# Properties

 Label 1925.2.b.d.1849.1 Level $1925$ Weight $2$ Character 1925.1849 Analytic conductor $15.371$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1925 = 5^{2} \cdot 7 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1925.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$15.3712023891$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 77) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 1849.1 Root $$-1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 1925.1849 Dual form 1925.2.b.d.1849.2

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000i q^{2} +2.00000i q^{3} +1.00000 q^{4} +2.00000 q^{6} +1.00000i q^{7} -3.00000i q^{8} -1.00000 q^{9} +O(q^{10})$$ $$q-1.00000i q^{2} +2.00000i q^{3} +1.00000 q^{4} +2.00000 q^{6} +1.00000i q^{7} -3.00000i q^{8} -1.00000 q^{9} +1.00000 q^{11} +2.00000i q^{12} +4.00000i q^{13} +1.00000 q^{14} -1.00000 q^{16} -4.00000i q^{17} +1.00000i q^{18} -2.00000 q^{21} -1.00000i q^{22} -4.00000i q^{23} +6.00000 q^{24} +4.00000 q^{26} +4.00000i q^{27} +1.00000i q^{28} +6.00000 q^{29} +10.0000 q^{31} -5.00000i q^{32} +2.00000i q^{33} -4.00000 q^{34} -1.00000 q^{36} +6.00000i q^{37} -8.00000 q^{39} +4.00000 q^{41} +2.00000i q^{42} +12.0000i q^{43} +1.00000 q^{44} -4.00000 q^{46} +10.0000i q^{47} -2.00000i q^{48} -1.00000 q^{49} +8.00000 q^{51} +4.00000i q^{52} -6.00000i q^{53} +4.00000 q^{54} +3.00000 q^{56} -6.00000i q^{58} -2.00000 q^{59} -10.0000i q^{62} -1.00000i q^{63} -7.00000 q^{64} +2.00000 q^{66} -8.00000i q^{67} -4.00000i q^{68} +8.00000 q^{69} -12.0000 q^{71} +3.00000i q^{72} -8.00000i q^{73} +6.00000 q^{74} +1.00000i q^{77} +8.00000i q^{78} -8.00000 q^{79} -11.0000 q^{81} -4.00000i q^{82} -2.00000 q^{84} +12.0000 q^{86} +12.0000i q^{87} -3.00000i q^{88} +6.00000 q^{89} -4.00000 q^{91} -4.00000i q^{92} +20.0000i q^{93} +10.0000 q^{94} +10.0000 q^{96} +10.0000i q^{97} +1.00000i q^{98} -1.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{4} + 4q^{6} - 2q^{9} + O(q^{10})$$ $$2q + 2q^{4} + 4q^{6} - 2q^{9} + 2q^{11} + 2q^{14} - 2q^{16} - 4q^{21} + 12q^{24} + 8q^{26} + 12q^{29} + 20q^{31} - 8q^{34} - 2q^{36} - 16q^{39} + 8q^{41} + 2q^{44} - 8q^{46} - 2q^{49} + 16q^{51} + 8q^{54} + 6q^{56} - 4q^{59} - 14q^{64} + 4q^{66} + 16q^{69} - 24q^{71} + 12q^{74} - 16q^{79} - 22q^{81} - 4q^{84} + 24q^{86} + 12q^{89} - 8q^{91} + 20q^{94} + 20q^{96} - 2q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1925\mathbb{Z}\right)^\times$$.

 $$n$$ $$276$$ $$1002$$ $$1751$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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 −0.935414 0.353553i $$-0.884973\pi$$
0.935414 0.353553i $$-0.115027\pi$$
$$3$$ 2.00000i 1.15470i 0.816497 + 0.577350i $$0.195913\pi$$
−0.816497 + 0.577350i $$0.804087\pi$$
$$4$$ 1.00000 0.500000
$$5$$ 0 0
$$6$$ 2.00000 0.816497
$$7$$ 1.00000i 0.377964i
$$8$$ − 3.00000i − 1.06066i
$$9$$ −1.00000 −0.333333
$$10$$ 0 0
$$11$$ 1.00000 0.301511
$$12$$ 2.00000i 0.577350i
$$13$$ 4.00000i 1.10940i 0.832050 + 0.554700i $$0.187167\pi$$
−0.832050 + 0.554700i $$0.812833\pi$$
$$14$$ 1.00000 0.267261
$$15$$ 0 0
$$16$$ −1.00000 −0.250000
$$17$$ − 4.00000i − 0.970143i −0.874475 0.485071i $$-0.838794\pi$$
0.874475 0.485071i $$-0.161206\pi$$
$$18$$ 1.00000i 0.235702i
$$19$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$20$$ 0 0
$$21$$ −2.00000 −0.436436
$$22$$ − 1.00000i − 0.213201i
$$23$$ − 4.00000i − 0.834058i −0.908893 0.417029i $$-0.863071\pi$$
0.908893 0.417029i $$-0.136929\pi$$
$$24$$ 6.00000 1.22474
$$25$$ 0 0
$$26$$ 4.00000 0.784465
$$27$$ 4.00000i 0.769800i
$$28$$ 1.00000i 0.188982i
$$29$$ 6.00000 1.11417 0.557086 0.830455i $$-0.311919\pi$$
0.557086 + 0.830455i $$0.311919\pi$$
$$30$$ 0 0
$$31$$ 10.0000 1.79605 0.898027 0.439941i $$-0.145001\pi$$
0.898027 + 0.439941i $$0.145001\pi$$
$$32$$ − 5.00000i − 0.883883i
$$33$$ 2.00000i 0.348155i
$$34$$ −4.00000 −0.685994
$$35$$ 0 0
$$36$$ −1.00000 −0.166667
$$37$$ 6.00000i 0.986394i 0.869918 + 0.493197i $$0.164172\pi$$
−0.869918 + 0.493197i $$0.835828\pi$$
$$38$$ 0 0
$$39$$ −8.00000 −1.28103
$$40$$ 0 0
$$41$$ 4.00000 0.624695 0.312348 0.949968i $$-0.398885\pi$$
0.312348 + 0.949968i $$0.398885\pi$$
$$42$$ 2.00000i 0.308607i
$$43$$ 12.0000i 1.82998i 0.403473 + 0.914991i $$0.367803\pi$$
−0.403473 + 0.914991i $$0.632197\pi$$
$$44$$ 1.00000 0.150756
$$45$$ 0 0
$$46$$ −4.00000 −0.589768
$$47$$ 10.0000i 1.45865i 0.684167 + 0.729325i $$0.260166\pi$$
−0.684167 + 0.729325i $$0.739834\pi$$
$$48$$ − 2.00000i − 0.288675i
$$49$$ −1.00000 −0.142857
$$50$$ 0 0
$$51$$ 8.00000 1.12022
$$52$$ 4.00000i 0.554700i
$$53$$ − 6.00000i − 0.824163i −0.911147 0.412082i $$-0.864802\pi$$
0.911147 0.412082i $$-0.135198\pi$$
$$54$$ 4.00000 0.544331
$$55$$ 0 0
$$56$$ 3.00000 0.400892
$$57$$ 0 0
$$58$$ − 6.00000i − 0.787839i
$$59$$ −2.00000 −0.260378 −0.130189 0.991489i $$-0.541558\pi$$
−0.130189 + 0.991489i $$0.541558\pi$$
$$60$$ 0 0
$$61$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$62$$ − 10.0000i − 1.27000i
$$63$$ − 1.00000i − 0.125988i
$$64$$ −7.00000 −0.875000
$$65$$ 0 0
$$66$$ 2.00000 0.246183
$$67$$ − 8.00000i − 0.977356i −0.872464 0.488678i $$-0.837479\pi$$
0.872464 0.488678i $$-0.162521\pi$$
$$68$$ − 4.00000i − 0.485071i
$$69$$ 8.00000 0.963087
$$70$$ 0 0
$$71$$ −12.0000 −1.42414 −0.712069 0.702109i $$-0.752242\pi$$
−0.712069 + 0.702109i $$0.752242\pi$$
$$72$$ 3.00000i 0.353553i
$$73$$ − 8.00000i − 0.936329i −0.883641 0.468165i $$-0.844915\pi$$
0.883641 0.468165i $$-0.155085\pi$$
$$74$$ 6.00000 0.697486
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 1.00000i 0.113961i
$$78$$ 8.00000i 0.905822i
$$79$$ −8.00000 −0.900070 −0.450035 0.893011i $$-0.648589\pi$$
−0.450035 + 0.893011i $$0.648589\pi$$
$$80$$ 0 0
$$81$$ −11.0000 −1.22222
$$82$$ − 4.00000i − 0.441726i
$$83$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$84$$ −2.00000 −0.218218
$$85$$ 0 0
$$86$$ 12.0000 1.29399
$$87$$ 12.0000i 1.28654i
$$88$$ − 3.00000i − 0.319801i
$$89$$ 6.00000 0.635999 0.317999 0.948091i $$-0.396989\pi$$
0.317999 + 0.948091i $$0.396989\pi$$
$$90$$ 0 0
$$91$$ −4.00000 −0.419314
$$92$$ − 4.00000i − 0.417029i
$$93$$ 20.0000i 2.07390i
$$94$$ 10.0000 1.03142
$$95$$ 0 0
$$96$$ 10.0000 1.02062
$$97$$ 10.0000i 1.01535i 0.861550 + 0.507673i $$0.169494\pi$$
−0.861550 + 0.507673i $$0.830506\pi$$
$$98$$ 1.00000i 0.101015i
$$99$$ −1.00000 −0.100504
$$100$$ 0 0
$$101$$ −4.00000 −0.398015 −0.199007 0.979998i $$-0.563772\pi$$
−0.199007 + 0.979998i $$0.563772\pi$$
$$102$$ − 8.00000i − 0.792118i
$$103$$ 14.0000i 1.37946i 0.724066 + 0.689730i $$0.242271\pi$$
−0.724066 + 0.689730i $$0.757729\pi$$
$$104$$ 12.0000 1.17670
$$105$$ 0 0
$$106$$ −6.00000 −0.582772
$$107$$ − 12.0000i − 1.16008i −0.814587 0.580042i $$-0.803036\pi$$
0.814587 0.580042i $$-0.196964\pi$$
$$108$$ 4.00000i 0.384900i
$$109$$ 14.0000 1.34096 0.670478 0.741929i $$-0.266089\pi$$
0.670478 + 0.741929i $$0.266089\pi$$
$$110$$ 0 0
$$111$$ −12.0000 −1.13899
$$112$$ − 1.00000i − 0.0944911i
$$113$$ 18.0000i 1.69330i 0.532152 + 0.846649i $$0.321383\pi$$
−0.532152 + 0.846649i $$0.678617\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 6.00000 0.557086
$$117$$ − 4.00000i − 0.369800i
$$118$$ 2.00000i 0.184115i
$$119$$ 4.00000 0.366679
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ 0 0
$$123$$ 8.00000i 0.721336i
$$124$$ 10.0000 0.898027
$$125$$ 0 0
$$126$$ −1.00000 −0.0890871
$$127$$ − 8.00000i − 0.709885i −0.934888 0.354943i $$-0.884500\pi$$
0.934888 0.354943i $$-0.115500\pi$$
$$128$$ − 3.00000i − 0.265165i
$$129$$ −24.0000 −2.11308
$$130$$ 0 0
$$131$$ 12.0000 1.04844 0.524222 0.851581i $$-0.324356\pi$$
0.524222 + 0.851581i $$0.324356\pi$$
$$132$$ 2.00000i 0.174078i
$$133$$ 0 0
$$134$$ −8.00000 −0.691095
$$135$$ 0 0
$$136$$ −12.0000 −1.02899
$$137$$ 10.0000i 0.854358i 0.904167 + 0.427179i $$0.140493\pi$$
−0.904167 + 0.427179i $$0.859507\pi$$
$$138$$ − 8.00000i − 0.681005i
$$139$$ 8.00000 0.678551 0.339276 0.940687i $$-0.389818\pi$$
0.339276 + 0.940687i $$0.389818\pi$$
$$140$$ 0 0
$$141$$ −20.0000 −1.68430
$$142$$ 12.0000i 1.00702i
$$143$$ 4.00000i 0.334497i
$$144$$ 1.00000 0.0833333
$$145$$ 0 0
$$146$$ −8.00000 −0.662085
$$147$$ − 2.00000i − 0.164957i
$$148$$ 6.00000i 0.493197i
$$149$$ 10.0000 0.819232 0.409616 0.912258i $$-0.365663\pi$$
0.409616 + 0.912258i $$0.365663\pi$$
$$150$$ 0 0
$$151$$ −16.0000 −1.30206 −0.651031 0.759051i $$-0.725663\pi$$
−0.651031 + 0.759051i $$0.725663\pi$$
$$152$$ 0 0
$$153$$ 4.00000i 0.323381i
$$154$$ 1.00000 0.0805823
$$155$$ 0 0
$$156$$ −8.00000 −0.640513
$$157$$ − 14.0000i − 1.11732i −0.829396 0.558661i $$-0.811315\pi$$
0.829396 0.558661i $$-0.188685\pi$$
$$158$$ 8.00000i 0.636446i
$$159$$ 12.0000 0.951662
$$160$$ 0 0
$$161$$ 4.00000 0.315244
$$162$$ 11.0000i 0.864242i
$$163$$ − 8.00000i − 0.626608i −0.949653 0.313304i $$-0.898564\pi$$
0.949653 0.313304i $$-0.101436\pi$$
$$164$$ 4.00000 0.312348
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$168$$ 6.00000i 0.462910i
$$169$$ −3.00000 −0.230769
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 12.0000i 0.914991i
$$173$$ 12.0000i 0.912343i 0.889892 + 0.456172i $$0.150780\pi$$
−0.889892 + 0.456172i $$0.849220\pi$$
$$174$$ 12.0000 0.909718
$$175$$ 0 0
$$176$$ −1.00000 −0.0753778
$$177$$ − 4.00000i − 0.300658i
$$178$$ − 6.00000i − 0.449719i
$$179$$ −12.0000 −0.896922 −0.448461 0.893802i $$-0.648028\pi$$
−0.448461 + 0.893802i $$0.648028\pi$$
$$180$$ 0 0
$$181$$ 10.0000 0.743294 0.371647 0.928374i $$-0.378793\pi$$
0.371647 + 0.928374i $$0.378793\pi$$
$$182$$ 4.00000i 0.296500i
$$183$$ 0 0
$$184$$ −12.0000 −0.884652
$$185$$ 0 0
$$186$$ 20.0000 1.46647
$$187$$ − 4.00000i − 0.292509i
$$188$$ 10.0000i 0.729325i
$$189$$ −4.00000 −0.290957
$$190$$ 0 0
$$191$$ 8.00000 0.578860 0.289430 0.957199i $$-0.406534\pi$$
0.289430 + 0.957199i $$0.406534\pi$$
$$192$$ − 14.0000i − 1.01036i
$$193$$ − 14.0000i − 1.00774i −0.863779 0.503871i $$-0.831909\pi$$
0.863779 0.503871i $$-0.168091\pi$$
$$194$$ 10.0000 0.717958
$$195$$ 0 0
$$196$$ −1.00000 −0.0714286
$$197$$ − 22.0000i − 1.56744i −0.621117 0.783718i $$-0.713321\pi$$
0.621117 0.783718i $$-0.286679\pi$$
$$198$$ 1.00000i 0.0710669i
$$199$$ 18.0000 1.27599 0.637993 0.770042i $$-0.279765\pi$$
0.637993 + 0.770042i $$0.279765\pi$$
$$200$$ 0 0
$$201$$ 16.0000 1.12855
$$202$$ 4.00000i 0.281439i
$$203$$ 6.00000i 0.421117i
$$204$$ 8.00000 0.560112
$$205$$ 0 0
$$206$$ 14.0000 0.975426
$$207$$ 4.00000i 0.278019i
$$208$$ − 4.00000i − 0.277350i
$$209$$ 0 0
$$210$$ 0 0
$$211$$ −12.0000 −0.826114 −0.413057 0.910705i $$-0.635539\pi$$
−0.413057 + 0.910705i $$0.635539\pi$$
$$212$$ − 6.00000i − 0.412082i
$$213$$ − 24.0000i − 1.64445i
$$214$$ −12.0000 −0.820303
$$215$$ 0 0
$$216$$ 12.0000 0.816497
$$217$$ 10.0000i 0.678844i
$$218$$ − 14.0000i − 0.948200i
$$219$$ 16.0000 1.08118
$$220$$ 0 0
$$221$$ 16.0000 1.07628
$$222$$ 12.0000i 0.805387i
$$223$$ 22.0000i 1.47323i 0.676313 + 0.736614i $$0.263577\pi$$
−0.676313 + 0.736614i $$0.736423\pi$$
$$224$$ 5.00000 0.334077
$$225$$ 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
$$229$$ −18.0000 −1.18947 −0.594737 0.803921i $$-0.702744\pi$$
−0.594737 + 0.803921i $$0.702744\pi$$
$$230$$ 0 0
$$231$$ −2.00000 −0.131590
$$232$$ − 18.0000i − 1.18176i
$$233$$ − 18.0000i − 1.17922i −0.807688 0.589610i $$-0.799282\pi$$
0.807688 0.589610i $$-0.200718\pi$$
$$234$$ −4.00000 −0.261488
$$235$$ 0 0
$$236$$ −2.00000 −0.130189
$$237$$ − 16.0000i − 1.03931i
$$238$$ − 4.00000i − 0.259281i
$$239$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$240$$ 0 0
$$241$$ −20.0000 −1.28831 −0.644157 0.764894i $$-0.722792\pi$$
−0.644157 + 0.764894i $$0.722792\pi$$
$$242$$ − 1.00000i − 0.0642824i
$$243$$ − 10.0000i − 0.641500i
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 8.00000 0.510061
$$247$$ 0 0
$$248$$ − 30.0000i − 1.90500i
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −2.00000 −0.126239 −0.0631194 0.998006i $$-0.520105\pi$$
−0.0631194 + 0.998006i $$0.520105\pi$$
$$252$$ − 1.00000i − 0.0629941i
$$253$$ − 4.00000i − 0.251478i
$$254$$ −8.00000 −0.501965
$$255$$ 0 0
$$256$$ −17.0000 −1.06250
$$257$$ 14.0000i 0.873296i 0.899632 + 0.436648i $$0.143834\pi$$
−0.899632 + 0.436648i $$0.856166\pi$$
$$258$$ 24.0000i 1.49417i
$$259$$ −6.00000 −0.372822
$$260$$ 0 0
$$261$$ −6.00000 −0.371391
$$262$$ − 12.0000i − 0.741362i
$$263$$ 8.00000i 0.493301i 0.969104 + 0.246651i $$0.0793300\pi$$
−0.969104 + 0.246651i $$0.920670\pi$$
$$264$$ 6.00000 0.369274
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 12.0000i 0.734388i
$$268$$ − 8.00000i − 0.488678i
$$269$$ −10.0000 −0.609711 −0.304855 0.952399i $$-0.598608\pi$$
−0.304855 + 0.952399i $$0.598608\pi$$
$$270$$ 0 0
$$271$$ −4.00000 −0.242983 −0.121491 0.992592i $$-0.538768\pi$$
−0.121491 + 0.992592i $$0.538768\pi$$
$$272$$ 4.00000i 0.242536i
$$273$$ − 8.00000i − 0.484182i
$$274$$ 10.0000 0.604122
$$275$$ 0 0
$$276$$ 8.00000 0.481543
$$277$$ − 22.0000i − 1.32185i −0.750451 0.660926i $$-0.770164\pi$$
0.750451 0.660926i $$-0.229836\pi$$
$$278$$ − 8.00000i − 0.479808i
$$279$$ −10.0000 −0.598684
$$280$$ 0 0
$$281$$ 6.00000 0.357930 0.178965 0.983855i $$-0.442725\pi$$
0.178965 + 0.983855i $$0.442725\pi$$
$$282$$ 20.0000i 1.19098i
$$283$$ 4.00000i 0.237775i 0.992908 + 0.118888i $$0.0379328\pi$$
−0.992908 + 0.118888i $$0.962067\pi$$
$$284$$ −12.0000 −0.712069
$$285$$ 0 0
$$286$$ 4.00000 0.236525
$$287$$ 4.00000i 0.236113i
$$288$$ 5.00000i 0.294628i
$$289$$ 1.00000 0.0588235
$$290$$ 0 0
$$291$$ −20.0000 −1.17242
$$292$$ − 8.00000i − 0.468165i
$$293$$ − 24.0000i − 1.40209i −0.713115 0.701047i $$-0.752716\pi$$
0.713115 0.701047i $$-0.247284\pi$$
$$294$$ −2.00000 −0.116642
$$295$$ 0 0
$$296$$ 18.0000 1.04623
$$297$$ 4.00000i 0.232104i
$$298$$ − 10.0000i − 0.579284i
$$299$$ 16.0000 0.925304
$$300$$ 0 0
$$301$$ −12.0000 −0.691669
$$302$$ 16.0000i 0.920697i
$$303$$ − 8.00000i − 0.459588i
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 4.00000 0.228665
$$307$$ − 20.0000i − 1.14146i −0.821138 0.570730i $$-0.806660\pi$$
0.821138 0.570730i $$-0.193340\pi$$
$$308$$ 1.00000i 0.0569803i
$$309$$ −28.0000 −1.59286
$$310$$ 0 0
$$311$$ −18.0000 −1.02069 −0.510343 0.859971i $$-0.670482\pi$$
−0.510343 + 0.859971i $$0.670482\pi$$
$$312$$ 24.0000i 1.35873i
$$313$$ 2.00000i 0.113047i 0.998401 + 0.0565233i $$0.0180015\pi$$
−0.998401 + 0.0565233i $$0.981998\pi$$
$$314$$ −14.0000 −0.790066
$$315$$ 0 0
$$316$$ −8.00000 −0.450035
$$317$$ 2.00000i 0.112331i 0.998421 + 0.0561656i $$0.0178875\pi$$
−0.998421 + 0.0561656i $$0.982113\pi$$
$$318$$ − 12.0000i − 0.672927i
$$319$$ 6.00000 0.335936
$$320$$ 0 0
$$321$$ 24.0000 1.33955
$$322$$ − 4.00000i − 0.222911i
$$323$$ 0 0
$$324$$ −11.0000 −0.611111
$$325$$ 0 0
$$326$$ −8.00000 −0.443079
$$327$$ 28.0000i 1.54840i
$$328$$ − 12.0000i − 0.662589i
$$329$$ −10.0000 −0.551318
$$330$$ 0 0
$$331$$ −20.0000 −1.09930 −0.549650 0.835395i $$-0.685239\pi$$
−0.549650 + 0.835395i $$0.685239\pi$$
$$332$$ 0 0
$$333$$ − 6.00000i − 0.328798i
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 2.00000 0.109109
$$337$$ − 14.0000i − 0.762629i −0.924445 0.381314i $$-0.875472\pi$$
0.924445 0.381314i $$-0.124528\pi$$
$$338$$ 3.00000i 0.163178i
$$339$$ −36.0000 −1.95525
$$340$$ 0 0
$$341$$ 10.0000 0.541530
$$342$$ 0 0
$$343$$ − 1.00000i − 0.0539949i
$$344$$ 36.0000 1.94099
$$345$$ 0 0
$$346$$ 12.0000 0.645124
$$347$$ − 4.00000i − 0.214731i −0.994220 0.107366i $$-0.965758\pi$$
0.994220 0.107366i $$-0.0342415\pi$$
$$348$$ 12.0000i 0.643268i
$$349$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$350$$ 0 0
$$351$$ −16.0000 −0.854017
$$352$$ − 5.00000i − 0.266501i
$$353$$ − 30.0000i − 1.59674i −0.602168 0.798369i $$-0.705696\pi$$
0.602168 0.798369i $$-0.294304\pi$$
$$354$$ −4.00000 −0.212598
$$355$$ 0 0
$$356$$ 6.00000 0.317999
$$357$$ 8.00000i 0.423405i
$$358$$ 12.0000i 0.634220i
$$359$$ −16.0000 −0.844448 −0.422224 0.906492i $$-0.638750\pi$$
−0.422224 + 0.906492i $$0.638750\pi$$
$$360$$ 0 0
$$361$$ −19.0000 −1.00000
$$362$$ − 10.0000i − 0.525588i
$$363$$ 2.00000i 0.104973i
$$364$$ −4.00000 −0.209657
$$365$$ 0 0
$$366$$ 0 0
$$367$$ − 22.0000i − 1.14839i −0.818718 0.574195i $$-0.805315\pi$$
0.818718 0.574195i $$-0.194685\pi$$
$$368$$ 4.00000i 0.208514i
$$369$$ −4.00000 −0.208232
$$370$$ 0 0
$$371$$ 6.00000 0.311504
$$372$$ 20.0000i 1.03695i
$$373$$ − 26.0000i − 1.34623i −0.739538 0.673114i $$-0.764956\pi$$
0.739538 0.673114i $$-0.235044\pi$$
$$374$$ −4.00000 −0.206835
$$375$$ 0 0
$$376$$ 30.0000 1.54713
$$377$$ 24.0000i 1.23606i
$$378$$ 4.00000i 0.205738i
$$379$$ 8.00000 0.410932 0.205466 0.978664i $$-0.434129\pi$$
0.205466 + 0.978664i $$0.434129\pi$$
$$380$$ 0 0
$$381$$ 16.0000 0.819705
$$382$$ − 8.00000i − 0.409316i
$$383$$ 2.00000i 0.102195i 0.998694 + 0.0510976i $$0.0162720\pi$$
−0.998694 + 0.0510976i $$0.983728\pi$$
$$384$$ 6.00000 0.306186
$$385$$ 0 0
$$386$$ −14.0000 −0.712581
$$387$$ − 12.0000i − 0.609994i
$$388$$ 10.0000i 0.507673i
$$389$$ −6.00000 −0.304212 −0.152106 0.988364i $$-0.548606\pi$$
−0.152106 + 0.988364i $$0.548606\pi$$
$$390$$ 0 0
$$391$$ −16.0000 −0.809155
$$392$$ 3.00000i 0.151523i
$$393$$ 24.0000i 1.21064i
$$394$$ −22.0000 −1.10834
$$395$$ 0 0
$$396$$ −1.00000 −0.0502519
$$397$$ − 22.0000i − 1.10415i −0.833795 0.552074i $$-0.813837\pi$$
0.833795 0.552074i $$-0.186163\pi$$
$$398$$ − 18.0000i − 0.902258i
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −22.0000 −1.09863 −0.549314 0.835616i $$-0.685111\pi$$
−0.549314 + 0.835616i $$0.685111\pi$$
$$402$$ − 16.0000i − 0.798007i
$$403$$ 40.0000i 1.99254i
$$404$$ −4.00000 −0.199007
$$405$$ 0 0
$$406$$ 6.00000 0.297775
$$407$$ 6.00000i 0.297409i
$$408$$ − 24.0000i − 1.18818i
$$409$$ −24.0000 −1.18672 −0.593362 0.804936i $$-0.702200\pi$$
−0.593362 + 0.804936i $$0.702200\pi$$
$$410$$ 0 0
$$411$$ −20.0000 −0.986527
$$412$$ 14.0000i 0.689730i
$$413$$ − 2.00000i − 0.0984136i
$$414$$ 4.00000 0.196589
$$415$$ 0 0
$$416$$ 20.0000 0.980581
$$417$$ 16.0000i 0.783523i
$$418$$ 0 0
$$419$$ −2.00000 −0.0977064 −0.0488532 0.998806i $$-0.515557\pi$$
−0.0488532 + 0.998806i $$0.515557\pi$$
$$420$$ 0 0
$$421$$ −14.0000 −0.682318 −0.341159 0.940006i $$-0.610819\pi$$
−0.341159 + 0.940006i $$0.610819\pi$$
$$422$$ 12.0000i 0.584151i
$$423$$ − 10.0000i − 0.486217i
$$424$$ −18.0000 −0.874157
$$425$$ 0 0
$$426$$ −24.0000 −1.16280
$$427$$ 0 0
$$428$$ − 12.0000i − 0.580042i
$$429$$ −8.00000 −0.386244
$$430$$ 0 0
$$431$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$432$$ − 4.00000i − 0.192450i
$$433$$ − 26.0000i − 1.24948i −0.780833 0.624740i $$-0.785205\pi$$
0.780833 0.624740i $$-0.214795\pi$$
$$434$$ 10.0000 0.480015
$$435$$ 0 0
$$436$$ 14.0000 0.670478
$$437$$ 0 0
$$438$$ − 16.0000i − 0.764510i
$$439$$ 20.0000 0.954548 0.477274 0.878755i $$-0.341625\pi$$
0.477274 + 0.878755i $$0.341625\pi$$
$$440$$ 0 0
$$441$$ 1.00000 0.0476190
$$442$$ − 16.0000i − 0.761042i
$$443$$ 4.00000i 0.190046i 0.995475 + 0.0950229i $$0.0302924\pi$$
−0.995475 + 0.0950229i $$0.969708\pi$$
$$444$$ −12.0000 −0.569495
$$445$$ 0 0
$$446$$ 22.0000 1.04173
$$447$$ 20.0000i 0.945968i
$$448$$ − 7.00000i − 0.330719i
$$449$$ 10.0000 0.471929 0.235965 0.971762i $$-0.424175\pi$$
0.235965 + 0.971762i $$0.424175\pi$$
$$450$$ 0 0
$$451$$ 4.00000 0.188353
$$452$$ 18.0000i 0.846649i
$$453$$ − 32.0000i − 1.50349i
$$454$$ −12.0000 −0.563188
$$455$$ 0 0
$$456$$ 0 0
$$457$$ − 18.0000i − 0.842004i −0.907060 0.421002i $$-0.861678\pi$$
0.907060 0.421002i $$-0.138322\pi$$
$$458$$ 18.0000i 0.841085i
$$459$$ 16.0000 0.746816
$$460$$ 0 0
$$461$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$462$$ 2.00000i 0.0930484i
$$463$$ 4.00000i 0.185896i 0.995671 + 0.0929479i $$0.0296290\pi$$
−0.995671 + 0.0929479i $$0.970371\pi$$
$$464$$ −6.00000 −0.278543
$$465$$ 0 0
$$466$$ −18.0000 −0.833834
$$467$$ − 30.0000i − 1.38823i −0.719862 0.694117i $$-0.755795\pi$$
0.719862 0.694117i $$-0.244205\pi$$
$$468$$ − 4.00000i − 0.184900i
$$469$$ 8.00000 0.369406
$$470$$ 0 0
$$471$$ 28.0000 1.29017
$$472$$ 6.00000i 0.276172i
$$473$$ 12.0000i 0.551761i
$$474$$ −16.0000 −0.734904
$$475$$ 0 0
$$476$$ 4.00000 0.183340
$$477$$ 6.00000i 0.274721i
$$478$$ 0 0
$$479$$ 4.00000 0.182765 0.0913823 0.995816i $$-0.470871\pi$$
0.0913823 + 0.995816i $$0.470871\pi$$
$$480$$ 0 0
$$481$$ −24.0000 −1.09431
$$482$$ 20.0000i 0.910975i
$$483$$ 8.00000i 0.364013i
$$484$$ 1.00000 0.0454545
$$485$$ 0 0
$$486$$ −10.0000 −0.453609
$$487$$ 28.0000i 1.26880i 0.773004 + 0.634401i $$0.218753\pi$$
−0.773004 + 0.634401i $$0.781247\pi$$
$$488$$ 0 0
$$489$$ 16.0000 0.723545
$$490$$ 0 0
$$491$$ 28.0000 1.26362 0.631811 0.775122i $$-0.282312\pi$$
0.631811 + 0.775122i $$0.282312\pi$$
$$492$$ 8.00000i 0.360668i
$$493$$ − 24.0000i − 1.08091i
$$494$$ 0 0
$$495$$ 0 0
$$496$$ −10.0000 −0.449013
$$497$$ − 12.0000i − 0.538274i
$$498$$ 0 0
$$499$$ 16.0000 0.716258 0.358129 0.933672i $$-0.383415\pi$$
0.358129 + 0.933672i $$0.383415\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 2.00000i 0.0892644i
$$503$$ 4.00000i 0.178351i 0.996016 + 0.0891756i $$0.0284232\pi$$
−0.996016 + 0.0891756i $$0.971577\pi$$
$$504$$ −3.00000 −0.133631
$$505$$ 0 0
$$506$$ −4.00000 −0.177822
$$507$$ − 6.00000i − 0.266469i
$$508$$ − 8.00000i − 0.354943i
$$509$$ −18.0000 −0.797836 −0.398918 0.916987i $$-0.630614\pi$$
−0.398918 + 0.916987i $$0.630614\pi$$
$$510$$ 0 0
$$511$$ 8.00000 0.353899
$$512$$ 11.0000i 0.486136i
$$513$$ 0 0
$$514$$ 14.0000 0.617514
$$515$$ 0 0
$$516$$ −24.0000 −1.05654
$$517$$ 10.0000i 0.439799i
$$518$$ 6.00000i 0.263625i
$$519$$ −24.0000 −1.05348
$$520$$ 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
$$523$$ 20.0000i 0.874539i 0.899331 + 0.437269i $$0.144054\pi$$
−0.899331 + 0.437269i $$0.855946\pi$$
$$524$$ 12.0000 0.524222
$$525$$ 0 0
$$526$$ 8.00000 0.348817
$$527$$ − 40.0000i − 1.74243i
$$528$$ − 2.00000i − 0.0870388i
$$529$$ 7.00000 0.304348
$$530$$ 0 0
$$531$$ 2.00000 0.0867926
$$532$$ 0 0
$$533$$ 16.0000i 0.693037i
$$534$$ 12.0000 0.519291
$$535$$ 0 0
$$536$$ −24.0000 −1.03664
$$537$$ − 24.0000i − 1.03568i
$$538$$ 10.0000i 0.431131i
$$539$$ −1.00000 −0.0430730
$$540$$ 0 0
$$541$$ −26.0000 −1.11783 −0.558914 0.829226i $$-0.688782\pi$$
−0.558914 + 0.829226i $$0.688782\pi$$
$$542$$ 4.00000i 0.171815i
$$543$$ 20.0000i 0.858282i
$$544$$ −20.0000 −0.857493
$$545$$ 0 0
$$546$$ −8.00000 −0.342368
$$547$$ 28.0000i 1.19719i 0.801050 + 0.598597i $$0.204275\pi$$
−0.801050 + 0.598597i $$0.795725\pi$$
$$548$$ 10.0000i 0.427179i
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 0 0
$$552$$ − 24.0000i − 1.02151i
$$553$$ − 8.00000i − 0.340195i
$$554$$ −22.0000 −0.934690
$$555$$ 0 0
$$556$$ 8.00000 0.339276
$$557$$ − 22.0000i − 0.932170i −0.884740 0.466085i $$-0.845664\pi$$
0.884740 0.466085i $$-0.154336\pi$$
$$558$$ 10.0000i 0.423334i
$$559$$ −48.0000 −2.03018
$$560$$ 0 0
$$561$$ 8.00000 0.337760
$$562$$ − 6.00000i − 0.253095i
$$563$$ − 32.0000i − 1.34864i −0.738440 0.674320i $$-0.764437\pi$$
0.738440 0.674320i $$-0.235563\pi$$
$$564$$ −20.0000 −0.842152
$$565$$ 0 0
$$566$$ 4.00000 0.168133
$$567$$ − 11.0000i − 0.461957i
$$568$$ 36.0000i 1.51053i
$$569$$ −30.0000 −1.25767 −0.628833 0.777541i $$-0.716467\pi$$
−0.628833 + 0.777541i $$0.716467\pi$$
$$570$$ 0 0
$$571$$ 20.0000 0.836974 0.418487 0.908223i $$-0.362561\pi$$
0.418487 + 0.908223i $$0.362561\pi$$
$$572$$ 4.00000i 0.167248i
$$573$$ 16.0000i 0.668410i
$$574$$ 4.00000 0.166957
$$575$$ 0 0
$$576$$ 7.00000 0.291667
$$577$$ 18.0000i 0.749350i 0.927156 + 0.374675i $$0.122246\pi$$
−0.927156 + 0.374675i $$0.877754\pi$$
$$578$$ − 1.00000i − 0.0415945i
$$579$$ 28.0000 1.16364
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 20.0000i 0.829027i
$$583$$ − 6.00000i − 0.248495i
$$584$$ −24.0000 −0.993127
$$585$$ 0 0
$$586$$ −24.0000 −0.991431
$$587$$ 2.00000i 0.0825488i 0.999148 + 0.0412744i $$0.0131418\pi$$
−0.999148 + 0.0412744i $$0.986858\pi$$
$$588$$ − 2.00000i − 0.0824786i
$$589$$ 0 0
$$590$$ 0 0
$$591$$ 44.0000 1.80992
$$592$$ − 6.00000i − 0.246598i
$$593$$ 32.0000i 1.31408i 0.753855 + 0.657041i $$0.228192\pi$$
−0.753855 + 0.657041i $$0.771808\pi$$
$$594$$ 4.00000 0.164122
$$595$$ 0 0
$$596$$ 10.0000 0.409616
$$597$$ 36.0000i 1.47338i
$$598$$ − 16.0000i − 0.654289i
$$599$$ 20.0000 0.817178 0.408589 0.912719i $$-0.366021\pi$$
0.408589 + 0.912719i $$0.366021\pi$$
$$600$$ 0 0
$$601$$ −28.0000 −1.14214 −0.571072 0.820900i $$-0.693472\pi$$
−0.571072 + 0.820900i $$0.693472\pi$$
$$602$$ 12.0000i 0.489083i
$$603$$ 8.00000i 0.325785i
$$604$$ −16.0000 −0.651031
$$605$$ 0 0
$$606$$ −8.00000 −0.324978
$$607$$ 40.0000i 1.62355i 0.583970 + 0.811775i $$0.301498\pi$$
−0.583970 + 0.811775i $$0.698502\pi$$
$$608$$ 0 0
$$609$$ −12.0000 −0.486265
$$610$$ 0 0
$$611$$ −40.0000 −1.61823
$$612$$ 4.00000i 0.161690i
$$613$$ 26.0000i 1.05013i 0.851062 + 0.525065i $$0.175959\pi$$
−0.851062 + 0.525065i $$0.824041\pi$$
$$614$$ −20.0000 −0.807134
$$615$$ 0 0
$$616$$ 3.00000 0.120873
$$617$$ − 6.00000i − 0.241551i −0.992680 0.120775i $$-0.961462\pi$$
0.992680 0.120775i $$-0.0385381\pi$$
$$618$$ 28.0000i 1.12633i
$$619$$ −14.0000 −0.562708 −0.281354 0.959604i $$-0.590783\pi$$
−0.281354 + 0.959604i $$0.590783\pi$$
$$620$$ 0 0
$$621$$ 16.0000 0.642058
$$622$$ 18.0000i 0.721734i
$$623$$ 6.00000i 0.240385i
$$624$$ 8.00000 0.320256
$$625$$ 0 0
$$626$$ 2.00000 0.0799361
$$627$$ 0 0
$$628$$ − 14.0000i − 0.558661i
$$629$$ 24.0000 0.956943
$$630$$ 0 0
$$631$$ −8.00000 −0.318475 −0.159237 0.987240i $$-0.550904\pi$$
−0.159237 + 0.987240i $$0.550904\pi$$
$$632$$ 24.0000i 0.954669i
$$633$$ − 24.0000i − 0.953914i
$$634$$ 2.00000 0.0794301
$$635$$ 0 0
$$636$$ 12.0000 0.475831
$$637$$ − 4.00000i − 0.158486i
$$638$$ − 6.00000i − 0.237542i
$$639$$ 12.0000 0.474713
$$640$$ 0 0
$$641$$ −18.0000 −0.710957 −0.355479 0.934684i $$-0.615682\pi$$
−0.355479 + 0.934684i $$0.615682\pi$$
$$642$$ − 24.0000i − 0.947204i
$$643$$ 14.0000i 0.552106i 0.961142 + 0.276053i $$0.0890266\pi$$
−0.961142 + 0.276053i $$0.910973\pi$$
$$644$$ 4.00000 0.157622
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 22.0000i 0.864909i 0.901656 + 0.432455i $$0.142352\pi$$
−0.901656 + 0.432455i $$0.857648\pi$$
$$648$$ 33.0000i 1.29636i
$$649$$ −2.00000 −0.0785069
$$650$$ 0 0
$$651$$ −20.0000 −0.783862
$$652$$ − 8.00000i − 0.313304i
$$653$$ − 26.0000i − 1.01746i −0.860927 0.508729i $$-0.830115\pi$$
0.860927 0.508729i $$-0.169885\pi$$
$$654$$ 28.0000 1.09489
$$655$$ 0 0
$$656$$ −4.00000 −0.156174
$$657$$ 8.00000i 0.312110i
$$658$$ 10.0000i 0.389841i
$$659$$ −4.00000 −0.155818 −0.0779089 0.996960i $$-0.524824\pi$$
−0.0779089 + 0.996960i $$0.524824\pi$$
$$660$$ 0 0
$$661$$ 22.0000 0.855701 0.427850 0.903850i $$-0.359271\pi$$
0.427850 + 0.903850i $$0.359271\pi$$
$$662$$ 20.0000i 0.777322i
$$663$$ 32.0000i 1.24278i
$$664$$ 0 0
$$665$$ 0 0
$$666$$ −6.00000 −0.232495
$$667$$ − 24.0000i − 0.929284i
$$668$$ 0 0
$$669$$ −44.0000 −1.70114
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 10.0000i 0.385758i
$$673$$ − 34.0000i − 1.31060i −0.755367 0.655302i $$-0.772541\pi$$
0.755367 0.655302i $$-0.227459\pi$$
$$674$$ −14.0000 −0.539260
$$675$$ 0 0
$$676$$ −3.00000 −0.115385
$$677$$ 12.0000i 0.461197i 0.973049 + 0.230599i $$0.0740685\pi$$
−0.973049 + 0.230599i $$0.925932\pi$$
$$678$$ 36.0000i 1.38257i
$$679$$ −10.0000 −0.383765
$$680$$ 0 0
$$681$$ 24.0000 0.919682
$$682$$ − 10.0000i − 0.382920i
$$683$$ − 4.00000i − 0.153056i −0.997067 0.0765279i $$-0.975617\pi$$
0.997067 0.0765279i $$-0.0243834\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ −1.00000 −0.0381802
$$687$$ − 36.0000i − 1.37349i
$$688$$ − 12.0000i − 0.457496i
$$689$$ 24.0000 0.914327
$$690$$ 0 0
$$691$$ −46.0000 −1.74992 −0.874961 0.484193i $$-0.839113\pi$$
−0.874961 + 0.484193i $$0.839113\pi$$
$$692$$ 12.0000i 0.456172i
$$693$$ − 1.00000i − 0.0379869i
$$694$$ −4.00000 −0.151838
$$695$$ 0 0
$$696$$ 36.0000 1.36458
$$697$$ − 16.0000i − 0.606043i
$$698$$ 0 0
$$699$$ 36.0000 1.36165
$$700$$ 0 0
$$701$$ −22.0000 −0.830929 −0.415464 0.909610i $$-0.636381\pi$$
−0.415464 + 0.909610i $$0.636381\pi$$
$$702$$ 16.0000i 0.603881i
$$703$$ 0 0
$$704$$ −7.00000 −0.263822
$$705$$ 0 0
$$706$$ −30.0000 −1.12906
$$707$$ − 4.00000i − 0.150435i
$$708$$ − 4.00000i − 0.150329i
$$709$$ 34.0000 1.27690 0.638448 0.769665i $$-0.279577\pi$$
0.638448 + 0.769665i $$0.279577\pi$$
$$710$$ 0 0
$$711$$ 8.00000 0.300023
$$712$$ − 18.0000i − 0.674579i
$$713$$ − 40.0000i − 1.49801i
$$714$$ 8.00000 0.299392
$$715$$ 0 0
$$716$$ −12.0000 −0.448461
$$717$$ 0 0
$$718$$ 16.0000i 0.597115i
$$719$$ 6.00000 0.223762 0.111881 0.993722i $$-0.464312\pi$$
0.111881 + 0.993722i $$0.464312\pi$$
$$720$$ 0 0
$$721$$ −14.0000 −0.521387
$$722$$ 19.0000i 0.707107i
$$723$$ − 40.0000i − 1.48762i
$$724$$ 10.0000 0.371647
$$725$$ 0 0
$$726$$ 2.00000 0.0742270
$$727$$ − 18.0000i − 0.667583i −0.942647 0.333792i $$-0.891672\pi$$
0.942647 0.333792i $$-0.108328\pi$$
$$728$$ 12.0000i 0.444750i
$$729$$ −13.0000 −0.481481
$$730$$ 0 0
$$731$$ 48.0000 1.77534
$$732$$ 0 0
$$733$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$734$$ −22.0000 −0.812035
$$735$$ 0 0
$$736$$ −20.0000 −0.737210
$$737$$ − 8.00000i − 0.294684i
$$738$$ 4.00000i 0.147242i
$$739$$ 4.00000 0.147142 0.0735712 0.997290i $$-0.476560\pi$$
0.0735712 + 0.997290i $$0.476560\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ − 6.00000i − 0.220267i
$$743$$ − 8.00000i − 0.293492i −0.989174 0.146746i $$-0.953120\pi$$
0.989174 0.146746i $$-0.0468799\pi$$
$$744$$ 60.0000 2.19971
$$745$$ 0 0
$$746$$ −26.0000 −0.951928
$$747$$ 0 0
$$748$$ − 4.00000i − 0.146254i
$$749$$ 12.0000 0.438470
$$750$$ 0 0
$$751$$ −20.0000 −0.729810 −0.364905 0.931045i $$-0.618899\pi$$
−0.364905 + 0.931045i $$0.618899\pi$$
$$752$$ − 10.0000i − 0.364662i
$$753$$ − 4.00000i − 0.145768i
$$754$$ 24.0000 0.874028
$$755$$ 0 0
$$756$$ −4.00000 −0.145479
$$757$$ 10.0000i 0.363456i 0.983349 + 0.181728i $$0.0581691\pi$$
−0.983349 + 0.181728i $$0.941831\pi$$
$$758$$ − 8.00000i − 0.290573i
$$759$$ 8.00000 0.290382
$$760$$ 0 0
$$761$$ 48.0000 1.74000 0.869999 0.493053i $$-0.164119\pi$$
0.869999 + 0.493053i $$0.164119\pi$$
$$762$$ − 16.0000i − 0.579619i
$$763$$ 14.0000i 0.506834i
$$764$$ 8.00000 0.289430
$$765$$ 0 0
$$766$$ 2.00000 0.0722629
$$767$$ − 8.00000i − 0.288863i
$$768$$ − 34.0000i − 1.22687i
$$769$$ −32.0000 −1.15395 −0.576975 0.816762i $$-0.695767\pi$$
−0.576975 + 0.816762i $$0.695767\pi$$
$$770$$ 0 0
$$771$$ −28.0000 −1.00840
$$772$$ − 14.0000i − 0.503871i
$$773$$ − 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
$$776$$ 30.0000 1.07694
$$777$$ − 12.0000i − 0.430498i
$$778$$ 6.00000i 0.215110i
$$779$$ 0 0
$$780$$ 0 0
$$781$$ −12.0000 −0.429394
$$782$$ 16.0000i 0.572159i
$$783$$ 24.0000i 0.857690i
$$784$$ 1.00000 0.0357143
$$785$$ 0 0
$$786$$ 24.0000 0.856052
$$787$$ − 16.0000i − 0.570338i −0.958477 0.285169i $$-0.907950\pi$$
0.958477 0.285169i $$-0.0920498\pi$$
$$788$$ − 22.0000i − 0.783718i
$$789$$ −16.0000 −0.569615
$$790$$ 0 0
$$791$$ −18.0000 −0.640006
$$792$$ 3.00000i 0.106600i
$$793$$ 0 0
$$794$$ −22.0000 −0.780751
$$795$$ 0 0
$$796$$ 18.0000 0.637993
$$797$$ − 14.0000i − 0.495905i −0.968772 0.247953i $$-0.920242\pi$$
0.968772 0.247953i $$-0.0797578\pi$$
$$798$$ 0 0
$$799$$ 40.0000 1.41510
$$800$$ 0 0
$$801$$ −6.00000 −0.212000
$$802$$ 22.0000i 0.776847i
$$803$$ − 8.00000i − 0.282314i
$$804$$ 16.0000 0.564276
$$805$$ 0 0
$$806$$ 40.0000 1.40894
$$807$$ − 20.0000i − 0.704033i
$$808$$ 12.0000i 0.422159i
$$809$$ 30.0000 1.05474 0.527372 0.849635i $$-0.323177\pi$$
0.527372 + 0.849635i $$0.323177\pi$$
$$810$$ 0 0
$$811$$ 28.0000 0.983213 0.491606 0.870817i $$-0.336410\pi$$
0.491606 + 0.870817i $$0.336410\pi$$
$$812$$ 6.00000i 0.210559i
$$813$$ − 8.00000i − 0.280572i
$$814$$ 6.00000 0.210300
$$815$$ 0 0
$$816$$ −8.00000 −0.280056
$$817$$ 0 0
$$818$$ 24.0000i 0.839140i
$$819$$ 4.00000 0.139771
$$820$$ 0 0
$$821$$ 46.0000 1.60541 0.802706 0.596376i $$-0.203393\pi$$
0.802706 + 0.596376i $$0.203393\pi$$
$$822$$ 20.0000i 0.697580i
$$823$$ 24.0000i 0.836587i 0.908312 + 0.418294i $$0.137372\pi$$
−0.908312 + 0.418294i $$0.862628\pi$$
$$824$$ 42.0000 1.46314
$$825$$ 0 0
$$826$$ −2.00000 −0.0695889
$$827$$ 28.0000i 0.973655i 0.873498 + 0.486828i $$0.161846\pi$$
−0.873498 + 0.486828i $$0.838154\pi$$
$$828$$ 4.00000i 0.139010i
$$829$$ 2.00000 0.0694629 0.0347314 0.999397i $$-0.488942\pi$$
0.0347314 + 0.999397i $$0.488942\pi$$
$$830$$ 0 0
$$831$$ 44.0000 1.52634
$$832$$ − 28.0000i − 0.970725i
$$833$$ 4.00000i 0.138592i
$$834$$ 16.0000 0.554035
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 40.0000i 1.38260i
$$838$$ 2.00000i 0.0690889i
$$839$$ −34.0000 −1.17381 −0.586905 0.809656i $$-0.699654\pi$$
−0.586905 + 0.809656i $$0.699654\pi$$
$$840$$ 0 0
$$841$$ 7.00000 0.241379
$$842$$ 14.0000i 0.482472i
$$843$$ 12.0000i 0.413302i
$$844$$ −12.0000 −0.413057
$$845$$ 0 0
$$846$$ −10.0000 −0.343807
$$847$$ 1.00000i 0.0343604i
$$848$$ 6.00000i 0.206041i
$$849$$ −8.00000 −0.274559
$$850$$ 0 0
$$851$$ 24.0000 0.822709
$$852$$ − 24.0000i − 0.822226i
$$853$$ 44.0000i 1.50653i 0.657716 + 0.753266i $$0.271523\pi$$
−0.657716 + 0.753266i $$0.728477\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ −36.0000 −1.23045
$$857$$ − 56.0000i − 1.91292i −0.291858 0.956462i $$-0.594273\pi$$
0.291858 0.956462i $$-0.405727\pi$$
$$858$$ 8.00000i 0.273115i
$$859$$ −6.00000 −0.204717 −0.102359 0.994748i $$-0.532639\pi$$
−0.102359 + 0.994748i $$0.532639\pi$$
$$860$$ 0 0
$$861$$ −8.00000 −0.272639
$$862$$ 0 0
$$863$$ 24.0000i 0.816970i 0.912765 + 0.408485i $$0.133943\pi$$
−0.912765 + 0.408485i $$0.866057\pi$$
$$864$$ 20.0000 0.680414
$$865$$ 0 0
$$866$$ −26.0000 −0.883516
$$867$$ 2.00000i 0.0679236i
$$868$$ 10.0000i 0.339422i
$$869$$ −8.00000 −0.271381
$$870$$ 0 0
$$871$$ 32.0000 1.08428
$$872$$ − 42.0000i − 1.42230i
$$873$$ − 10.0000i − 0.338449i
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 16.0000 0.540590
$$877$$ − 42.0000i − 1.41824i −0.705088 0.709120i $$-0.749093\pi$$
0.705088 0.709120i $$-0.250907\pi$$
$$878$$ − 20.0000i − 0.674967i
$$879$$ 48.0000 1.61900
$$880$$ 0 0
$$881$$ −34.0000 −1.14549 −0.572745 0.819734i $$-0.694121\pi$$
−0.572745 + 0.819734i $$0.694121\pi$$
$$882$$ − 1.00000i − 0.0336718i
$$883$$ 28.0000i 0.942275i 0.882060 + 0.471138i $$0.156156\pi$$
−0.882060 + 0.471138i $$0.843844\pi$$
$$884$$ 16.0000 0.538138
$$885$$ 0 0
$$886$$ 4.00000 0.134383
$$887$$ 28.0000i 0.940148i 0.882627 + 0.470074i $$0.155773\pi$$
−0.882627 + 0.470074i $$0.844227\pi$$
$$888$$ 36.0000i 1.20808i
$$889$$ 8.00000 0.268311
$$890$$ 0 0
$$891$$ −11.0000 −0.368514
$$892$$ 22.0000i 0.736614i
$$893$$ 0 0
$$894$$ 20.0000 0.668900
$$895$$ 0 0
$$896$$ 3.00000 0.100223
$$897$$ 32.0000i 1.06845i
$$898$$ − 10.0000i − 0.333704i
$$899$$ 60.0000 2.00111
$$900$$ 0 0
$$901$$ −24.0000 −0.799556
$$902$$ − 4.00000i − 0.133185i
$$903$$ − 24.0000i − 0.798670i
$$904$$ 54.0000 1.79601
$$905$$ 0 0
$$906$$ −32.0000 −1.06313
$$907$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$908$$ − 12.0000i − 0.398234i
$$909$$ 4.00000 0.132672
$$910$$ 0 0
$$911$$ 36.0000 1.19273 0.596367 0.802712i $$-0.296610\pi$$
0.596367 + 0.802712i $$0.296610\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ −18.0000 −0.595387
$$915$$ 0 0
$$916$$ −18.0000 −0.594737
$$917$$ 12.0000i 0.396275i
$$918$$ − 16.0000i − 0.528079i
$$919$$ −40.0000 −1.31948 −0.659739 0.751495i $$-0.729333\pi$$
−0.659739 + 0.751495i $$0.729333\pi$$
$$920$$ 0 0
$$921$$ 40.0000 1.31804
$$922$$ 0 0
$$923$$ − 48.0000i − 1.57994i
$$924$$ −2.00000 −0.0657952
$$925$$ 0 0
$$926$$ 4.00000 0.131448
$$927$$ − 14.0000i − 0.459820i
$$928$$ − 30.0000i − 0.984798i
$$929$$ −6.00000 −0.196854 −0.0984268 0.995144i $$-0.531381\pi$$
−0.0984268 + 0.995144i $$0.531381\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ − 18.0000i − 0.589610i
$$933$$ − 36.0000i − 1.17859i
$$934$$ −30.0000 −0.981630
$$935$$ 0 0
$$936$$ −12.0000 −0.392232
$$937$$ 16.0000i 0.522697i 0.965244 + 0.261349i $$0.0841672\pi$$
−0.965244 + 0.261349i $$0.915833\pi$$
$$938$$ − 8.00000i − 0.261209i
$$939$$ −4.00000 −0.130535
$$940$$ 0 0
$$941$$ −24.0000 −0.782378 −0.391189 0.920310i $$-0.627936\pi$$
−0.391189 + 0.920310i $$0.627936\pi$$
$$942$$ − 28.0000i − 0.912289i
$$943$$ − 16.0000i − 0.521032i
$$944$$ 2.00000 0.0650945
$$945$$ 0 0
$$946$$ 12.0000 0.390154
$$947$$ 36.0000i 1.16984i 0.811090 + 0.584921i $$0.198875\pi$$
−0.811090 + 0.584921i $$0.801125\pi$$
$$948$$ − 16.0000i − 0.519656i
$$949$$ 32.0000 1.03876
$$950$$ 0 0
$$951$$ −4.00000 −0.129709
$$952$$ − 12.0000i − 0.388922i
$$953$$ 34.0000i 1.10137i 0.834714 + 0.550684i $$0.185633\pi$$
−0.834714 + 0.550684i $$0.814367\pi$$
$$954$$ 6.00000 0.194257
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 12.0000i 0.387905i
$$958$$ − 4.00000i − 0.129234i
$$959$$ −10.0000 −0.322917
$$960$$ 0 0
$$961$$ 69.0000 2.22581
$$962$$ 24.0000i 0.773791i
$$963$$ 12.0000i 0.386695i
$$964$$ −20.0000 −0.644157
$$965$$ 0 0
$$966$$ 8.00000 0.257396
$$967$$ − 40.0000i − 1.28631i −0.765735 0.643157i $$-0.777624\pi$$
0.765735 0.643157i $$-0.222376\pi$$
$$968$$ − 3.00000i − 0.0964237i
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 14.0000 0.449281 0.224641 0.974442i $$-0.427879\pi$$
0.224641 + 0.974442i $$0.427879\pi$$
$$972$$ − 10.0000i − 0.320750i
$$973$$ 8.00000i 0.256468i
$$974$$ 28.0000 0.897178
$$975$$ 0 0
$$976$$ 0 0
$$977$$ − 42.0000i − 1.34370i −0.740688 0.671850i $$-0.765500\pi$$
0.740688 0.671850i $$-0.234500\pi$$
$$978$$ − 16.0000i − 0.511624i
$$979$$ 6.00000 0.191761
$$980$$ 0 0
$$981$$ −14.0000 −0.446986
$$982$$ − 28.0000i − 0.893516i
$$983$$ 54.0000i 1.72233i 0.508323 + 0.861166i $$0.330265\pi$$
−0.508323 + 0.861166i $$0.669735\pi$$
$$984$$ 24.0000 0.765092
$$985$$ 0 0
$$986$$ −24.0000 −0.764316
$$987$$ − 20.0000i − 0.636607i
$$988$$ 0 0
$$989$$ 48.0000 1.52631
$$990$$ 0 0
$$991$$ 52.0000 1.65183 0.825917 0.563791i $$-0.190658\pi$$
0.825917 + 0.563791i $$0.190658\pi$$
$$992$$ − 50.0000i − 1.58750i
$$993$$ − 40.0000i − 1.26936i
$$994$$ −12.0000 −0.380617
$$995$$ 0 0
$$996$$ 0 0
$$997$$ − 20.0000i − 0.633406i −0.948525 0.316703i $$-0.897424\pi$$
0.948525 0.316703i $$-0.102576\pi$$
$$998$$ − 16.0000i − 0.506471i
$$999$$ −24.0000 −0.759326
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 1925.2.b.d.1849.1 2
5.2 odd 4 77.2.a.c.1.1 1
5.3 odd 4 1925.2.a.c.1.1 1
5.4 even 2 inner 1925.2.b.d.1849.2 2
15.2 even 4 693.2.a.a.1.1 1
20.7 even 4 1232.2.a.a.1.1 1
35.2 odd 12 539.2.e.a.67.1 2
35.12 even 12 539.2.e.b.67.1 2
35.17 even 12 539.2.e.b.177.1 2
35.27 even 4 539.2.a.d.1.1 1
35.32 odd 12 539.2.e.a.177.1 2
40.27 even 4 4928.2.a.bi.1.1 1
40.37 odd 4 4928.2.a.g.1.1 1
55.2 even 20 847.2.f.k.323.1 4
55.7 even 20 847.2.f.k.148.1 4
55.17 even 20 847.2.f.k.729.1 4
55.27 odd 20 847.2.f.e.729.1 4
55.32 even 4 847.2.a.a.1.1 1
55.37 odd 20 847.2.f.e.148.1 4
55.42 odd 20 847.2.f.e.323.1 4
55.47 odd 20 847.2.f.e.372.1 4
55.52 even 20 847.2.f.k.372.1 4
105.62 odd 4 4851.2.a.a.1.1 1
140.27 odd 4 8624.2.a.bc.1.1 1
165.32 odd 4 7623.2.a.n.1.1 1
385.307 odd 4 5929.2.a.b.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
77.2.a.c.1.1 1 5.2 odd 4
539.2.a.d.1.1 1 35.27 even 4
539.2.e.a.67.1 2 35.2 odd 12
539.2.e.a.177.1 2 35.32 odd 12
539.2.e.b.67.1 2 35.12 even 12
539.2.e.b.177.1 2 35.17 even 12
693.2.a.a.1.1 1 15.2 even 4
847.2.a.a.1.1 1 55.32 even 4
847.2.f.e.148.1 4 55.37 odd 20
847.2.f.e.323.1 4 55.42 odd 20
847.2.f.e.372.1 4 55.47 odd 20
847.2.f.e.729.1 4 55.27 odd 20
847.2.f.k.148.1 4 55.7 even 20
847.2.f.k.323.1 4 55.2 even 20
847.2.f.k.372.1 4 55.52 even 20
847.2.f.k.729.1 4 55.17 even 20
1232.2.a.a.1.1 1 20.7 even 4
1925.2.a.c.1.1 1 5.3 odd 4
1925.2.b.d.1849.1 2 1.1 even 1 trivial
1925.2.b.d.1849.2 2 5.4 even 2 inner
4851.2.a.a.1.1 1 105.62 odd 4
4928.2.a.g.1.1 1 40.37 odd 4
4928.2.a.bi.1.1 1 40.27 even 4
5929.2.a.b.1.1 1 385.307 odd 4
7623.2.a.n.1.1 1 165.32 odd 4
8624.2.a.bc.1.1 1 140.27 odd 4