# Properties

 Label 825.2.c.c.199.1 Level $825$ Weight $2$ Character 825.199 Analytic conductor $6.588$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$825 = 3 \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 825.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.58765816676$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 165) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 199.1 Root $$0.866025 - 0.500000i$$ of defining polynomial Character $$\chi$$ $$=$$ 825.199 Dual form 825.2.c.c.199.4

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.73205i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.73205 q^{6} +2.00000i q^{7} -1.73205i q^{8} -1.00000 q^{9} +O(q^{10})$$ $$q-1.73205i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.73205 q^{6} +2.00000i q^{7} -1.73205i q^{8} -1.00000 q^{9} -1.00000 q^{11} +1.00000i q^{12} -5.46410i q^{13} +3.46410 q^{14} -5.00000 q^{16} +1.73205i q^{18} -5.46410 q^{19} +2.00000 q^{21} +1.73205i q^{22} -6.92820i q^{23} -1.73205 q^{24} -9.46410 q^{26} +1.00000i q^{27} -2.00000i q^{28} +3.46410 q^{29} -10.9282 q^{31} +5.19615i q^{32} +1.00000i q^{33} +1.00000 q^{36} -4.92820i q^{37} +9.46410i q^{38} -5.46410 q^{39} +3.46410 q^{41} -3.46410i q^{42} +4.92820i q^{43} +1.00000 q^{44} -12.0000 q^{46} -6.92820i q^{47} +5.00000i q^{48} +3.00000 q^{49} +5.46410i q^{52} -0.928203i q^{53} +1.73205 q^{54} +3.46410 q^{56} +5.46410i q^{57} -6.00000i q^{58} +6.92820 q^{59} +2.00000 q^{61} +18.9282i q^{62} -2.00000i q^{63} -1.00000 q^{64} +1.73205 q^{66} +8.00000i q^{67} -6.92820 q^{69} +13.8564 q^{71} +1.73205i q^{72} +8.39230i q^{73} -8.53590 q^{74} +5.46410 q^{76} -2.00000i q^{77} +9.46410i q^{78} +6.53590 q^{79} +1.00000 q^{81} -6.00000i q^{82} -8.53590i q^{83} -2.00000 q^{84} +8.53590 q^{86} -3.46410i q^{87} +1.73205i q^{88} -0.928203 q^{89} +10.9282 q^{91} +6.92820i q^{92} +10.9282i q^{93} -12.0000 q^{94} +5.19615 q^{96} -10.0000i q^{97} -5.19615i q^{98} +1.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{4} - 4q^{9} + O(q^{10})$$ $$4q - 4q^{4} - 4q^{9} - 4q^{11} - 20q^{16} - 8q^{19} + 8q^{21} - 24q^{26} - 16q^{31} + 4q^{36} - 8q^{39} + 4q^{44} - 48q^{46} + 12q^{49} + 8q^{61} - 4q^{64} - 48q^{74} + 8q^{76} + 40q^{79} + 4q^{81} - 8q^{84} + 48q^{86} + 24q^{89} + 16q^{91} - 48q^{94} + 4q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/825\mathbb{Z}\right)^\times$$.

 $$n$$ $$376$$ $$551$$ $$727$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ − 1.73205i − 1.22474i −0.790569 0.612372i $$-0.790215\pi$$
0.790569 0.612372i $$-0.209785\pi$$
$$3$$ − 1.00000i − 0.577350i
$$4$$ −1.00000 −0.500000
$$5$$ 0 0
$$6$$ −1.73205 −0.707107
$$7$$ 2.00000i 0.755929i 0.925820 + 0.377964i $$0.123376\pi$$
−0.925820 + 0.377964i $$0.876624\pi$$
$$8$$ − 1.73205i − 0.612372i
$$9$$ −1.00000 −0.333333
$$10$$ 0 0
$$11$$ −1.00000 −0.301511
$$12$$ 1.00000i 0.288675i
$$13$$ − 5.46410i − 1.51547i −0.652563 0.757735i $$-0.726306\pi$$
0.652563 0.757735i $$-0.273694\pi$$
$$14$$ 3.46410 0.925820
$$15$$ 0 0
$$16$$ −5.00000 −1.25000
$$17$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$18$$ 1.73205i 0.408248i
$$19$$ −5.46410 −1.25355 −0.626775 0.779200i $$-0.715626\pi$$
−0.626775 + 0.779200i $$0.715626\pi$$
$$20$$ 0 0
$$21$$ 2.00000 0.436436
$$22$$ 1.73205i 0.369274i
$$23$$ − 6.92820i − 1.44463i −0.691564 0.722315i $$-0.743078\pi$$
0.691564 0.722315i $$-0.256922\pi$$
$$24$$ −1.73205 −0.353553
$$25$$ 0 0
$$26$$ −9.46410 −1.85606
$$27$$ 1.00000i 0.192450i
$$28$$ − 2.00000i − 0.377964i
$$29$$ 3.46410 0.643268 0.321634 0.946864i $$-0.395768\pi$$
0.321634 + 0.946864i $$0.395768\pi$$
$$30$$ 0 0
$$31$$ −10.9282 −1.96276 −0.981382 0.192068i $$-0.938481\pi$$
−0.981382 + 0.192068i $$0.938481\pi$$
$$32$$ 5.19615i 0.918559i
$$33$$ 1.00000i 0.174078i
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ − 4.92820i − 0.810192i −0.914274 0.405096i $$-0.867238\pi$$
0.914274 0.405096i $$-0.132762\pi$$
$$38$$ 9.46410i 1.53528i
$$39$$ −5.46410 −0.874957
$$40$$ 0 0
$$41$$ 3.46410 0.541002 0.270501 0.962720i $$-0.412811\pi$$
0.270501 + 0.962720i $$0.412811\pi$$
$$42$$ − 3.46410i − 0.534522i
$$43$$ 4.92820i 0.751544i 0.926712 + 0.375772i $$0.122622\pi$$
−0.926712 + 0.375772i $$0.877378\pi$$
$$44$$ 1.00000 0.150756
$$45$$ 0 0
$$46$$ −12.0000 −1.76930
$$47$$ − 6.92820i − 1.01058i −0.862949 0.505291i $$-0.831385\pi$$
0.862949 0.505291i $$-0.168615\pi$$
$$48$$ 5.00000i 0.721688i
$$49$$ 3.00000 0.428571
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 5.46410i 0.757735i
$$53$$ − 0.928203i − 0.127499i −0.997966 0.0637493i $$-0.979694\pi$$
0.997966 0.0637493i $$-0.0203058\pi$$
$$54$$ 1.73205 0.235702
$$55$$ 0 0
$$56$$ 3.46410 0.462910
$$57$$ 5.46410i 0.723738i
$$58$$ − 6.00000i − 0.787839i
$$59$$ 6.92820 0.901975 0.450988 0.892530i $$-0.351072\pi$$
0.450988 + 0.892530i $$0.351072\pi$$
$$60$$ 0 0
$$61$$ 2.00000 0.256074 0.128037 0.991769i $$-0.459132\pi$$
0.128037 + 0.991769i $$0.459132\pi$$
$$62$$ 18.9282i 2.40388i
$$63$$ − 2.00000i − 0.251976i
$$64$$ −1.00000 −0.125000
$$65$$ 0 0
$$66$$ 1.73205 0.213201
$$67$$ 8.00000i 0.977356i 0.872464 + 0.488678i $$0.162521\pi$$
−0.872464 + 0.488678i $$0.837479\pi$$
$$68$$ 0 0
$$69$$ −6.92820 −0.834058
$$70$$ 0 0
$$71$$ 13.8564 1.64445 0.822226 0.569160i $$-0.192732\pi$$
0.822226 + 0.569160i $$0.192732\pi$$
$$72$$ 1.73205i 0.204124i
$$73$$ 8.39230i 0.982245i 0.871091 + 0.491122i $$0.163413\pi$$
−0.871091 + 0.491122i $$0.836587\pi$$
$$74$$ −8.53590 −0.992278
$$75$$ 0 0
$$76$$ 5.46410 0.626775
$$77$$ − 2.00000i − 0.227921i
$$78$$ 9.46410i 1.07160i
$$79$$ 6.53590 0.735346 0.367673 0.929955i $$-0.380155\pi$$
0.367673 + 0.929955i $$0.380155\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ − 6.00000i − 0.662589i
$$83$$ − 8.53590i − 0.936937i −0.883480 0.468468i $$-0.844806\pi$$
0.883480 0.468468i $$-0.155194\pi$$
$$84$$ −2.00000 −0.218218
$$85$$ 0 0
$$86$$ 8.53590 0.920450
$$87$$ − 3.46410i − 0.371391i
$$88$$ 1.73205i 0.184637i
$$89$$ −0.928203 −0.0983893 −0.0491947 0.998789i $$-0.515665\pi$$
−0.0491947 + 0.998789i $$0.515665\pi$$
$$90$$ 0 0
$$91$$ 10.9282 1.14559
$$92$$ 6.92820i 0.722315i
$$93$$ 10.9282i 1.13320i
$$94$$ −12.0000 −1.23771
$$95$$ 0 0
$$96$$ 5.19615 0.530330
$$97$$ − 10.0000i − 1.01535i −0.861550 0.507673i $$-0.830506\pi$$
0.861550 0.507673i $$-0.169494\pi$$
$$98$$ − 5.19615i − 0.524891i
$$99$$ 1.00000 0.100504
$$100$$ 0 0
$$101$$ −10.3923 −1.03407 −0.517036 0.855963i $$-0.672965\pi$$
−0.517036 + 0.855963i $$0.672965\pi$$
$$102$$ 0 0
$$103$$ − 8.00000i − 0.788263i −0.919054 0.394132i $$-0.871045\pi$$
0.919054 0.394132i $$-0.128955\pi$$
$$104$$ −9.46410 −0.928032
$$105$$ 0 0
$$106$$ −1.60770 −0.156153
$$107$$ 8.53590i 0.825196i 0.910913 + 0.412598i $$0.135379\pi$$
−0.910913 + 0.412598i $$0.864621\pi$$
$$108$$ − 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
$$111$$ −4.92820 −0.467764
$$112$$ − 10.0000i − 0.944911i
$$113$$ 12.9282i 1.21618i 0.793867 + 0.608092i $$0.208065\pi$$
−0.793867 + 0.608092i $$0.791935\pi$$
$$114$$ 9.46410 0.886394
$$115$$ 0 0
$$116$$ −3.46410 −0.321634
$$117$$ 5.46410i 0.505156i
$$118$$ − 12.0000i − 1.10469i
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ − 3.46410i − 0.313625i
$$123$$ − 3.46410i − 0.312348i
$$124$$ 10.9282 0.981382
$$125$$ 0 0
$$126$$ −3.46410 −0.308607
$$127$$ 8.92820i 0.792250i 0.918197 + 0.396125i $$0.129645\pi$$
−0.918197 + 0.396125i $$0.870355\pi$$
$$128$$ 12.1244i 1.07165i
$$129$$ 4.92820 0.433904
$$130$$ 0 0
$$131$$ 18.9282 1.65376 0.826882 0.562375i $$-0.190112\pi$$
0.826882 + 0.562375i $$0.190112\pi$$
$$132$$ − 1.00000i − 0.0870388i
$$133$$ − 10.9282i − 0.947595i
$$134$$ 13.8564 1.19701
$$135$$ 0 0
$$136$$ 0 0
$$137$$ − 18.0000i − 1.53784i −0.639343 0.768922i $$-0.720793\pi$$
0.639343 0.768922i $$-0.279207\pi$$
$$138$$ 12.0000i 1.02151i
$$139$$ −12.3923 −1.05110 −0.525551 0.850762i $$-0.676141\pi$$
−0.525551 + 0.850762i $$0.676141\pi$$
$$140$$ 0 0
$$141$$ −6.92820 −0.583460
$$142$$ − 24.0000i − 2.01404i
$$143$$ 5.46410i 0.456931i
$$144$$ 5.00000 0.416667
$$145$$ 0 0
$$146$$ 14.5359 1.20300
$$147$$ − 3.00000i − 0.247436i
$$148$$ 4.92820i 0.405096i
$$149$$ 15.4641 1.26687 0.633434 0.773796i $$-0.281645\pi$$
0.633434 + 0.773796i $$0.281645\pi$$
$$150$$ 0 0
$$151$$ −20.3923 −1.65950 −0.829751 0.558134i $$-0.811518\pi$$
−0.829751 + 0.558134i $$0.811518\pi$$
$$152$$ 9.46410i 0.767640i
$$153$$ 0 0
$$154$$ −3.46410 −0.279145
$$155$$ 0 0
$$156$$ 5.46410 0.437478
$$157$$ − 3.07180i − 0.245156i −0.992459 0.122578i $$-0.960884\pi$$
0.992459 0.122578i $$-0.0391162\pi$$
$$158$$ − 11.3205i − 0.900611i
$$159$$ −0.928203 −0.0736113
$$160$$ 0 0
$$161$$ 13.8564 1.09204
$$162$$ − 1.73205i − 0.136083i
$$163$$ − 9.85641i − 0.772013i −0.922496 0.386007i $$-0.873854\pi$$
0.922496 0.386007i $$-0.126146\pi$$
$$164$$ −3.46410 −0.270501
$$165$$ 0 0
$$166$$ −14.7846 −1.14751
$$167$$ − 10.3923i − 0.804181i −0.915600 0.402090i $$-0.868284\pi$$
0.915600 0.402090i $$-0.131716\pi$$
$$168$$ − 3.46410i − 0.267261i
$$169$$ −16.8564 −1.29665
$$170$$ 0 0
$$171$$ 5.46410 0.417850
$$172$$ − 4.92820i − 0.375772i
$$173$$ 12.0000i 0.912343i 0.889892 + 0.456172i $$0.150780\pi$$
−0.889892 + 0.456172i $$0.849220\pi$$
$$174$$ −6.00000 −0.454859
$$175$$ 0 0
$$176$$ 5.00000 0.376889
$$177$$ − 6.92820i − 0.520756i
$$178$$ 1.60770i 0.120502i
$$179$$ −6.92820 −0.517838 −0.258919 0.965899i $$-0.583366\pi$$
−0.258919 + 0.965899i $$0.583366\pi$$
$$180$$ 0 0
$$181$$ 15.8564 1.17860 0.589299 0.807915i $$-0.299404\pi$$
0.589299 + 0.807915i $$0.299404\pi$$
$$182$$ − 18.9282i − 1.40305i
$$183$$ − 2.00000i − 0.147844i
$$184$$ −12.0000 −0.884652
$$185$$ 0 0
$$186$$ 18.9282 1.38788
$$187$$ 0 0
$$188$$ 6.92820i 0.505291i
$$189$$ −2.00000 −0.145479
$$190$$ 0 0
$$191$$ −18.9282 −1.36960 −0.684798 0.728733i $$-0.740110\pi$$
−0.684798 + 0.728733i $$0.740110\pi$$
$$192$$ 1.00000i 0.0721688i
$$193$$ − 24.3923i − 1.75580i −0.478847 0.877898i $$-0.658945\pi$$
0.478847 0.877898i $$-0.341055\pi$$
$$194$$ −17.3205 −1.24354
$$195$$ 0 0
$$196$$ −3.00000 −0.214286
$$197$$ − 12.0000i − 0.854965i −0.904024 0.427482i $$-0.859401\pi$$
0.904024 0.427482i $$-0.140599\pi$$
$$198$$ − 1.73205i − 0.123091i
$$199$$ 24.7846 1.75693 0.878467 0.477803i $$-0.158567\pi$$
0.878467 + 0.477803i $$0.158567\pi$$
$$200$$ 0 0
$$201$$ 8.00000 0.564276
$$202$$ 18.0000i 1.26648i
$$203$$ 6.92820i 0.486265i
$$204$$ 0 0
$$205$$ 0 0
$$206$$ −13.8564 −0.965422
$$207$$ 6.92820i 0.481543i
$$208$$ 27.3205i 1.89434i
$$209$$ 5.46410 0.377960
$$210$$ 0 0
$$211$$ −8.39230 −0.577750 −0.288875 0.957367i $$-0.593281\pi$$
−0.288875 + 0.957367i $$0.593281\pi$$
$$212$$ 0.928203i 0.0637493i
$$213$$ − 13.8564i − 0.949425i
$$214$$ 14.7846 1.01066
$$215$$ 0 0
$$216$$ 1.73205 0.117851
$$217$$ − 21.8564i − 1.48371i
$$218$$ − 17.3205i − 1.17309i
$$219$$ 8.39230 0.567099
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 8.53590i 0.572892i
$$223$$ − 9.85641i − 0.660034i −0.943975 0.330017i $$-0.892946\pi$$
0.943975 0.330017i $$-0.107054\pi$$
$$224$$ −10.3923 −0.694365
$$225$$ 0 0
$$226$$ 22.3923 1.48951
$$227$$ 15.4641i 1.02639i 0.858272 + 0.513194i $$0.171538\pi$$
−0.858272 + 0.513194i $$0.828462\pi$$
$$228$$ − 5.46410i − 0.361869i
$$229$$ 23.8564 1.57648 0.788238 0.615371i $$-0.210994\pi$$
0.788238 + 0.615371i $$0.210994\pi$$
$$230$$ 0 0
$$231$$ −2.00000 −0.131590
$$232$$ − 6.00000i − 0.393919i
$$233$$ − 12.0000i − 0.786146i −0.919507 0.393073i $$-0.871412\pi$$
0.919507 0.393073i $$-0.128588\pi$$
$$234$$ 9.46410 0.618688
$$235$$ 0 0
$$236$$ −6.92820 −0.450988
$$237$$ − 6.53590i − 0.424552i
$$238$$ 0 0
$$239$$ 12.0000 0.776215 0.388108 0.921614i $$-0.373129\pi$$
0.388108 + 0.921614i $$0.373129\pi$$
$$240$$ 0 0
$$241$$ 0.143594 0.00924967 0.00462484 0.999989i $$-0.498528\pi$$
0.00462484 + 0.999989i $$0.498528\pi$$
$$242$$ − 1.73205i − 0.111340i
$$243$$ − 1.00000i − 0.0641500i
$$244$$ −2.00000 −0.128037
$$245$$ 0 0
$$246$$ −6.00000 −0.382546
$$247$$ 29.8564i 1.89972i
$$248$$ 18.9282i 1.20194i
$$249$$ −8.53590 −0.540941
$$250$$ 0 0
$$251$$ −1.85641 −0.117175 −0.0585877 0.998282i $$-0.518660\pi$$
−0.0585877 + 0.998282i $$0.518660\pi$$
$$252$$ 2.00000i 0.125988i
$$253$$ 6.92820i 0.435572i
$$254$$ 15.4641 0.970304
$$255$$ 0 0
$$256$$ 19.0000 1.18750
$$257$$ − 19.8564i − 1.23861i −0.785151 0.619304i $$-0.787415\pi$$
0.785151 0.619304i $$-0.212585\pi$$
$$258$$ − 8.53590i − 0.531422i
$$259$$ 9.85641 0.612447
$$260$$ 0 0
$$261$$ −3.46410 −0.214423
$$262$$ − 32.7846i − 2.02544i
$$263$$ − 20.5359i − 1.26630i −0.774030 0.633149i $$-0.781762\pi$$
0.774030 0.633149i $$-0.218238\pi$$
$$264$$ 1.73205 0.106600
$$265$$ 0 0
$$266$$ −18.9282 −1.16056
$$267$$ 0.928203i 0.0568051i
$$268$$ − 8.00000i − 0.488678i
$$269$$ −19.8564 −1.21067 −0.605333 0.795972i $$-0.706960\pi$$
−0.605333 + 0.795972i $$0.706960\pi$$
$$270$$ 0 0
$$271$$ −11.6077 −0.705117 −0.352559 0.935790i $$-0.614688\pi$$
−0.352559 + 0.935790i $$0.614688\pi$$
$$272$$ 0 0
$$273$$ − 10.9282i − 0.661405i
$$274$$ −31.1769 −1.88347
$$275$$ 0 0
$$276$$ 6.92820 0.417029
$$277$$ 29.4641i 1.77033i 0.465281 + 0.885163i $$0.345953\pi$$
−0.465281 + 0.885163i $$0.654047\pi$$
$$278$$ 21.4641i 1.28733i
$$279$$ 10.9282 0.654254
$$280$$ 0 0
$$281$$ 3.46410 0.206651 0.103325 0.994648i $$-0.467052\pi$$
0.103325 + 0.994648i $$0.467052\pi$$
$$282$$ 12.0000i 0.714590i
$$283$$ 4.92820i 0.292951i 0.989214 + 0.146476i $$0.0467930\pi$$
−0.989214 + 0.146476i $$0.953207\pi$$
$$284$$ −13.8564 −0.822226
$$285$$ 0 0
$$286$$ 9.46410 0.559624
$$287$$ 6.92820i 0.408959i
$$288$$ − 5.19615i − 0.306186i
$$289$$ 17.0000 1.00000
$$290$$ 0 0
$$291$$ −10.0000 −0.586210
$$292$$ − 8.39230i − 0.491122i
$$293$$ 13.8564i 0.809500i 0.914427 + 0.404750i $$0.132641\pi$$
−0.914427 + 0.404750i $$0.867359\pi$$
$$294$$ −5.19615 −0.303046
$$295$$ 0 0
$$296$$ −8.53590 −0.496139
$$297$$ − 1.00000i − 0.0580259i
$$298$$ − 26.7846i − 1.55159i
$$299$$ −37.8564 −2.18929
$$300$$ 0 0
$$301$$ −9.85641 −0.568114
$$302$$ 35.3205i 2.03247i
$$303$$ 10.3923i 0.597022i
$$304$$ 27.3205 1.56694
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 14.0000i 0.799022i 0.916728 + 0.399511i $$0.130820\pi$$
−0.916728 + 0.399511i $$0.869180\pi$$
$$308$$ 2.00000i 0.113961i
$$309$$ −8.00000 −0.455104
$$310$$ 0 0
$$311$$ 5.07180 0.287595 0.143798 0.989607i $$-0.454069\pi$$
0.143798 + 0.989607i $$0.454069\pi$$
$$312$$ 9.46410i 0.535799i
$$313$$ − 20.9282i − 1.18293i −0.806330 0.591466i $$-0.798549\pi$$
0.806330 0.591466i $$-0.201451\pi$$
$$314$$ −5.32051 −0.300254
$$315$$ 0 0
$$316$$ −6.53590 −0.367673
$$317$$ − 24.9282i − 1.40011i −0.714090 0.700054i $$-0.753159\pi$$
0.714090 0.700054i $$-0.246841\pi$$
$$318$$ 1.60770i 0.0901551i
$$319$$ −3.46410 −0.193952
$$320$$ 0 0
$$321$$ 8.53590 0.476427
$$322$$ − 24.0000i − 1.33747i
$$323$$ 0 0
$$324$$ −1.00000 −0.0555556
$$325$$ 0 0
$$326$$ −17.0718 −0.945519
$$327$$ − 10.0000i − 0.553001i
$$328$$ − 6.00000i − 0.331295i
$$329$$ 13.8564 0.763928
$$330$$ 0 0
$$331$$ 9.85641 0.541757 0.270879 0.962614i $$-0.412686\pi$$
0.270879 + 0.962614i $$0.412686\pi$$
$$332$$ 8.53590i 0.468468i
$$333$$ 4.92820i 0.270064i
$$334$$ −18.0000 −0.984916
$$335$$ 0 0
$$336$$ −10.0000 −0.545545
$$337$$ 33.1769i 1.80726i 0.428312 + 0.903631i $$0.359108\pi$$
−0.428312 + 0.903631i $$0.640892\pi$$
$$338$$ 29.1962i 1.58806i
$$339$$ 12.9282 0.702164
$$340$$ 0 0
$$341$$ 10.9282 0.591795
$$342$$ − 9.46410i − 0.511760i
$$343$$ 20.0000i 1.07990i
$$344$$ 8.53590 0.460225
$$345$$ 0 0
$$346$$ 20.7846 1.11739
$$347$$ 22.3923i 1.20208i 0.799218 + 0.601041i $$0.205247\pi$$
−0.799218 + 0.601041i $$0.794753\pi$$
$$348$$ 3.46410i 0.185695i
$$349$$ 8.14359 0.435917 0.217958 0.975958i $$-0.430060\pi$$
0.217958 + 0.975958i $$0.430060\pi$$
$$350$$ 0 0
$$351$$ 5.46410 0.291652
$$352$$ − 5.19615i − 0.276956i
$$353$$ − 12.9282i − 0.688099i −0.938952 0.344049i $$-0.888201\pi$$
0.938952 0.344049i $$-0.111799\pi$$
$$354$$ −12.0000 −0.637793
$$355$$ 0 0
$$356$$ 0.928203 0.0491947
$$357$$ 0 0
$$358$$ 12.0000i 0.634220i
$$359$$ 20.7846 1.09697 0.548485 0.836160i $$-0.315205\pi$$
0.548485 + 0.836160i $$0.315205\pi$$
$$360$$ 0 0
$$361$$ 10.8564 0.571390
$$362$$ − 27.4641i − 1.44348i
$$363$$ − 1.00000i − 0.0524864i
$$364$$ −10.9282 −0.572793
$$365$$ 0 0
$$366$$ −3.46410 −0.181071
$$367$$ 20.0000i 1.04399i 0.852948 + 0.521996i $$0.174812\pi$$
−0.852948 + 0.521996i $$0.825188\pi$$
$$368$$ 34.6410i 1.80579i
$$369$$ −3.46410 −0.180334
$$370$$ 0 0
$$371$$ 1.85641 0.0963798
$$372$$ − 10.9282i − 0.566601i
$$373$$ 20.3923i 1.05587i 0.849284 + 0.527937i $$0.177034\pi$$
−0.849284 + 0.527937i $$0.822966\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ −12.0000 −0.618853
$$377$$ − 18.9282i − 0.974852i
$$378$$ 3.46410i 0.178174i
$$379$$ 17.8564 0.917222 0.458611 0.888637i $$-0.348347\pi$$
0.458611 + 0.888637i $$0.348347\pi$$
$$380$$ 0 0
$$381$$ 8.92820 0.457406
$$382$$ 32.7846i 1.67741i
$$383$$ 13.8564i 0.708029i 0.935240 + 0.354015i $$0.115184\pi$$
−0.935240 + 0.354015i $$0.884816\pi$$
$$384$$ 12.1244 0.618718
$$385$$ 0 0
$$386$$ −42.2487 −2.15040
$$387$$ − 4.92820i − 0.250515i
$$388$$ 10.0000i 0.507673i
$$389$$ −11.0718 −0.561362 −0.280681 0.959801i $$-0.590560\pi$$
−0.280681 + 0.959801i $$0.590560\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ − 5.19615i − 0.262445i
$$393$$ − 18.9282i − 0.954802i
$$394$$ −20.7846 −1.04711
$$395$$ 0 0
$$396$$ −1.00000 −0.0502519
$$397$$ 2.00000i 0.100377i 0.998740 + 0.0501886i $$0.0159822\pi$$
−0.998740 + 0.0501886i $$0.984018\pi$$
$$398$$ − 42.9282i − 2.15180i
$$399$$ −10.9282 −0.547094
$$400$$ 0 0
$$401$$ −7.85641 −0.392330 −0.196165 0.980571i $$-0.562849\pi$$
−0.196165 + 0.980571i $$0.562849\pi$$
$$402$$ − 13.8564i − 0.691095i
$$403$$ 59.7128i 2.97451i
$$404$$ 10.3923 0.517036
$$405$$ 0 0
$$406$$ 12.0000 0.595550
$$407$$ 4.92820i 0.244282i
$$408$$ 0 0
$$409$$ 6.78461 0.335477 0.167739 0.985831i $$-0.446354\pi$$
0.167739 + 0.985831i $$0.446354\pi$$
$$410$$ 0 0
$$411$$ −18.0000 −0.887875
$$412$$ 8.00000i 0.394132i
$$413$$ 13.8564i 0.681829i
$$414$$ 12.0000 0.589768
$$415$$ 0 0
$$416$$ 28.3923 1.39205
$$417$$ 12.3923i 0.606854i
$$418$$ − 9.46410i − 0.462904i
$$419$$ −30.9282 −1.51094 −0.755471 0.655182i $$-0.772592\pi$$
−0.755471 + 0.655182i $$0.772592\pi$$
$$420$$ 0 0
$$421$$ 2.00000 0.0974740 0.0487370 0.998812i $$-0.484480\pi$$
0.0487370 + 0.998812i $$0.484480\pi$$
$$422$$ 14.5359i 0.707596i
$$423$$ 6.92820i 0.336861i
$$424$$ −1.60770 −0.0780766
$$425$$ 0 0
$$426$$ −24.0000 −1.16280
$$427$$ 4.00000i 0.193574i
$$428$$ − 8.53590i − 0.412598i
$$429$$ 5.46410 0.263809
$$430$$ 0 0
$$431$$ 8.78461 0.423140 0.211570 0.977363i $$-0.432142\pi$$
0.211570 + 0.977363i $$0.432142\pi$$
$$432$$ − 5.00000i − 0.240563i
$$433$$ − 0.143594i − 0.00690067i −0.999994 0.00345033i $$-0.998902\pi$$
0.999994 0.00345033i $$-0.00109828\pi$$
$$434$$ −37.8564 −1.81717
$$435$$ 0 0
$$436$$ −10.0000 −0.478913
$$437$$ 37.8564i 1.81092i
$$438$$ − 14.5359i − 0.694552i
$$439$$ −33.1769 −1.58345 −0.791724 0.610879i $$-0.790816\pi$$
−0.791724 + 0.610879i $$0.790816\pi$$
$$440$$ 0 0
$$441$$ −3.00000 −0.142857
$$442$$ 0 0
$$443$$ − 12.0000i − 0.570137i −0.958507 0.285069i $$-0.907984\pi$$
0.958507 0.285069i $$-0.0920164\pi$$
$$444$$ 4.92820 0.233882
$$445$$ 0 0
$$446$$ −17.0718 −0.808373
$$447$$ − 15.4641i − 0.731427i
$$448$$ − 2.00000i − 0.0944911i
$$449$$ 26.7846 1.26404 0.632022 0.774950i $$-0.282225\pi$$
0.632022 + 0.774950i $$0.282225\pi$$
$$450$$ 0 0
$$451$$ −3.46410 −0.163118
$$452$$ − 12.9282i − 0.608092i
$$453$$ 20.3923i 0.958114i
$$454$$ 26.7846 1.25706
$$455$$ 0 0
$$456$$ 9.46410 0.443197
$$457$$ 12.3923i 0.579688i 0.957074 + 0.289844i $$0.0936034\pi$$
−0.957074 + 0.289844i $$0.906397\pi$$
$$458$$ − 41.3205i − 1.93078i
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 36.2487 1.68827 0.844135 0.536130i $$-0.180114\pi$$
0.844135 + 0.536130i $$0.180114\pi$$
$$462$$ 3.46410i 0.161165i
$$463$$ 28.0000i 1.30127i 0.759390 + 0.650635i $$0.225497\pi$$
−0.759390 + 0.650635i $$0.774503\pi$$
$$464$$ −17.3205 −0.804084
$$465$$ 0 0
$$466$$ −20.7846 −0.962828
$$467$$ 5.07180i 0.234695i 0.993091 + 0.117347i $$0.0374391\pi$$
−0.993091 + 0.117347i $$0.962561\pi$$
$$468$$ − 5.46410i − 0.252578i
$$469$$ −16.0000 −0.738811
$$470$$ 0 0
$$471$$ −3.07180 −0.141541
$$472$$ − 12.0000i − 0.552345i
$$473$$ − 4.92820i − 0.226599i
$$474$$ −11.3205 −0.519968
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 0.928203i 0.0424995i
$$478$$ − 20.7846i − 0.950666i
$$479$$ −12.0000 −0.548294 −0.274147 0.961688i $$-0.588395\pi$$
−0.274147 + 0.961688i $$0.588395\pi$$
$$480$$ 0 0
$$481$$ −26.9282 −1.22782
$$482$$ − 0.248711i − 0.0113285i
$$483$$ − 13.8564i − 0.630488i
$$484$$ −1.00000 −0.0454545
$$485$$ 0 0
$$486$$ −1.73205 −0.0785674
$$487$$ − 31.7128i − 1.43704i −0.695504 0.718522i $$-0.744819\pi$$
0.695504 0.718522i $$-0.255181\pi$$
$$488$$ − 3.46410i − 0.156813i
$$489$$ −9.85641 −0.445722
$$490$$ 0 0
$$491$$ −30.9282 −1.39577 −0.697885 0.716210i $$-0.745875\pi$$
−0.697885 + 0.716210i $$0.745875\pi$$
$$492$$ 3.46410i 0.156174i
$$493$$ 0 0
$$494$$ 51.7128 2.32667
$$495$$ 0 0
$$496$$ 54.6410 2.45345
$$497$$ 27.7128i 1.24309i
$$498$$ 14.7846i 0.662514i
$$499$$ −28.7846 −1.28858 −0.644288 0.764783i $$-0.722846\pi$$
−0.644288 + 0.764783i $$0.722846\pi$$
$$500$$ 0 0
$$501$$ −10.3923 −0.464294
$$502$$ 3.21539i 0.143510i
$$503$$ − 31.1769i − 1.39011i −0.718957 0.695055i $$-0.755380\pi$$
0.718957 0.695055i $$-0.244620\pi$$
$$504$$ −3.46410 −0.154303
$$505$$ 0 0
$$506$$ 12.0000 0.533465
$$507$$ 16.8564i 0.748619i
$$508$$ − 8.92820i − 0.396125i
$$509$$ −19.8564 −0.880120 −0.440060 0.897968i $$-0.645043\pi$$
−0.440060 + 0.897968i $$0.645043\pi$$
$$510$$ 0 0
$$511$$ −16.7846 −0.742507
$$512$$ − 8.66025i − 0.382733i
$$513$$ − 5.46410i − 0.241246i
$$514$$ −34.3923 −1.51698
$$515$$ 0 0
$$516$$ −4.92820 −0.216952
$$517$$ 6.92820i 0.304702i
$$518$$ − 17.0718i − 0.750092i
$$519$$ 12.0000 0.526742
$$520$$ 0 0
$$521$$ 6.00000 0.262865 0.131432 0.991325i $$-0.458042\pi$$
0.131432 + 0.991325i $$0.458042\pi$$
$$522$$ 6.00000i 0.262613i
$$523$$ 22.0000i 0.961993i 0.876723 + 0.480996i $$0.159725\pi$$
−0.876723 + 0.480996i $$0.840275\pi$$
$$524$$ −18.9282 −0.826882
$$525$$ 0 0
$$526$$ −35.5692 −1.55089
$$527$$ 0 0
$$528$$ − 5.00000i − 0.217597i
$$529$$ −25.0000 −1.08696
$$530$$ 0 0
$$531$$ −6.92820 −0.300658
$$532$$ 10.9282i 0.473798i
$$533$$ − 18.9282i − 0.819871i
$$534$$ 1.60770 0.0695718
$$535$$ 0 0
$$536$$ 13.8564 0.598506
$$537$$ 6.92820i 0.298974i
$$538$$ 34.3923i 1.48276i
$$539$$ −3.00000 −0.129219
$$540$$ 0 0
$$541$$ 27.8564 1.19764 0.598820 0.800883i $$-0.295636\pi$$
0.598820 + 0.800883i $$0.295636\pi$$
$$542$$ 20.1051i 0.863589i
$$543$$ − 15.8564i − 0.680464i
$$544$$ 0 0
$$545$$ 0 0
$$546$$ −18.9282 −0.810052
$$547$$ 2.00000i 0.0855138i 0.999086 + 0.0427569i $$0.0136141\pi$$
−0.999086 + 0.0427569i $$0.986386\pi$$
$$548$$ 18.0000i 0.768922i
$$549$$ −2.00000 −0.0853579
$$550$$ 0 0
$$551$$ −18.9282 −0.806369
$$552$$ 12.0000i 0.510754i
$$553$$ 13.0718i 0.555869i
$$554$$ 51.0333 2.16820
$$555$$ 0 0
$$556$$ 12.3923 0.525551
$$557$$ 3.21539i 0.136240i 0.997677 + 0.0681202i $$0.0217001\pi$$
−0.997677 + 0.0681202i $$0.978300\pi$$
$$558$$ − 18.9282i − 0.801295i
$$559$$ 26.9282 1.13894
$$560$$ 0 0
$$561$$ 0 0
$$562$$ − 6.00000i − 0.253095i
$$563$$ − 10.3923i − 0.437983i −0.975727 0.218992i $$-0.929723\pi$$
0.975727 0.218992i $$-0.0702768\pi$$
$$564$$ 6.92820 0.291730
$$565$$ 0 0
$$566$$ 8.53590 0.358791
$$567$$ 2.00000i 0.0839921i
$$568$$ − 24.0000i − 1.00702i
$$569$$ −5.32051 −0.223047 −0.111524 0.993762i $$-0.535573\pi$$
−0.111524 + 0.993762i $$0.535573\pi$$
$$570$$ 0 0
$$571$$ 3.60770 0.150977 0.0754887 0.997147i $$-0.475948\pi$$
0.0754887 + 0.997147i $$0.475948\pi$$
$$572$$ − 5.46410i − 0.228466i
$$573$$ 18.9282i 0.790737i
$$574$$ 12.0000 0.500870
$$575$$ 0 0
$$576$$ 1.00000 0.0416667
$$577$$ − 18.7846i − 0.782014i −0.920388 0.391007i $$-0.872127\pi$$
0.920388 0.391007i $$-0.127873\pi$$
$$578$$ − 29.4449i − 1.22474i
$$579$$ −24.3923 −1.01371
$$580$$ 0 0
$$581$$ 17.0718 0.708257
$$582$$ 17.3205i 0.717958i
$$583$$ 0.928203i 0.0384422i
$$584$$ 14.5359 0.601500
$$585$$ 0 0
$$586$$ 24.0000 0.991431
$$587$$ − 18.9282i − 0.781251i −0.920550 0.390625i $$-0.872259\pi$$
0.920550 0.390625i $$-0.127741\pi$$
$$588$$ 3.00000i 0.123718i
$$589$$ 59.7128 2.46042
$$590$$ 0 0
$$591$$ −12.0000 −0.493614
$$592$$ 24.6410i 1.01274i
$$593$$ − 8.78461i − 0.360741i −0.983599 0.180370i $$-0.942270\pi$$
0.983599 0.180370i $$-0.0577296\pi$$
$$594$$ −1.73205 −0.0710669
$$595$$ 0 0
$$596$$ −15.4641 −0.633434
$$597$$ − 24.7846i − 1.01437i
$$598$$ 65.5692i 2.68132i
$$599$$ 37.8564 1.54677 0.773385 0.633936i $$-0.218562\pi$$
0.773385 + 0.633936i $$0.218562\pi$$
$$600$$ 0 0
$$601$$ −32.6410 −1.33145 −0.665727 0.746195i $$-0.731879\pi$$
−0.665727 + 0.746195i $$0.731879\pi$$
$$602$$ 17.0718i 0.695794i
$$603$$ − 8.00000i − 0.325785i
$$604$$ 20.3923 0.829751
$$605$$ 0 0
$$606$$ 18.0000 0.731200
$$607$$ − 18.7846i − 0.762444i −0.924484 0.381222i $$-0.875503\pi$$
0.924484 0.381222i $$-0.124497\pi$$
$$608$$ − 28.3923i − 1.15146i
$$609$$ 6.92820 0.280745
$$610$$ 0 0
$$611$$ −37.8564 −1.53151
$$612$$ 0 0
$$613$$ 20.3923i 0.823637i 0.911266 + 0.411819i $$0.135106\pi$$
−0.911266 + 0.411819i $$0.864894\pi$$
$$614$$ 24.2487 0.978598
$$615$$ 0 0
$$616$$ −3.46410 −0.139573
$$617$$ − 36.9282i − 1.48667i −0.668917 0.743337i $$-0.733242\pi$$
0.668917 0.743337i $$-0.266758\pi$$
$$618$$ 13.8564i 0.557386i
$$619$$ −20.0000 −0.803868 −0.401934 0.915669i $$-0.631662\pi$$
−0.401934 + 0.915669i $$0.631662\pi$$
$$620$$ 0 0
$$621$$ 6.92820 0.278019
$$622$$ − 8.78461i − 0.352231i
$$623$$ − 1.85641i − 0.0743754i
$$624$$ 27.3205 1.09370
$$625$$ 0 0
$$626$$ −36.2487 −1.44879
$$627$$ − 5.46410i − 0.218215i
$$628$$ 3.07180i 0.122578i
$$629$$ 0 0
$$630$$ 0 0
$$631$$ −21.0718 −0.838855 −0.419427 0.907789i $$-0.637769\pi$$
−0.419427 + 0.907789i $$0.637769\pi$$
$$632$$ − 11.3205i − 0.450306i
$$633$$ 8.39230i 0.333564i
$$634$$ −43.1769 −1.71477
$$635$$ 0 0
$$636$$ 0.928203 0.0368057
$$637$$ − 16.3923i − 0.649487i
$$638$$ 6.00000i 0.237542i
$$639$$ −13.8564 −0.548151
$$640$$ 0 0
$$641$$ −12.9282 −0.510633 −0.255317 0.966857i $$-0.582180\pi$$
−0.255317 + 0.966857i $$0.582180\pi$$
$$642$$ − 14.7846i − 0.583502i
$$643$$ − 37.5692i − 1.48159i −0.671734 0.740793i $$-0.734450\pi$$
0.671734 0.740793i $$-0.265550\pi$$
$$644$$ −13.8564 −0.546019
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 27.7128i 1.08950i 0.838597 + 0.544752i $$0.183376\pi$$
−0.838597 + 0.544752i $$0.816624\pi$$
$$648$$ − 1.73205i − 0.0680414i
$$649$$ −6.92820 −0.271956
$$650$$ 0 0
$$651$$ −21.8564 −0.856620
$$652$$ 9.85641i 0.386007i
$$653$$ − 19.8564i − 0.777041i −0.921440 0.388521i $$-0.872986\pi$$
0.921440 0.388521i $$-0.127014\pi$$
$$654$$ −17.3205 −0.677285
$$655$$ 0 0
$$656$$ −17.3205 −0.676252
$$657$$ − 8.39230i − 0.327415i
$$658$$ − 24.0000i − 0.935617i
$$659$$ 15.7128 0.612084 0.306042 0.952018i $$-0.400995\pi$$
0.306042 + 0.952018i $$0.400995\pi$$
$$660$$ 0 0
$$661$$ 14.0000 0.544537 0.272268 0.962221i $$-0.412226\pi$$
0.272268 + 0.962221i $$0.412226\pi$$
$$662$$ − 17.0718i − 0.663514i
$$663$$ 0 0
$$664$$ −14.7846 −0.573754
$$665$$ 0 0
$$666$$ 8.53590 0.330759
$$667$$ − 24.0000i − 0.929284i
$$668$$ 10.3923i 0.402090i
$$669$$ −9.85641 −0.381071
$$670$$ 0 0
$$671$$ −2.00000 −0.0772091
$$672$$ 10.3923i 0.400892i
$$673$$ 3.32051i 0.127996i 0.997950 + 0.0639981i $$0.0203852\pi$$
−0.997950 + 0.0639981i $$0.979615\pi$$
$$674$$ 57.4641 2.21343
$$675$$ 0 0
$$676$$ 16.8564 0.648323
$$677$$ 8.78461i 0.337620i 0.985649 + 0.168810i $$0.0539924\pi$$
−0.985649 + 0.168810i $$0.946008\pi$$
$$678$$ − 22.3923i − 0.859971i
$$679$$ 20.0000 0.767530
$$680$$ 0 0
$$681$$ 15.4641 0.592586
$$682$$ − 18.9282i − 0.724798i
$$683$$ 32.7846i 1.25447i 0.778831 + 0.627234i $$0.215813\pi$$
−0.778831 + 0.627234i $$0.784187\pi$$
$$684$$ −5.46410 −0.208925
$$685$$ 0 0
$$686$$ 34.6410 1.32260
$$687$$ − 23.8564i − 0.910179i
$$688$$ − 24.6410i − 0.939430i
$$689$$ −5.07180 −0.193220
$$690$$ 0 0
$$691$$ 47.7128 1.81508 0.907540 0.419965i $$-0.137958\pi$$
0.907540 + 0.419965i $$0.137958\pi$$
$$692$$ − 12.0000i − 0.456172i
$$693$$ 2.00000i 0.0759737i
$$694$$ 38.7846 1.47224
$$695$$ 0 0
$$696$$ −6.00000 −0.227429
$$697$$ 0 0
$$698$$ − 14.1051i − 0.533887i
$$699$$ −12.0000 −0.453882
$$700$$ 0 0
$$701$$ 39.4641 1.49054 0.745269 0.666764i $$-0.232321\pi$$
0.745269 + 0.666764i $$0.232321\pi$$
$$702$$ − 9.46410i − 0.357199i
$$703$$ 26.9282i 1.01562i
$$704$$ 1.00000 0.0376889
$$705$$ 0 0
$$706$$ −22.3923 −0.842746
$$707$$ − 20.7846i − 0.781686i
$$708$$ 6.92820i 0.260378i
$$709$$ 11.8564 0.445277 0.222638 0.974901i $$-0.428533\pi$$
0.222638 + 0.974901i $$0.428533\pi$$
$$710$$ 0 0
$$711$$ −6.53590 −0.245115
$$712$$ 1.60770i 0.0602509i
$$713$$ 75.7128i 2.83547i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 6.92820 0.258919
$$717$$ − 12.0000i − 0.448148i
$$718$$ − 36.0000i − 1.34351i
$$719$$ 5.07180 0.189146 0.0945731 0.995518i $$-0.469851\pi$$
0.0945731 + 0.995518i $$0.469851\pi$$
$$720$$ 0 0
$$721$$ 16.0000 0.595871
$$722$$ − 18.8038i − 0.699807i
$$723$$ − 0.143594i − 0.00534030i
$$724$$ −15.8564 −0.589299
$$725$$ 0 0
$$726$$ −1.73205 −0.0642824
$$727$$ 8.00000i 0.296704i 0.988935 + 0.148352i $$0.0473968\pi$$
−0.988935 + 0.148352i $$0.952603\pi$$
$$728$$ − 18.9282i − 0.701526i
$$729$$ −1.00000 −0.0370370
$$730$$ 0 0
$$731$$ 0 0
$$732$$ 2.00000i 0.0739221i
$$733$$ − 53.9615i − 1.99311i −0.0829082 0.996557i $$-0.526421\pi$$
0.0829082 0.996557i $$-0.473579\pi$$
$$734$$ 34.6410 1.27862
$$735$$ 0 0
$$736$$ 36.0000 1.32698
$$737$$ − 8.00000i − 0.294684i
$$738$$ 6.00000i 0.220863i
$$739$$ −17.4641 −0.642427 −0.321214 0.947007i $$-0.604091\pi$$
−0.321214 + 0.947007i $$0.604091\pi$$
$$740$$ 0 0
$$741$$ 29.8564 1.09680
$$742$$ − 3.21539i − 0.118041i
$$743$$ − 25.6077i − 0.939455i −0.882811 0.469728i $$-0.844352\pi$$
0.882811 0.469728i $$-0.155648\pi$$
$$744$$ 18.9282 0.693942
$$745$$ 0 0
$$746$$ 35.3205 1.29318
$$747$$ 8.53590i 0.312312i
$$748$$ 0 0
$$749$$ −17.0718 −0.623790
$$750$$ 0 0
$$751$$ 26.9282 0.982624 0.491312 0.870984i $$-0.336517\pi$$
0.491312 + 0.870984i $$0.336517\pi$$
$$752$$ 34.6410i 1.26323i
$$753$$ 1.85641i 0.0676512i
$$754$$ −32.7846 −1.19395
$$755$$ 0 0
$$756$$ 2.00000 0.0727393
$$757$$ 34.7846i 1.26427i 0.774859 + 0.632134i $$0.217821\pi$$
−0.774859 + 0.632134i $$0.782179\pi$$
$$758$$ − 30.9282i − 1.12336i
$$759$$ 6.92820 0.251478
$$760$$ 0 0
$$761$$ −32.5359 −1.17943 −0.589713 0.807613i $$-0.700759\pi$$
−0.589713 + 0.807613i $$0.700759\pi$$
$$762$$ − 15.4641i − 0.560205i
$$763$$ 20.0000i 0.724049i
$$764$$ 18.9282 0.684798
$$765$$ 0 0
$$766$$ 24.0000 0.867155
$$767$$ − 37.8564i − 1.36692i
$$768$$ − 19.0000i − 0.685603i
$$769$$ −50.4974 −1.82098 −0.910492 0.413527i $$-0.864297\pi$$
−0.910492 + 0.413527i $$0.864297\pi$$
$$770$$ 0 0
$$771$$ −19.8564 −0.715111
$$772$$ 24.3923i 0.877898i
$$773$$ 4.14359i 0.149035i 0.997220 + 0.0745174i $$0.0237416\pi$$
−0.997220 + 0.0745174i $$0.976258\pi$$
$$774$$ −8.53590 −0.306817
$$775$$ 0 0
$$776$$ −17.3205 −0.621770
$$777$$ − 9.85641i − 0.353597i
$$778$$ 19.1769i 0.687526i
$$779$$ −18.9282 −0.678173
$$780$$ 0 0
$$781$$ −13.8564 −0.495821
$$782$$ 0 0
$$783$$ 3.46410i 0.123797i
$$784$$ −15.0000 −0.535714
$$785$$ 0 0
$$786$$ −32.7846 −1.16939
$$787$$ 22.7846i 0.812184i 0.913832 + 0.406092i $$0.133109\pi$$
−0.913832 + 0.406092i $$0.866891\pi$$
$$788$$ 12.0000i 0.427482i
$$789$$ −20.5359 −0.731097
$$790$$ 0 0
$$791$$ −25.8564 −0.919348
$$792$$ − 1.73205i − 0.0615457i
$$793$$ − 10.9282i − 0.388072i
$$794$$ 3.46410 0.122936
$$795$$ 0 0
$$796$$ −24.7846 −0.878467
$$797$$ − 52.6410i − 1.86464i −0.361634 0.932320i $$-0.617781\pi$$
0.361634 0.932320i $$-0.382219\pi$$
$$798$$ 18.9282i 0.670051i
$$799$$ 0 0
$$800$$ 0 0
$$801$$ 0.928203 0.0327964
$$802$$ 13.6077i 0.480504i
$$803$$ − 8.39230i − 0.296158i
$$804$$ −8.00000 −0.282138
$$805$$ 0 0
$$806$$ 103.426 3.64301
$$807$$ 19.8564i 0.698979i
$$808$$ 18.0000i 0.633238i
$$809$$ 15.4641 0.543689 0.271844 0.962341i $$-0.412366\pi$$
0.271844 + 0.962341i $$0.412366\pi$$
$$810$$ 0 0
$$811$$ 12.3923 0.435153 0.217576 0.976043i $$-0.430185\pi$$
0.217576 + 0.976043i $$0.430185\pi$$
$$812$$ − 6.92820i − 0.243132i
$$813$$ 11.6077i 0.407100i
$$814$$ 8.53590 0.299183
$$815$$ 0 0
$$816$$ 0 0
$$817$$ − 26.9282i − 0.942099i
$$818$$ − 11.7513i − 0.410874i
$$819$$ −10.9282 −0.381862
$$820$$ 0 0
$$821$$ 20.5359 0.716708 0.358354 0.933586i $$-0.383338\pi$$
0.358354 + 0.933586i $$0.383338\pi$$
$$822$$ 31.1769i 1.08742i
$$823$$ 33.5692i 1.17015i 0.810979 + 0.585075i $$0.198935\pi$$
−0.810979 + 0.585075i $$0.801065\pi$$
$$824$$ −13.8564 −0.482711
$$825$$ 0 0
$$826$$ 24.0000 0.835067
$$827$$ 22.3923i 0.778657i 0.921099 + 0.389328i $$0.127293\pi$$
−0.921099 + 0.389328i $$0.872707\pi$$
$$828$$ − 6.92820i − 0.240772i
$$829$$ −29.7128 −1.03197 −0.515984 0.856598i $$-0.672574\pi$$
−0.515984 + 0.856598i $$0.672574\pi$$
$$830$$ 0 0
$$831$$ 29.4641 1.02210
$$832$$ 5.46410i 0.189434i
$$833$$ 0 0
$$834$$ 21.4641 0.743241
$$835$$ 0 0
$$836$$ −5.46410 −0.188980
$$837$$ − 10.9282i − 0.377734i
$$838$$ 53.5692i 1.85052i
$$839$$ −56.7846 −1.96042 −0.980211 0.197954i $$-0.936570\pi$$
−0.980211 + 0.197954i $$0.936570\pi$$
$$840$$ 0 0
$$841$$ −17.0000 −0.586207
$$842$$ − 3.46410i − 0.119381i
$$843$$ − 3.46410i − 0.119310i
$$844$$ 8.39230 0.288875
$$845$$ 0 0
$$846$$ 12.0000 0.412568
$$847$$ 2.00000i 0.0687208i
$$848$$ 4.64102i 0.159373i
$$849$$ 4.92820 0.169135
$$850$$ 0 0
$$851$$ −34.1436 −1.17043
$$852$$ 13.8564i 0.474713i
$$853$$ − 3.60770i − 0.123525i −0.998091 0.0617626i $$-0.980328\pi$$
0.998091 0.0617626i $$-0.0196722\pi$$
$$854$$ 6.92820 0.237078
$$855$$ 0 0
$$856$$ 14.7846 0.505328
$$857$$ − 37.8564i − 1.29315i −0.762850 0.646575i $$-0.776201\pi$$
0.762850 0.646575i $$-0.223799\pi$$
$$858$$ − 9.46410i − 0.323099i
$$859$$ 7.71281 0.263158 0.131579 0.991306i $$-0.457995\pi$$
0.131579 + 0.991306i $$0.457995\pi$$
$$860$$ 0 0
$$861$$ 6.92820 0.236113
$$862$$ − 15.2154i − 0.518238i
$$863$$ 37.8564i 1.28865i 0.764753 + 0.644324i $$0.222861\pi$$
−0.764753 + 0.644324i $$0.777139\pi$$
$$864$$ −5.19615 −0.176777
$$865$$ 0 0
$$866$$ −0.248711 −0.00845155
$$867$$ − 17.0000i − 0.577350i
$$868$$ 21.8564i 0.741855i
$$869$$ −6.53590 −0.221715
$$870$$ 0 0
$$871$$ 43.7128 1.48115
$$872$$ − 17.3205i − 0.586546i
$$873$$ 10.0000i 0.338449i
$$874$$ 65.5692 2.21791
$$875$$ 0 0
$$876$$ −8.39230 −0.283550
$$877$$ − 34.2487i − 1.15650i −0.815861 0.578248i $$-0.803736\pi$$
0.815861 0.578248i $$-0.196264\pi$$
$$878$$ 57.4641i 1.93932i
$$879$$ 13.8564 0.467365
$$880$$ 0 0
$$881$$ −0.928203 −0.0312720 −0.0156360 0.999878i $$-0.504977\pi$$
−0.0156360 + 0.999878i $$0.504977\pi$$
$$882$$ 5.19615i 0.174964i
$$883$$ 4.00000i 0.134611i 0.997732 + 0.0673054i $$0.0214402\pi$$
−0.997732 + 0.0673054i $$0.978560\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ −20.7846 −0.698273
$$887$$ 12.2487i 0.411271i 0.978629 + 0.205636i $$0.0659262\pi$$
−0.978629 + 0.205636i $$0.934074\pi$$
$$888$$ 8.53590i 0.286446i
$$889$$ −17.8564 −0.598885
$$890$$ 0 0
$$891$$ −1.00000 −0.0335013
$$892$$ 9.85641i 0.330017i
$$893$$ 37.8564i 1.26682i
$$894$$ −26.7846 −0.895811
$$895$$ 0 0
$$896$$ −24.2487 −0.810093
$$897$$ 37.8564i 1.26399i
$$898$$ − 46.3923i − 1.54813i
$$899$$ −37.8564 −1.26258
$$900$$ 0 0
$$901$$ 0 0
$$902$$ 6.00000i 0.199778i
$$903$$ 9.85641i 0.328001i
$$904$$ 22.3923 0.744757
$$905$$ 0 0
$$906$$ 35.3205 1.17345
$$907$$ 18.1436i 0.602448i 0.953553 + 0.301224i $$0.0973952\pi$$
−0.953553 + 0.301224i $$0.902605\pi$$
$$908$$ − 15.4641i − 0.513194i
$$909$$ 10.3923 0.344691
$$910$$ 0 0
$$911$$ 18.9282 0.627119 0.313560 0.949568i $$-0.398478\pi$$
0.313560 + 0.949568i $$0.398478\pi$$
$$912$$ − 27.3205i − 0.904672i
$$913$$ 8.53590i 0.282497i
$$914$$ 21.4641 0.709969
$$915$$ 0 0
$$916$$ −23.8564 −0.788238
$$917$$ 37.8564i 1.25013i
$$918$$ 0 0
$$919$$ 32.3923 1.06852 0.534262 0.845319i $$-0.320590\pi$$
0.534262 + 0.845319i $$0.320590\pi$$
$$920$$ 0 0
$$921$$ 14.0000 0.461316
$$922$$ − 62.7846i − 2.06770i
$$923$$ − 75.7128i − 2.49212i
$$924$$ 2.00000 0.0657952
$$925$$ 0 0
$$926$$ 48.4974 1.59372
$$927$$ 8.00000i 0.262754i
$$928$$ 18.0000i 0.590879i
$$929$$ −2.78461 −0.0913601 −0.0456800 0.998956i $$-0.514545\pi$$
−0.0456800 + 0.998956i $$0.514545\pi$$
$$930$$ 0 0
$$931$$ −16.3923 −0.537236
$$932$$ 12.0000i 0.393073i
$$933$$ − 5.07180i − 0.166043i
$$934$$ 8.78461 0.287441
$$935$$ 0 0
$$936$$ 9.46410 0.309344
$$937$$ − 20.3923i − 0.666188i −0.942894 0.333094i $$-0.891907\pi$$
0.942894 0.333094i $$-0.108093\pi$$
$$938$$ 27.7128i 0.904855i
$$939$$ −20.9282 −0.682966
$$940$$ 0 0
$$941$$ 27.4641 0.895304 0.447652 0.894208i $$-0.352260\pi$$
0.447652 + 0.894208i $$0.352260\pi$$
$$942$$ 5.32051i 0.173352i
$$943$$ − 24.0000i − 0.781548i
$$944$$ −34.6410 −1.12747
$$945$$ 0 0
$$946$$ −8.53590 −0.277526
$$947$$ − 18.9282i − 0.615084i −0.951535 0.307542i $$-0.900494\pi$$
0.951535 0.307542i $$-0.0995065\pi$$
$$948$$ 6.53590i 0.212276i
$$949$$ 45.8564 1.48856
$$950$$ 0 0
$$951$$ −24.9282 −0.808352
$$952$$ 0 0
$$953$$ 3.21539i 0.104157i 0.998643 + 0.0520784i $$0.0165846\pi$$
−0.998643 + 0.0520784i $$0.983415\pi$$
$$954$$ 1.60770 0.0520511
$$955$$ 0 0
$$956$$ −12.0000 −0.388108
$$957$$ 3.46410i 0.111979i
$$958$$ 20.7846i 0.671520i
$$959$$ 36.0000 1.16250
$$960$$ 0 0
$$961$$ 88.4256 2.85244
$$962$$ 46.6410i 1.50377i
$$963$$ − 8.53590i − 0.275065i
$$964$$ −0.143594 −0.00462484
$$965$$ 0 0
$$966$$ −24.0000 −0.772187
$$967$$ 22.7846i 0.732704i 0.930476 + 0.366352i $$0.119393\pi$$
−0.930476 + 0.366352i $$0.880607\pi$$
$$968$$ − 1.73205i − 0.0556702i
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 25.8564 0.829772 0.414886 0.909873i $$-0.363822\pi$$
0.414886 + 0.909873i $$0.363822\pi$$
$$972$$ 1.00000i 0.0320750i
$$973$$ − 24.7846i − 0.794558i
$$974$$ −54.9282 −1.76001
$$975$$ 0 0
$$976$$ −10.0000 −0.320092
$$977$$ 47.5692i 1.52187i 0.648826 + 0.760937i $$0.275260\pi$$
−0.648826 + 0.760937i $$0.724740\pi$$
$$978$$ 17.0718i 0.545896i
$$979$$ 0.928203 0.0296655
$$980$$ 0 0
$$981$$ −10.0000 −0.319275
$$982$$ 53.5692i 1.70946i
$$983$$ − 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
$$986$$ 0 0
$$987$$ − 13.8564i − 0.441054i
$$988$$ − 29.8564i − 0.949859i
$$989$$ 34.1436 1.08570
$$990$$ 0 0
$$991$$ −7.21539 −0.229204 −0.114602 0.993411i $$-0.536559\pi$$
−0.114602 + 0.993411i $$0.536559\pi$$
$$992$$ − 56.7846i − 1.80291i
$$993$$ − 9.85641i − 0.312784i
$$994$$ 48.0000 1.52247
$$995$$ 0 0
$$996$$ 8.53590 0.270470
$$997$$ 27.6077i 0.874344i 0.899378 + 0.437172i $$0.144020\pi$$
−0.899378 + 0.437172i $$0.855980\pi$$
$$998$$ 49.8564i 1.57818i
$$999$$ 4.92820 0.155921
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 825.2.c.c.199.1 4
3.2 odd 2 2475.2.c.n.199.4 4
5.2 odd 4 825.2.a.e.1.2 2
5.3 odd 4 165.2.a.b.1.1 2
5.4 even 2 inner 825.2.c.c.199.4 4
15.2 even 4 2475.2.a.r.1.1 2
15.8 even 4 495.2.a.c.1.2 2
15.14 odd 2 2475.2.c.n.199.1 4
20.3 even 4 2640.2.a.x.1.2 2
35.13 even 4 8085.2.a.bd.1.1 2
55.32 even 4 9075.2.a.bh.1.1 2
55.43 even 4 1815.2.a.i.1.2 2
60.23 odd 4 7920.2.a.bz.1.2 2
165.98 odd 4 5445.2.a.s.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
165.2.a.b.1.1 2 5.3 odd 4
495.2.a.c.1.2 2 15.8 even 4
825.2.a.e.1.2 2 5.2 odd 4
825.2.c.c.199.1 4 1.1 even 1 trivial
825.2.c.c.199.4 4 5.4 even 2 inner
1815.2.a.i.1.2 2 55.43 even 4
2475.2.a.r.1.1 2 15.2 even 4
2475.2.c.n.199.1 4 15.14 odd 2
2475.2.c.n.199.4 4 3.2 odd 2
2640.2.a.x.1.2 2 20.3 even 4
5445.2.a.s.1.1 2 165.98 odd 4
7920.2.a.bz.1.2 2 60.23 odd 4
8085.2.a.bd.1.1 2 35.13 even 4
9075.2.a.bh.1.1 2 55.32 even 4