000 −1.25345 −0.626726 0.779240i $$-0.715605\pi$$
−0.626726 + 0.779240i $$0.715605\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ − 20.0000i − 0.892644i
$$503$$ 8.00000i 0.356702i 0.983967 + 0.178351i $$0.0570763\pi$$
−0.983967 + 0.178351i $$0.942924\pi$$
$$504$$ −3.00000 −0.133631
$$505$$ 0 0
$$506$$ −4.00000 −0.177822
$$507$$ 0 0
$$508$$ 8.00000i 0.354943i
$$509$$ 2.00000 0.0886484 0.0443242 0.999017i $$-0.485887\pi$$
0.0443242 + 0.999017i $$0.485887\pi$$
$$510$$ 0 0
$$511$$ −10.0000 −0.442374
$$512$$ − 1.00000i − 0.0441942i
$$513$$ 0 0
$$514$$ 18.0000 0.793946
$$515$$ 0 0
$$516$$ 0 0
$$517$$ − 4.00000i − 0.175920i
$$518$$ − 2.00000i − 0.0878750i
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −22.0000 −0.963837 −0.481919 0.876216i $$-0.660060\pi$$
−0.481919 + 0.876216i $$0.660060\pi$$
$$522$$ 18.0000i 0.787839i
$$523$$ 12.0000i 0.524723i 0.964970 + 0.262362i $$0.0845013\pi$$
−0.964970 + 0.262362i $$0.915499\pi$$
$$524$$ 12.0000 0.524222
$$525$$ 0 0
$$526$$ −16.0000 −0.697633
$$527$$ 0 0
$$528$$ 0 0
$$529$$ 7.00000 0.304348
$$530$$ 0 0
$$531$$ −36.0000 −1.56227
$$532$$ − 4.00000i − 0.173422i
$$533$$ − 36.0000i − 1.55933i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 8.00000 0.345547
$$537$$ 0 0
$$538$$ − 2.00000i − 0.0862261i
$$539$$ 1.00000 0.0430730
$$540$$ 0 0
$$541$$ −18.0000 −0.773880 −0.386940 0.922105i $$-0.626468\pi$$
−0.386940 + 0.922105i $$0.626468\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ −2.00000 −0.0857493
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 36.0000i 1.53925i 0.638497 + 0.769624i $$0.279557\pi$$
−0.638497 + 0.769624i $$0.720443\pi$$
$$548$$ − 18.0000i − 0.768922i
$$549$$ −6.00000 −0.256074
$$550$$ 0 0
$$551$$ −24.0000 −1.02243
$$552$$ 0 0
$$553$$ 8.00000i 0.340195i
$$554$$ 26.0000 1.10463
$$555$$ 0 0
$$556$$ −20.0000 −0.848189
$$557$$ − 6.00000i − 0.254228i −0.991888 0.127114i $$-0.959429\pi$$
0.991888 0.127114i $$-0.0405714\pi$$
$$558$$ 0 0
$$559$$ −24.0000 −1.01509
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 30.0000i 1.26547i
$$563$$ 4.00000i 0.168580i 0.996441 + 0.0842900i $$0.0268622\pi$$
−0.996441 + 0.0842900i $$0.973138\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 20.0000 0.840663
$$567$$ 9.00000i 0.377964i
$$568$$ − 8.00000i − 0.335673i
$$569$$ −42.0000 −1.76073 −0.880366 0.474295i $$-0.842703\pi$$
−0.880366 + 0.474295i $$0.842703\pi$$
$$570$$ 0 0
$$571$$ 4.00000 0.167395 0.0836974 0.996491i $$-0.473327\pi$$
0.0836974 + 0.996491i $$0.473327\pi$$
$$572$$ 6.00000i 0.250873i
$$573$$ 0 0
$$574$$ −6.00000 −0.250435
$$575$$ 0 0
$$576$$ −3.00000 −0.125000
$$577$$ − 38.0000i − 1.58196i −0.611842 0.790980i $$-0.709571\pi$$
0.611842 0.790980i $$-0.290429\pi$$
$$578$$ − 13.0000i − 0.540729i
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −12.0000 −0.497844
$$582$$ 0 0
$$583$$ − 2.00000i − 0.0828315i
$$584$$ −10.0000 −0.413803
$$585$$ 0 0
$$586$$ 14.0000 0.578335
$$587$$ 8.00000i 0.330195i 0.986277 + 0.165098i $$0.0527939\pi$$
−0.986277 + 0.165098i $$0.947206\pi$$
$$588$$ 0 0
$$589$$ 0 0
$$590$$ 0 0
$$591$$ 0 0
$$592$$ − 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
$$595$$ 0 0
$$596$$ 6.00000 0.245770
$$597$$ 0 0
$$598$$ 24.0000i 0.981433i
$$599$$ 40.0000 1.63436 0.817178 0.576386i $$-0.195537\pi$$
0.817178 + 0.576386i $$0.195537\pi$$
$$600$$ 0 0
$$601$$ 26.0000 1.06056 0.530281 0.847822i $$-0.322086\pi$$
0.530281 + 0.847822i $$0.322086\pi$$
$$602$$ 4.00000i 0.163028i
$$603$$ − 24.0000i − 0.977356i
$$604$$ −8.00000 −0.325515
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 40.0000i 1.62355i 0.583970 + 0.811775i $$0.301498\pi$$
−0.583970 + 0.811775i $$0.698502\pi$$
$$608$$ − 4.00000i − 0.162221i
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −24.0000 −0.970936
$$612$$ 6.00000i 0.242536i
$$613$$ − 34.0000i − 1.37325i −0.727013 0.686624i $$-0.759092\pi$$
0.727013 0.686624i $$-0.240908\pi$$
$$614$$ −20.0000 −0.807134
$$615$$ 0 0
$$616$$ 1.00000 0.0402911
$$617$$ 34.0000i 1.36879i 0.729112 + 0.684394i $$0.239933\pi$$
−0.729112 + 0.684394i $$0.760067\pi$$
$$618$$ 0 0
$$619$$ −4.00000 −0.160774 −0.0803868 0.996764i $$-0.525616\pi$$
−0.0803868 + 0.996764i $$0.525616\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ − 10.0000i − 0.400642i
$$624$$ 0 0
$$625$$ 0 0
$$626$$ −10.0000 −0.399680
$$627$$ 0 0
$$628$$ 2.00000i 0.0798087i
$$629$$ −4.00000 −0.159490
$$630$$ 0 0
$$631$$ 8.00000 0.318475 0.159237 0.987240i $$-0.449096\pi$$
0.159237 + 0.987240i $$0.449096\pi$$
$$632$$ 8.00000i 0.318223i
$$633$$ 0 0
$$634$$ 14.0000 0.556011
$$635$$ 0 0
$$636$$ 0 0
$$637$$ − 6.00000i − 0.237729i
$$638$$ − 6.00000i − 0.237542i
$$639$$ −24.0000 −0.949425
$$640$$ 0 0
$$641$$ −30.0000 −1.18493 −0.592464 0.805597i $$-0.701845\pi$$
−0.592464 + 0.805597i $$0.701845\pi$$
$$642$$ 0 0
$$643$$ − 32.0000i − 1.26196i −0.775800 0.630978i $$-0.782654\pi$$
0.775800 0.630978i $$-0.217346\pi$$
$$644$$ 4.00000 0.157622
$$645$$ 0 0
$$646$$ −8.00000 −0.314756
$$647$$ − 12.0000i − 0.471769i −0.971781 0.235884i $$-0.924201\pi$$
0.971781 0.235884i $$-0.0757987\pi$$
$$648$$ 9.00000i 0.353553i
$$649$$ 12.0000 0.471041
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 8.00000i 0.313304i
$$653$$ 42.0000i 1.64359i 0.569785 + 0.821794i $$0.307026\pi$$
−0.569785 + 0.821794i $$0.692974\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ −6.00000 −0.234261
$$657$$ 30.0000i 1.17041i
$$658$$ 4.00000i 0.155936i
$$659$$ −36.0000 −1.40236 −0.701180 0.712984i $$-0.747343\pi$$
−0.701180 + 0.712984i $$0.747343\pi$$
$$660$$ 0 0
$$661$$ 22.0000 0.855701 0.427850 0.903850i $$-0.359271\pi$$
0.427850 + 0.903850i $$0.359271\pi$$
$$662$$ 28.0000i 1.08825i
$$663$$ 0 0
$$664$$ −12.0000 −0.465690
$$665$$ 0 0
$$666$$ −6.00000 −0.232495
$$667$$ − 24.0000i − 0.929284i
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 2.00000 0.0772091
$$672$$ 0 0
$$673$$ 10.0000i 0.385472i 0.981251 + 0.192736i $$0.0617360\pi$$
−0.981251 + 0.192736i $$0.938264\pi$$
$$674$$ 22.0000 0.847408
$$675$$ 0 0
$$676$$ 23.0000 0.884615
$$677$$ − 14.0000i − 0.538064i −0.963131 0.269032i $$-0.913296\pi$$
0.963131 0.269032i $$-0.0867037\pi$$
$$678$$ 0 0
$$679$$ 6.00000 0.230259
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ − 24.0000i − 0.918334i −0.888350 0.459167i $$-0.848148\pi$$
0.888350 0.459167i $$-0.151852\pi$$
$$684$$ −12.0000 −0.458831
$$685$$ 0 0
$$686$$ −1.00000 −0.0381802
$$687$$ 0 0
$$688$$ 4.00000i 0.152499i
$$689$$ −12.0000 −0.457164
$$690$$ 0 0
$$691$$ −36.0000 −1.36950 −0.684752 0.728776i $$-0.740090\pi$$
−0.684752 + 0.728776i $$0.740090\pi$$
$$692$$ − 14.0000i − 0.532200i
$$693$$ − 3.00000i − 0.113961i
$$694$$ −28.0000 −1.06287
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 12.0000i 0.454532i
$$698$$ − 34.0000i − 1.28692i
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −42.0000 −1.58632 −0.793159 0.609015i $$-0.791565\pi$$
−0.793159 + 0.609015i $$0.791565\pi$$
$$702$$ 0 0
$$703$$ − 8.00000i − 0.301726i
$$704$$ 1.00000 0.0376889
$$705$$ 0 0
$$706$$ 14.0000 0.526897
$$707$$ 14.0000i 0.526524i
$$708$$ 0 0
$$709$$ −22.0000 −0.826227 −0.413114 0.910679i $$-0.635559\pi$$
−0.413114 + 0.910679i $$0.635559\pi$$
$$710$$ 0 0
$$711$$ 24.0000 0.900070
$$712$$ − 10.0000i − 0.374766i
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 20.0000 0.747435
$$717$$ 0 0
$$718$$ − 16.0000i − 0.597115i
$$719$$ 40.0000 1.49175 0.745874 0.666087i $$-0.232032\pi$$
0.745874 + 0.666087i $$0.232032\pi$$
$$720$$ 0 0
$$721$$ −4.00000 −0.148968
$$722$$ 3.00000i 0.111648i
$$723$$ 0 0
$$724$$ −6.00000 −0.222988
$$725$$ 0 0
$$726$$ 0 0
$$727$$ − 12.0000i − 0.445055i −0.974926 0.222528i $$-0.928569\pi$$
0.974926 0.222528i $$-0.0714308\pi$$
$$728$$ − 6.00000i − 0.222375i
$$729$$ 27.0000 1.00000
$$730$$ 0 0
$$731$$ 8.00000 0.295891
$$732$$ 0 0
$$733$$ 46.0000i 1.69905i 0.527549 + 0.849524i $$0.323111\pi$$
−0.527549 + 0.849524i $$0.676889\pi$$
$$734$$ 20.0000 0.738213
$$735$$ 0 0
$$736$$ 4.00000 0.147442
$$737$$ 8.00000i 0.294684i
$$738$$ 18.0000i 0.662589i
$$739$$ 12.0000 0.441427 0.220714 0.975339i $$-0.429161\pi$$
0.220714 + 0.975339i $$0.429161\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 2.00000i 0.0734223i
$$743$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ −34.0000 −1.24483
$$747$$ 36.0000i 1.31717i
$$748$$ − 2.00000i − 0.0731272i
$$749$$ −4.00000 −0.146157
$$750$$ 0 0
$$751$$ −32.0000 −1.16770 −0.583848 0.811863i $$-0.698454\pi$$
−0.583848 + 0.811863i $$0.698454\pi$$
$$752$$ 4.00000i 0.145865i
$$753$$ 0 0
$$754$$ −36.0000 −1.31104
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 6.00000i 0.218074i 0.994038 + 0.109037i $$0.0347767\pi$$
−0.994038 + 0.109037i $$0.965223\pi$$
$$758$$ − 12.0000i − 0.435860i
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −14.0000 −0.507500 −0.253750 0.967270i $$-0.581664\pi$$
−0.253750 + 0.967270i $$0.581664\pi$$
$$762$$ 0 0
$$763$$ − 14.0000i − 0.506834i
$$764$$ −8.00000 −0.289430
$$765$$ 0 0
$$766$$ 28.0000 1.01168
$$767$$ − 72.0000i − 2.59977i
$$768$$ 0 0
$$769$$ 14.0000 0.504853 0.252426 0.967616i $$-0.418771\pi$$
0.252426 + 0.967616i $$0.418771\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 6.00000i 0.215945i
$$773$$ − 30.0000i − 1.07903i −0.841978 0.539513i $$-0.818609\pi$$
0.841978 0.539513i $$-0.181391\pi$$
$$774$$ 12.0000 0.431331
$$775$$ 0 0
$$776$$ 6.00000 0.215387
$$777$$ 0 0
$$778$$ 6.00000i 0.215110i
$$779$$ −24.0000 −0.859889
$$780$$ 0 0
$$781$$ 8.00000 0.286263
$$782$$ − 8.00000i − 0.286079i
$$783$$ 0 0
$$784$$ −1.00000 −0.0357143
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 28.0000i 0.998092i 0.866575 + 0.499046i $$0.166316\pi$$
−0.866575 + 0.499046i $$0.833684\pi$$
$$788$$ − 18.0000i − 0.641223i
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −6.00000 −0.213335
$$792$$ − 3.00000i − 0.106600i
$$793$$ − 12.0000i − 0.426132i
$$794$$ −34.0000 −1.20661
$$795$$ 0 0
$$796$$ −24.0000 −0.850657
$$797$$ − 34.0000i − 1.20434i −0.798367 0.602171i $$-0.794303\pi$$
0.798367 0.602171i $$-0.205697\pi$$
$$798$$ 0 0
$$799$$ 8.00000 0.283020
$$800$$ 0 0
$$801$$ −30.0000 −1.06000
$$802$$ − 34.0000i − 1.20058i
$$803$$ − 10.0000i − 0.352892i
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 14.0000i 0.492518i
$$809$$ 46.0000 1.61727 0.808637 0.588308i $$-0.200206\pi$$
0.808637 + 0.588308i $$0.200206\pi$$
$$810$$ 0 0
$$811$$ 52.0000 1.82597 0.912983 0.407997i $$-0.133772\pi$$
0.912983 + 0.407997i $$0.133772\pi$$
$$812$$ 6.00000i 0.210559i
$$813$$ 0 0
$$814$$ 2.00000 0.0701000
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 16.0000i 0.559769i
$$818$$ 34.0000i 1.18878i
$$819$$ −18.0000 −0.628971
$$820$$ 0 0
$$821$$ 30.0000 1.04701 0.523504 0.852023i $$-0.324625\pi$$
0.523504 + 0.852023i $$0.324625\pi$$
$$822$$ 0 0
$$823$$ − 4.00000i − 0.139431i −0.997567 0.0697156i $$-0.977791\pi$$
0.997567 0.0697156i $$-0.0222092\pi$$
$$824$$ −4.00000 −0.139347
$$825$$ 0 0
$$826$$ −12.0000 −0.417533
$$827$$ − 12.0000i − 0.417281i −0.977992 0.208640i $$-0.933096\pi$$
0.977992 0.208640i $$-0.0669038\pi$$
$$828$$ − 12.0000i − 0.417029i
$$829$$ −30.0000 −1.04194 −0.520972 0.853574i $$-0.674430\pi$$
−0.520972 + 0.853574i $$0.674430\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ − 6.00000i − 0.208013i
$$833$$ 2.00000i 0.0692959i
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 4.00000 0.138343
$$837$$ 0 0
$$838$$ 12.0000i 0.414533i
$$839$$ 40.0000 1.38095 0.690477 0.723355i $$-0.257401\pi$$
0.690477 + 0.723355i $$0.257401\pi$$
$$840$$ 0 0
$$841$$ 7.00000 0.241379
$$842$$ 26.0000i 0.896019i
$$843$$ 0 0
$$844$$ −4.00000 −0.137686
$$845$$ 0 0
$$846$$ 12.0000 0.412568
$$847$$ 1.00000i 0.0343604i
$$848$$ 2.00000i 0.0686803i
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 8.00000 0.274236
$$852$$ 0 0
$$853$$ 6.00000i 0.205436i 0.994711 + 0.102718i $$0.0327539\pi$$
−0.994711 + 0.102718i $$0.967246\pi$$
$$854$$ −2.00000 −0.0684386
$$855$$ 0 0
$$856$$ −4.00000 −0.136717
$$857$$ − 42.0000i − 1.43469i −0.696717 0.717346i $$-0.745357\pi$$
0.696717 0.717346i $$-0.254643\pi$$
$$858$$ 0 0
$$859$$ 44.0000 1.50126 0.750630 0.660722i $$-0.229750\pi$$
0.750630 + 0.660722i $$0.229750\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ − 8.00000i − 0.272481i
$$863$$ 52.0000i 1.77010i 0.465495 + 0.885050i $$0.345876\pi$$
−0.465495 + 0.885050i $$0.654124\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 6.00000 0.203888
$$867$$ 0 0
$$868$$ 0 0
$$869$$ −8.00000 −0.271381
$$870$$ 0 0
$$871$$ 48.0000 1.62642
$$872$$ − 14.0000i − 0.474100i
$$873$$ − 18.0000i − 0.609208i
$$874$$ 16.0000 0.541208
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 10.0000i 0.337676i 0.985644 + 0.168838i $$0.0540015\pi$$
−0.985644 + 0.168838i $$0.945999\pi$$
$$878$$ − 24.0000i − 0.809961i
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 2.00000 0.0673817 0.0336909 0.999432i $$-0.489274\pi$$
0.0336909 + 0.999432i $$0.489274\pi$$
$$882$$ 3.00000i 0.101015i
$$883$$ − 56.0000i − 1.88455i −0.334840 0.942275i $$-0.608682\pi$$
0.334840 0.942275i $$-0.391318\pi$$
$$884$$ −12.0000 −0.403604
$$885$$ 0 0
$$886$$ −16.0000 −0.537531
$$887$$ − 48.0000i − 1.61168i −0.592132 0.805841i $$-0.701714\pi$$
0.592132 0.805841i $$-0.298286\pi$$
$$888$$ 0 0
$$889$$ 8.00000 0.268311
$$890$$ 0 0
$$891$$ −9.00000 −0.301511
$$892$$ − 4.00000i − 0.133930i
$$893$$ 16.0000i 0.535420i
$$894$$ 0 0
$$895$$ 0 0
$$896$$ −1.00000 −0.0334077
$$897$$ 0 0
$$898$$ 2.00000i 0.0667409i
$$899$$ 0 0
$$900$$ 0 0
$$901$$ 4.00000 0.133259
$$902$$ − 6.00000i − 0.199778i
$$903$$ 0 0
$$904$$ −6.00000 −0.199557
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 32.0000i 1.06254i 0.847202 + 0.531271i $$0.178286\pi$$
−0.847202 + 0.531271i $$0.821714\pi$$
$$908$$ − 4.00000i − 0.132745i
$$909$$ 42.0000 1.39305
$$910$$ 0 0
$$911$$ −40.0000 −1.32526 −0.662630 0.748947i $$-0.730560\pi$$
−0.662630 + 0.748947i $$0.730560\pi$$
$$912$$ 0 0
$$913$$ − 12.0000i − 0.397142i
$$914$$ −26.0000 −0.860004
$$915$$ 0 0
$$916$$ 6.00000 0.198246
$$917$$ − 12.0000i − 0.396275i
$$918$$ 0 0
$$919$$ 40.0000 1.31948 0.659739 0.751495i $$-0.270667\pi$$
0.659739 + 0.751495i $$0.270667\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ − 6.00000i − 0.197599i
$$923$$ − 48.0000i − 1.57994i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ −20.0000 −0.657241
$$927$$ 12.0000i 0.394132i
$$928$$ 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
$$931$$ −4.00000 −0.131095
$$932$$ − 10.0000i − 0.327561i
$$933$$ 0 0
$$934$$ 0 0
$$935$$ 0 0
$$936$$ −18.0000 −0.588348
$$937$$ − 50.0000i − 1.63343i −0.577042 0.816714i $$-0.695793\pi$$
0.577042 0.816714i $$-0.304207\pi$$
$$938$$ − 8.00000i − 0.261209i
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 38.0000 1.23876 0.619382 0.785090i $$-0.287383\pi$$
0.619382 + 0.785090i $$0.287383\pi$$
$$942$$ 0 0
$$943$$ − 24.0000i − 0.781548i
$$944$$ −12.0000 −0.390567
$$945$$ 0 0
$$946$$ −4.00000 −0.130051
$$947$$ 24.0000i 0.779895i 0.920837 + 0.389948i $$0.127507\pi$$
−0.920837 + 0.389948i $$0.872493\pi$$
$$948$$ 0 0
$$949$$ −60.0000 −1.94768
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 2.00000i 0.0648204i
$$953$$ − 22.0000i − 0.712650i −0.934362 0.356325i $$-0.884030\pi$$
0.934362 0.356325i $$-0.115970\pi$$
$$954$$ 6.00000 0.194257
$$955$$ 0 0
$$956$$ −24.0000 −0.776215
$$957$$ 0 0
$$958$$ − 24.0000i − 0.775405i
$$959$$ −18.0000 −0.581250
$$960$$ 0 0
$$961$$ −31.0000 −1.00000
$$962$$ − 12.0000i − 0.386896i
$$963$$ 12.0000i 0.386695i
$$964$$ −2.00000 −0.0644157
$$965$$ 0 0
$$966$$ 0 0
$$967$$ − 32.0000i − 1.02905i −0.857475 0.514525i $$-0.827968\pi$$
0.857475 0.514525i $$-0.172032\pi$$
$$968$$ 1.00000i 0.0321412i
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −52.0000 −1.66876 −0.834380 0.551190i $$-0.814174\pi$$
−0.834380 + 0.551190i $$0.814174\pi$$
$$972$$ 0 0
$$973$$ 20.0000i 0.641171i
$$974$$ −28.0000 −0.897178
$$975$$ 0 0
$$976$$ −2.00000 −0.0640184
$$977$$ − 54.0000i − 1.72761i −0.503824 0.863807i $$-0.668074\pi$$
0.503824 0.863807i $$-0.331926\pi$$
$$978$$ 0 0
$$979$$ 10.0000 0.319601
$$980$$ 0 0
$$981$$ −42.0000 −1.34096
$$982$$ − 20.0000i − 0.638226i
$$983$$ 4.00000i 0.127580i 0.997963 + 0.0637901i $$0.0203188\pi$$
−0.997963 + 0.0637901i $$0.979681\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 12.0000 0.382158
$$987$$ 0 0
$$988$$ − 24.0000i − 0.763542i
$$989$$ −16.0000 −0.508770
$$990$$ 0 0
$$991$$ −40.0000 −1.27064 −0.635321 0.772248i $$-0.719132\pi$$
−0.635321 + 0.772248i $$0.719132\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ −8.00000 −0.253745
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 10.0000i 0.316703i 0.987383 + 0.158352i $$0.0506179\pi$$
−0.987383 + 0.158352i $$0.949382\pi$$
$$998$$ 28.0000i 0.886325i
$$999$$ 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 3850.2.c.m.1849.1 2
5.2 odd 4 3850.2.a.s.1.1 1
5.3 odd 4 770.2.a.d.1.1 1
5.4 even 2 inner 3850.2.c.m.1849.2 2
15.8 even 4 6930.2.a.x.1.1 1
20.3 even 4 6160.2.a.e.1.1 1
35.13 even 4 5390.2.a.j.1.1 1
55.43 even 4 8470.2.a.z.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
770.2.a.d.1.1 1 5.3 odd 4
3850.2.a.s.1.1 1 5.2 odd 4
3850.2.c.m.1849.1 2 1.1 even 1 trivial
3850.2.c.m.1849.2 2 5.4 even 2 inner
5390.2.a.j.1.1 1 35.13 even 4
6160.2.a.e.1.1 1 20.3 even 4
6930.2.a.x.1.1 1 15.8 even 4
8470.2.a.z.1.1 1 55.43 even 4