# Properties

 Label 1875.2.b.a.1249.4 Level $1875$ Weight $2$ Character 1875.1249 Analytic conductor $14.972$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1875 = 3 \cdot 5^{4}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1875.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$14.9719503790$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{5})$$ Defining polynomial: $$x^{4} + 3x^{2} + 1$$ x^4 + 3*x^2 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 1249.4 Root $$1.61803i$$ of defining polynomial Character $$\chi$$ $$=$$ 1875.1249 Dual form 1875.2.b.a.1249.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.61803i q^{2} -1.00000i q^{3} -0.618034 q^{4} +1.61803 q^{6} -2.00000i q^{7} +2.23607i q^{8} -1.00000 q^{9} +O(q^{10})$$ $$q+1.61803i q^{2} -1.00000i q^{3} -0.618034 q^{4} +1.61803 q^{6} -2.00000i q^{7} +2.23607i q^{8} -1.00000 q^{9} -3.00000 q^{11} +0.618034i q^{12} +1.00000i q^{13} +3.23607 q^{14} -4.85410 q^{16} -4.23607i q^{17} -1.61803i q^{18} +6.70820 q^{19} -2.00000 q^{21} -4.85410i q^{22} -5.38197i q^{23} +2.23607 q^{24} -1.61803 q^{26} +1.00000i q^{27} +1.23607i q^{28} +3.61803 q^{29} +8.70820 q^{31} -3.38197i q^{32} +3.00000i q^{33} +6.85410 q^{34} +0.618034 q^{36} -2.00000i q^{37} +10.8541i q^{38} +1.00000 q^{39} -9.38197 q^{41} -3.23607i q^{42} +7.38197i q^{43} +1.85410 q^{44} +8.70820 q^{46} -4.76393i q^{47} +4.85410i q^{48} +3.00000 q^{49} -4.23607 q^{51} -0.618034i q^{52} -11.2361i q^{53} -1.61803 q^{54} +4.47214 q^{56} -6.70820i q^{57} +5.85410i q^{58} -3.94427 q^{59} +8.70820 q^{61} +14.0902i q^{62} +2.00000i q^{63} -4.23607 q^{64} -4.85410 q^{66} -13.1803i q^{67} +2.61803i q^{68} -5.38197 q^{69} +10.0902 q^{71} -2.23607i q^{72} -15.7082i q^{73} +3.23607 q^{74} -4.14590 q^{76} +6.00000i q^{77} +1.61803i q^{78} +9.14590 q^{79} +1.00000 q^{81} -15.1803i q^{82} -9.00000i q^{83} +1.23607 q^{84} -11.9443 q^{86} -3.61803i q^{87} -6.70820i q^{88} +11.1803 q^{89} +2.00000 q^{91} +3.32624i q^{92} -8.70820i q^{93} +7.70820 q^{94} -3.38197 q^{96} +3.85410i q^{97} +4.85410i q^{98} +3.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{4} + 2 q^{6} - 4 q^{9}+O(q^{10})$$ 4 * q + 2 * q^4 + 2 * q^6 - 4 * q^9 $$4 q + 2 q^{4} + 2 q^{6} - 4 q^{9} - 12 q^{11} + 4 q^{14} - 6 q^{16} - 8 q^{21} - 2 q^{26} + 10 q^{29} + 8 q^{31} + 14 q^{34} - 2 q^{36} + 4 q^{39} - 42 q^{41} - 6 q^{44} + 8 q^{46} + 12 q^{49} - 8 q^{51} - 2 q^{54} + 20 q^{59} + 8 q^{61} - 8 q^{64} - 6 q^{66} - 26 q^{69} + 18 q^{71} + 4 q^{74} - 30 q^{76} + 50 q^{79} + 4 q^{81} - 4 q^{84} - 12 q^{86} + 8 q^{91} + 4 q^{94} - 18 q^{96} + 12 q^{99}+O(q^{100})$$ 4 * q + 2 * q^4 + 2 * q^6 - 4 * q^9 - 12 * q^11 + 4 * q^14 - 6 * q^16 - 8 * q^21 - 2 * q^26 + 10 * q^29 + 8 * q^31 + 14 * q^34 - 2 * q^36 + 4 * q^39 - 42 * q^41 - 6 * q^44 + 8 * q^46 + 12 * q^49 - 8 * q^51 - 2 * q^54 + 20 * q^59 + 8 * q^61 - 8 * q^64 - 6 * q^66 - 26 * q^69 + 18 * q^71 + 4 * q^74 - 30 * q^76 + 50 * q^79 + 4 * q^81 - 4 * q^84 - 12 * q^86 + 8 * q^91 + 4 * q^94 - 18 * q^96 + 12 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1875\mathbb{Z}\right)^\times$$.

 $$n$$ $$626$$ $$1252$$ $$\chi(n)$$ $$1$$ $$-1$$

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.61803i 1.14412i 0.820211 + 0.572061i $$0.193856\pi$$
−0.820211 + 0.572061i $$0.806144\pi$$
$$3$$ − 1.00000i − 0.577350i
$$4$$ −0.618034 −0.309017
$$5$$ 0 0
$$6$$ 1.61803 0.660560
$$7$$ − 2.00000i − 0.755929i −0.925820 0.377964i $$-0.876624\pi$$
0.925820 0.377964i $$-0.123376\pi$$
$$8$$ 2.23607i 0.790569i
$$9$$ −1.00000 −0.333333
$$10$$ 0 0
$$11$$ −3.00000 −0.904534 −0.452267 0.891883i $$-0.649385\pi$$
−0.452267 + 0.891883i $$0.649385\pi$$
$$12$$ 0.618034i 0.178411i
$$13$$ 1.00000i 0.277350i 0.990338 + 0.138675i $$0.0442844\pi$$
−0.990338 + 0.138675i $$0.955716\pi$$
$$14$$ 3.23607 0.864876
$$15$$ 0 0
$$16$$ −4.85410 −1.21353
$$17$$ − 4.23607i − 1.02740i −0.857971 0.513699i $$-0.828275\pi$$
0.857971 0.513699i $$-0.171725\pi$$
$$18$$ − 1.61803i − 0.381374i
$$19$$ 6.70820 1.53897 0.769484 0.638666i $$-0.220514\pi$$
0.769484 + 0.638666i $$0.220514\pi$$
$$20$$ 0 0
$$21$$ −2.00000 −0.436436
$$22$$ − 4.85410i − 1.03490i
$$23$$ − 5.38197i − 1.12222i −0.827742 0.561109i $$-0.810375\pi$$
0.827742 0.561109i $$-0.189625\pi$$
$$24$$ 2.23607 0.456435
$$25$$ 0 0
$$26$$ −1.61803 −0.317323
$$27$$ 1.00000i 0.192450i
$$28$$ 1.23607i 0.233595i
$$29$$ 3.61803 0.671852 0.335926 0.941888i $$-0.390951\pi$$
0.335926 + 0.941888i $$0.390951\pi$$
$$30$$ 0 0
$$31$$ 8.70820 1.56404 0.782020 0.623254i $$-0.214190\pi$$
0.782020 + 0.623254i $$0.214190\pi$$
$$32$$ − 3.38197i − 0.597853i
$$33$$ 3.00000i 0.522233i
$$34$$ 6.85410 1.17547
$$35$$ 0 0
$$36$$ 0.618034 0.103006
$$37$$ − 2.00000i − 0.328798i −0.986394 0.164399i $$-0.947432\pi$$
0.986394 0.164399i $$-0.0525685\pi$$
$$38$$ 10.8541i 1.76077i
$$39$$ 1.00000 0.160128
$$40$$ 0 0
$$41$$ −9.38197 −1.46522 −0.732608 0.680650i $$-0.761697\pi$$
−0.732608 + 0.680650i $$0.761697\pi$$
$$42$$ − 3.23607i − 0.499336i
$$43$$ 7.38197i 1.12574i 0.826546 + 0.562870i $$0.190303\pi$$
−0.826546 + 0.562870i $$0.809697\pi$$
$$44$$ 1.85410 0.279516
$$45$$ 0 0
$$46$$ 8.70820 1.28395
$$47$$ − 4.76393i − 0.694891i −0.937700 0.347445i $$-0.887049\pi$$
0.937700 0.347445i $$-0.112951\pi$$
$$48$$ 4.85410i 0.700629i
$$49$$ 3.00000 0.428571
$$50$$ 0 0
$$51$$ −4.23607 −0.593168
$$52$$ − 0.618034i − 0.0857059i
$$53$$ − 11.2361i − 1.54339i −0.635991 0.771696i $$-0.719409\pi$$
0.635991 0.771696i $$-0.280591\pi$$
$$54$$ −1.61803 −0.220187
$$55$$ 0 0
$$56$$ 4.47214 0.597614
$$57$$ − 6.70820i − 0.888523i
$$58$$ 5.85410i 0.768681i
$$59$$ −3.94427 −0.513500 −0.256750 0.966478i $$-0.582652\pi$$
−0.256750 + 0.966478i $$0.582652\pi$$
$$60$$ 0 0
$$61$$ 8.70820 1.11497 0.557486 0.830187i $$-0.311766\pi$$
0.557486 + 0.830187i $$0.311766\pi$$
$$62$$ 14.0902i 1.78945i
$$63$$ 2.00000i 0.251976i
$$64$$ −4.23607 −0.529508
$$65$$ 0 0
$$66$$ −4.85410 −0.597499
$$67$$ − 13.1803i − 1.61023i −0.593115 0.805117i $$-0.702102\pi$$
0.593115 0.805117i $$-0.297898\pi$$
$$68$$ 2.61803i 0.317483i
$$69$$ −5.38197 −0.647913
$$70$$ 0 0
$$71$$ 10.0902 1.19748 0.598741 0.800942i $$-0.295668\pi$$
0.598741 + 0.800942i $$0.295668\pi$$
$$72$$ − 2.23607i − 0.263523i
$$73$$ − 15.7082i − 1.83851i −0.393667 0.919253i $$-0.628794\pi$$
0.393667 0.919253i $$-0.371206\pi$$
$$74$$ 3.23607 0.376185
$$75$$ 0 0
$$76$$ −4.14590 −0.475567
$$77$$ 6.00000i 0.683763i
$$78$$ 1.61803i 0.183206i
$$79$$ 9.14590 1.02899 0.514497 0.857492i $$-0.327979\pi$$
0.514497 + 0.857492i $$0.327979\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ − 15.1803i − 1.67639i
$$83$$ − 9.00000i − 0.987878i −0.869496 0.493939i $$-0.835557\pi$$
0.869496 0.493939i $$-0.164443\pi$$
$$84$$ 1.23607 0.134866
$$85$$ 0 0
$$86$$ −11.9443 −1.28798
$$87$$ − 3.61803i − 0.387894i
$$88$$ − 6.70820i − 0.715097i
$$89$$ 11.1803 1.18511 0.592557 0.805529i $$-0.298119\pi$$
0.592557 + 0.805529i $$0.298119\pi$$
$$90$$ 0 0
$$91$$ 2.00000 0.209657
$$92$$ 3.32624i 0.346784i
$$93$$ − 8.70820i − 0.902999i
$$94$$ 7.70820 0.795041
$$95$$ 0 0
$$96$$ −3.38197 −0.345170
$$97$$ 3.85410i 0.391325i 0.980671 + 0.195662i $$0.0626857\pi$$
−0.980671 + 0.195662i $$0.937314\pi$$
$$98$$ 4.85410i 0.490338i
$$99$$ 3.00000 0.301511
$$100$$ 0 0
$$101$$ −9.38197 −0.933541 −0.466770 0.884379i $$-0.654583\pi$$
−0.466770 + 0.884379i $$0.654583\pi$$
$$102$$ − 6.85410i − 0.678657i
$$103$$ 14.4164i 1.42049i 0.703954 + 0.710245i $$0.251416\pi$$
−0.703954 + 0.710245i $$0.748584\pi$$
$$104$$ −2.23607 −0.219265
$$105$$ 0 0
$$106$$ 18.1803 1.76583
$$107$$ − 1.14590i − 0.110778i −0.998465 0.0553891i $$-0.982360\pi$$
0.998465 0.0553891i $$-0.0176399\pi$$
$$108$$ − 0.618034i − 0.0594703i
$$109$$ 4.14590 0.397105 0.198553 0.980090i $$-0.436376\pi$$
0.198553 + 0.980090i $$0.436376\pi$$
$$110$$ 0 0
$$111$$ −2.00000 −0.189832
$$112$$ 9.70820i 0.917339i
$$113$$ 3.76393i 0.354081i 0.984204 + 0.177040i $$0.0566523\pi$$
−0.984204 + 0.177040i $$0.943348\pi$$
$$114$$ 10.8541 1.01658
$$115$$ 0 0
$$116$$ −2.23607 −0.207614
$$117$$ − 1.00000i − 0.0924500i
$$118$$ − 6.38197i − 0.587508i
$$119$$ −8.47214 −0.776639
$$120$$ 0 0
$$121$$ −2.00000 −0.181818
$$122$$ 14.0902i 1.27566i
$$123$$ 9.38197i 0.845943i
$$124$$ −5.38197 −0.483315
$$125$$ 0 0
$$126$$ −3.23607 −0.288292
$$127$$ 13.6525i 1.21146i 0.795670 + 0.605731i $$0.207119\pi$$
−0.795670 + 0.605731i $$0.792881\pi$$
$$128$$ − 13.6180i − 1.20368i
$$129$$ 7.38197 0.649946
$$130$$ 0 0
$$131$$ −14.1803 −1.23894 −0.619471 0.785020i $$-0.712653\pi$$
−0.619471 + 0.785020i $$0.712653\pi$$
$$132$$ − 1.85410i − 0.161379i
$$133$$ − 13.4164i − 1.16335i
$$134$$ 21.3262 1.84231
$$135$$ 0 0
$$136$$ 9.47214 0.812229
$$137$$ 0.437694i 0.0373947i 0.999825 + 0.0186974i $$0.00595190\pi$$
−0.999825 + 0.0186974i $$0.994048\pi$$
$$138$$ − 8.70820i − 0.741292i
$$139$$ −13.4164 −1.13796 −0.568982 0.822350i $$-0.692663\pi$$
−0.568982 + 0.822350i $$0.692663\pi$$
$$140$$ 0 0
$$141$$ −4.76393 −0.401195
$$142$$ 16.3262i 1.37007i
$$143$$ − 3.00000i − 0.250873i
$$144$$ 4.85410 0.404508
$$145$$ 0 0
$$146$$ 25.4164 2.10348
$$147$$ − 3.00000i − 0.247436i
$$148$$ 1.23607i 0.101604i
$$149$$ 13.0902 1.07239 0.536194 0.844095i $$-0.319861\pi$$
0.536194 + 0.844095i $$0.319861\pi$$
$$150$$ 0 0
$$151$$ −6.61803 −0.538568 −0.269284 0.963061i $$-0.586787\pi$$
−0.269284 + 0.963061i $$0.586787\pi$$
$$152$$ 15.0000i 1.21666i
$$153$$ 4.23607i 0.342466i
$$154$$ −9.70820 −0.782309
$$155$$ 0 0
$$156$$ −0.618034 −0.0494823
$$157$$ − 2.85410i − 0.227782i −0.993493 0.113891i $$-0.963669\pi$$
0.993493 0.113891i $$-0.0363315\pi$$
$$158$$ 14.7984i 1.17730i
$$159$$ −11.2361 −0.891078
$$160$$ 0 0
$$161$$ −10.7639 −0.848317
$$162$$ 1.61803i 0.127125i
$$163$$ − 18.2705i − 1.43106i −0.698584 0.715528i $$-0.746186\pi$$
0.698584 0.715528i $$-0.253814\pi$$
$$164$$ 5.79837 0.452777
$$165$$ 0 0
$$166$$ 14.5623 1.13025
$$167$$ 17.7984i 1.37728i 0.725104 + 0.688640i $$0.241792\pi$$
−0.725104 + 0.688640i $$0.758208\pi$$
$$168$$ − 4.47214i − 0.345033i
$$169$$ 12.0000 0.923077
$$170$$ 0 0
$$171$$ −6.70820 −0.512989
$$172$$ − 4.56231i − 0.347873i
$$173$$ − 0.909830i − 0.0691731i −0.999402 0.0345865i $$-0.988989\pi$$
0.999402 0.0345865i $$-0.0110114\pi$$
$$174$$ 5.85410 0.443798
$$175$$ 0 0
$$176$$ 14.5623 1.09768
$$177$$ 3.94427i 0.296470i
$$178$$ 18.0902i 1.35592i
$$179$$ 15.6525 1.16992 0.584960 0.811062i $$-0.301110\pi$$
0.584960 + 0.811062i $$0.301110\pi$$
$$180$$ 0 0
$$181$$ −12.4721 −0.927047 −0.463523 0.886085i $$-0.653415\pi$$
−0.463523 + 0.886085i $$0.653415\pi$$
$$182$$ 3.23607i 0.239873i
$$183$$ − 8.70820i − 0.643729i
$$184$$ 12.0344 0.887191
$$185$$ 0 0
$$186$$ 14.0902 1.03314
$$187$$ 12.7082i 0.929316i
$$188$$ 2.94427i 0.214733i
$$189$$ 2.00000 0.145479
$$190$$ 0 0
$$191$$ 17.3262 1.25368 0.626841 0.779147i $$-0.284347\pi$$
0.626841 + 0.779147i $$0.284347\pi$$
$$192$$ 4.23607i 0.305712i
$$193$$ 11.0000i 0.791797i 0.918294 + 0.395899i $$0.129567\pi$$
−0.918294 + 0.395899i $$0.870433\pi$$
$$194$$ −6.23607 −0.447724
$$195$$ 0 0
$$196$$ −1.85410 −0.132436
$$197$$ − 0.0901699i − 0.00642434i −0.999995 0.00321217i $$-0.998978\pi$$
0.999995 0.00321217i $$-0.00102247\pi$$
$$198$$ 4.85410i 0.344966i
$$199$$ −11.7082 −0.829973 −0.414986 0.909828i $$-0.636214\pi$$
−0.414986 + 0.909828i $$0.636214\pi$$
$$200$$ 0 0
$$201$$ −13.1803 −0.929669
$$202$$ − 15.1803i − 1.06808i
$$203$$ − 7.23607i − 0.507872i
$$204$$ 2.61803 0.183299
$$205$$ 0 0
$$206$$ −23.3262 −1.62522
$$207$$ 5.38197i 0.374072i
$$208$$ − 4.85410i − 0.336571i
$$209$$ −20.1246 −1.39205
$$210$$ 0 0
$$211$$ −3.00000 −0.206529 −0.103264 0.994654i $$-0.532929\pi$$
−0.103264 + 0.994654i $$0.532929\pi$$
$$212$$ 6.94427i 0.476935i
$$213$$ − 10.0902i − 0.691367i
$$214$$ 1.85410 0.126744
$$215$$ 0 0
$$216$$ −2.23607 −0.152145
$$217$$ − 17.4164i − 1.18230i
$$218$$ 6.70820i 0.454337i
$$219$$ −15.7082 −1.06146
$$220$$ 0 0
$$221$$ 4.23607 0.284949
$$222$$ − 3.23607i − 0.217191i
$$223$$ 10.1459i 0.679420i 0.940530 + 0.339710i $$0.110329\pi$$
−0.940530 + 0.339710i $$0.889671\pi$$
$$224$$ −6.76393 −0.451934
$$225$$ 0 0
$$226$$ −6.09017 −0.405112
$$227$$ 5.76393i 0.382566i 0.981535 + 0.191283i $$0.0612648\pi$$
−0.981535 + 0.191283i $$0.938735\pi$$
$$228$$ 4.14590i 0.274569i
$$229$$ 16.1803 1.06923 0.534613 0.845097i $$-0.320457\pi$$
0.534613 + 0.845097i $$0.320457\pi$$
$$230$$ 0 0
$$231$$ 6.00000 0.394771
$$232$$ 8.09017i 0.531146i
$$233$$ − 10.1803i − 0.666936i −0.942761 0.333468i $$-0.891781\pi$$
0.942761 0.333468i $$-0.108219\pi$$
$$234$$ 1.61803 0.105774
$$235$$ 0 0
$$236$$ 2.43769 0.158680
$$237$$ − 9.14590i − 0.594090i
$$238$$ − 13.7082i − 0.888571i
$$239$$ −21.3820 −1.38308 −0.691542 0.722336i $$-0.743068\pi$$
−0.691542 + 0.722336i $$0.743068\pi$$
$$240$$ 0 0
$$241$$ 7.32624 0.471924 0.235962 0.971762i $$-0.424176\pi$$
0.235962 + 0.971762i $$0.424176\pi$$
$$242$$ − 3.23607i − 0.208022i
$$243$$ − 1.00000i − 0.0641500i
$$244$$ −5.38197 −0.344545
$$245$$ 0 0
$$246$$ −15.1803 −0.967863
$$247$$ 6.70820i 0.426833i
$$248$$ 19.4721i 1.23648i
$$249$$ −9.00000 −0.570352
$$250$$ 0 0
$$251$$ −18.9787 −1.19793 −0.598963 0.800777i $$-0.704420\pi$$
−0.598963 + 0.800777i $$0.704420\pi$$
$$252$$ − 1.23607i − 0.0778650i
$$253$$ 16.1459i 1.01508i
$$254$$ −22.0902 −1.38606
$$255$$ 0 0
$$256$$ 13.5623 0.847644
$$257$$ 31.2148i 1.94712i 0.228422 + 0.973562i $$0.426644\pi$$
−0.228422 + 0.973562i $$0.573356\pi$$
$$258$$ 11.9443i 0.743618i
$$259$$ −4.00000 −0.248548
$$260$$ 0 0
$$261$$ −3.61803 −0.223951
$$262$$ − 22.9443i − 1.41750i
$$263$$ 12.5066i 0.771189i 0.922668 + 0.385594i $$0.126004\pi$$
−0.922668 + 0.385594i $$0.873996\pi$$
$$264$$ −6.70820 −0.412861
$$265$$ 0 0
$$266$$ 21.7082 1.33102
$$267$$ − 11.1803i − 0.684226i
$$268$$ 8.14590i 0.497590i
$$269$$ 20.5279 1.25161 0.625803 0.779981i $$-0.284771\pi$$
0.625803 + 0.779981i $$0.284771\pi$$
$$270$$ 0 0
$$271$$ −11.4164 −0.693497 −0.346749 0.937958i $$-0.612714\pi$$
−0.346749 + 0.937958i $$0.612714\pi$$
$$272$$ 20.5623i 1.24677i
$$273$$ − 2.00000i − 0.121046i
$$274$$ −0.708204 −0.0427842
$$275$$ 0 0
$$276$$ 3.32624 0.200216
$$277$$ − 13.0557i − 0.784443i −0.919871 0.392221i $$-0.871707\pi$$
0.919871 0.392221i $$-0.128293\pi$$
$$278$$ − 21.7082i − 1.30197i
$$279$$ −8.70820 −0.521347
$$280$$ 0 0
$$281$$ −14.1803 −0.845928 −0.422964 0.906146i $$-0.639010\pi$$
−0.422964 + 0.906146i $$0.639010\pi$$
$$282$$ − 7.70820i − 0.459017i
$$283$$ − 2.29180i − 0.136233i −0.997677 0.0681166i $$-0.978301\pi$$
0.997677 0.0681166i $$-0.0216990\pi$$
$$284$$ −6.23607 −0.370043
$$285$$ 0 0
$$286$$ 4.85410 0.287029
$$287$$ 18.7639i 1.10760i
$$288$$ 3.38197i 0.199284i
$$289$$ −0.944272 −0.0555454
$$290$$ 0 0
$$291$$ 3.85410 0.225931
$$292$$ 9.70820i 0.568130i
$$293$$ 6.32624i 0.369583i 0.982778 + 0.184791i $$0.0591609\pi$$
−0.982778 + 0.184791i $$0.940839\pi$$
$$294$$ 4.85410 0.283097
$$295$$ 0 0
$$296$$ 4.47214 0.259938
$$297$$ − 3.00000i − 0.174078i
$$298$$ 21.1803i 1.22694i
$$299$$ 5.38197 0.311247
$$300$$ 0 0
$$301$$ 14.7639 0.850979
$$302$$ − 10.7082i − 0.616188i
$$303$$ 9.38197i 0.538980i
$$304$$ −32.5623 −1.86758
$$305$$ 0 0
$$306$$ −6.85410 −0.391823
$$307$$ 8.85410i 0.505330i 0.967554 + 0.252665i $$0.0813071\pi$$
−0.967554 + 0.252665i $$0.918693\pi$$
$$308$$ − 3.70820i − 0.211295i
$$309$$ 14.4164 0.820121
$$310$$ 0 0
$$311$$ −13.5279 −0.767095 −0.383547 0.923521i $$-0.625298\pi$$
−0.383547 + 0.923521i $$0.625298\pi$$
$$312$$ 2.23607i 0.126592i
$$313$$ − 2.29180i − 0.129540i −0.997900 0.0647700i $$-0.979369\pi$$
0.997900 0.0647700i $$-0.0206314\pi$$
$$314$$ 4.61803 0.260611
$$315$$ 0 0
$$316$$ −5.65248 −0.317977
$$317$$ 20.5623i 1.15489i 0.816428 + 0.577447i $$0.195951\pi$$
−0.816428 + 0.577447i $$0.804049\pi$$
$$318$$ − 18.1803i − 1.01950i
$$319$$ −10.8541 −0.607713
$$320$$ 0 0
$$321$$ −1.14590 −0.0639578
$$322$$ − 17.4164i − 0.970578i
$$323$$ − 28.4164i − 1.58113i
$$324$$ −0.618034 −0.0343352
$$325$$ 0 0
$$326$$ 29.5623 1.63730
$$327$$ − 4.14590i − 0.229269i
$$328$$ − 20.9787i − 1.15836i
$$329$$ −9.52786 −0.525288
$$330$$ 0 0
$$331$$ −30.6869 −1.68671 −0.843353 0.537360i $$-0.819422\pi$$
−0.843353 + 0.537360i $$0.819422\pi$$
$$332$$ 5.56231i 0.305271i
$$333$$ 2.00000i 0.109599i
$$334$$ −28.7984 −1.57578
$$335$$ 0 0
$$336$$ 9.70820 0.529626
$$337$$ − 33.1803i − 1.80745i −0.428115 0.903724i $$-0.640822\pi$$
0.428115 0.903724i $$-0.359178\pi$$
$$338$$ 19.4164i 1.05611i
$$339$$ 3.76393 0.204429
$$340$$ 0 0
$$341$$ −26.1246 −1.41473
$$342$$ − 10.8541i − 0.586923i
$$343$$ − 20.0000i − 1.07990i
$$344$$ −16.5066 −0.889975
$$345$$ 0 0
$$346$$ 1.47214 0.0791425
$$347$$ 12.2705i 0.658715i 0.944205 + 0.329358i $$0.106832\pi$$
−0.944205 + 0.329358i $$0.893168\pi$$
$$348$$ 2.23607i 0.119866i
$$349$$ −7.23607 −0.387338 −0.193669 0.981067i $$-0.562039\pi$$
−0.193669 + 0.981067i $$0.562039\pi$$
$$350$$ 0 0
$$351$$ −1.00000 −0.0533761
$$352$$ 10.1459i 0.540778i
$$353$$ 12.3820i 0.659026i 0.944151 + 0.329513i $$0.106884\pi$$
−0.944151 + 0.329513i $$0.893116\pi$$
$$354$$ −6.38197 −0.339198
$$355$$ 0 0
$$356$$ −6.90983 −0.366220
$$357$$ 8.47214i 0.448393i
$$358$$ 25.3262i 1.33853i
$$359$$ −23.9443 −1.26373 −0.631865 0.775078i $$-0.717710\pi$$
−0.631865 + 0.775078i $$0.717710\pi$$
$$360$$ 0 0
$$361$$ 26.0000 1.36842
$$362$$ − 20.1803i − 1.06066i
$$363$$ 2.00000i 0.104973i
$$364$$ −1.23607 −0.0647876
$$365$$ 0 0
$$366$$ 14.0902 0.736505
$$367$$ − 12.5279i − 0.653949i −0.945033 0.326975i $$-0.893971\pi$$
0.945033 0.326975i $$-0.106029\pi$$
$$368$$ 26.1246i 1.36184i
$$369$$ 9.38197 0.488406
$$370$$ 0 0
$$371$$ −22.4721 −1.16670
$$372$$ 5.38197i 0.279042i
$$373$$ − 17.4164i − 0.901787i −0.892578 0.450894i $$-0.851105\pi$$
0.892578 0.450894i $$-0.148895\pi$$
$$374$$ −20.5623 −1.06325
$$375$$ 0 0
$$376$$ 10.6525 0.549359
$$377$$ 3.61803i 0.186338i
$$378$$ 3.23607i 0.166445i
$$379$$ 13.6180 0.699511 0.349756 0.936841i $$-0.386265\pi$$
0.349756 + 0.936841i $$0.386265\pi$$
$$380$$ 0 0
$$381$$ 13.6525 0.699438
$$382$$ 28.0344i 1.43437i
$$383$$ − 5.05573i − 0.258336i −0.991623 0.129168i $$-0.958769\pi$$
0.991623 0.129168i $$-0.0412306\pi$$
$$384$$ −13.6180 −0.694942
$$385$$ 0 0
$$386$$ −17.7984 −0.905913
$$387$$ − 7.38197i − 0.375246i
$$388$$ − 2.38197i − 0.120926i
$$389$$ −0.652476 −0.0330818 −0.0165409 0.999863i $$-0.505265\pi$$
−0.0165409 + 0.999863i $$0.505265\pi$$
$$390$$ 0 0
$$391$$ −22.7984 −1.15296
$$392$$ 6.70820i 0.338815i
$$393$$ 14.1803i 0.715304i
$$394$$ 0.145898 0.00735024
$$395$$ 0 0
$$396$$ −1.85410 −0.0931721
$$397$$ − 2.52786i − 0.126870i −0.997986 0.0634349i $$-0.979794\pi$$
0.997986 0.0634349i $$-0.0202055\pi$$
$$398$$ − 18.9443i − 0.949591i
$$399$$ −13.4164 −0.671660
$$400$$ 0 0
$$401$$ 36.2705 1.81126 0.905631 0.424066i $$-0.139397\pi$$
0.905631 + 0.424066i $$0.139397\pi$$
$$402$$ − 21.3262i − 1.06366i
$$403$$ 8.70820i 0.433787i
$$404$$ 5.79837 0.288480
$$405$$ 0 0
$$406$$ 11.7082 0.581068
$$407$$ 6.00000i 0.297409i
$$408$$ − 9.47214i − 0.468941i
$$409$$ 5.12461 0.253396 0.126698 0.991941i $$-0.459562\pi$$
0.126698 + 0.991941i $$0.459562\pi$$
$$410$$ 0 0
$$411$$ 0.437694 0.0215899
$$412$$ − 8.90983i − 0.438956i
$$413$$ 7.88854i 0.388170i
$$414$$ −8.70820 −0.427985
$$415$$ 0 0
$$416$$ 3.38197 0.165815
$$417$$ 13.4164i 0.657004i
$$418$$ − 32.5623i − 1.59267i
$$419$$ −0.326238 −0.0159378 −0.00796888 0.999968i $$-0.502537\pi$$
−0.00796888 + 0.999968i $$0.502537\pi$$
$$420$$ 0 0
$$421$$ −30.3607 −1.47969 −0.739844 0.672778i $$-0.765101\pi$$
−0.739844 + 0.672778i $$0.765101\pi$$
$$422$$ − 4.85410i − 0.236294i
$$423$$ 4.76393i 0.231630i
$$424$$ 25.1246 1.22016
$$425$$ 0 0
$$426$$ 16.3262 0.791009
$$427$$ − 17.4164i − 0.842839i
$$428$$ 0.708204i 0.0342323i
$$429$$ −3.00000 −0.144841
$$430$$ 0 0
$$431$$ 29.7639 1.43368 0.716839 0.697239i $$-0.245588\pi$$
0.716839 + 0.697239i $$0.245588\pi$$
$$432$$ − 4.85410i − 0.233543i
$$433$$ − 3.47214i − 0.166860i −0.996514 0.0834301i $$-0.973412\pi$$
0.996514 0.0834301i $$-0.0265875\pi$$
$$434$$ 28.1803 1.35270
$$435$$ 0 0
$$436$$ −2.56231 −0.122712
$$437$$ − 36.1033i − 1.72706i
$$438$$ − 25.4164i − 1.21444i
$$439$$ 32.0344 1.52892 0.764460 0.644671i $$-0.223006\pi$$
0.764460 + 0.644671i $$0.223006\pi$$
$$440$$ 0 0
$$441$$ −3.00000 −0.142857
$$442$$ 6.85410i 0.326016i
$$443$$ 19.4164i 0.922501i 0.887270 + 0.461251i $$0.152599\pi$$
−0.887270 + 0.461251i $$0.847401\pi$$
$$444$$ 1.23607 0.0586612
$$445$$ 0 0
$$446$$ −16.4164 −0.777339
$$447$$ − 13.0902i − 0.619144i
$$448$$ 8.47214i 0.400271i
$$449$$ −16.5066 −0.778994 −0.389497 0.921028i $$-0.627351\pi$$
−0.389497 + 0.921028i $$0.627351\pi$$
$$450$$ 0 0
$$451$$ 28.1459 1.32534
$$452$$ − 2.32624i − 0.109417i
$$453$$ 6.61803i 0.310942i
$$454$$ −9.32624 −0.437702
$$455$$ 0 0
$$456$$ 15.0000 0.702439
$$457$$ − 9.88854i − 0.462567i −0.972886 0.231283i $$-0.925708\pi$$
0.972886 0.231283i $$-0.0742924\pi$$
$$458$$ 26.1803i 1.22333i
$$459$$ 4.23607 0.197723
$$460$$ 0 0
$$461$$ −19.1803 −0.893317 −0.446659 0.894704i $$-0.647386\pi$$
−0.446659 + 0.894704i $$0.647386\pi$$
$$462$$ 9.70820i 0.451667i
$$463$$ 33.6869i 1.56556i 0.622296 + 0.782782i $$0.286200\pi$$
−0.622296 + 0.782782i $$0.713800\pi$$
$$464$$ −17.5623 −0.815310
$$465$$ 0 0
$$466$$ 16.4721 0.763057
$$467$$ − 10.4164i − 0.482014i −0.970523 0.241007i $$-0.922522\pi$$
0.970523 0.241007i $$-0.0774777\pi$$
$$468$$ 0.618034i 0.0285686i
$$469$$ −26.3607 −1.21722
$$470$$ 0 0
$$471$$ −2.85410 −0.131510
$$472$$ − 8.81966i − 0.405958i
$$473$$ − 22.1459i − 1.01827i
$$474$$ 14.7984 0.679712
$$475$$ 0 0
$$476$$ 5.23607 0.239995
$$477$$ 11.2361i 0.514464i
$$478$$ − 34.5967i − 1.58242i
$$479$$ −4.79837 −0.219243 −0.109622 0.993973i $$-0.534964\pi$$
−0.109622 + 0.993973i $$0.534964\pi$$
$$480$$ 0 0
$$481$$ 2.00000 0.0911922
$$482$$ 11.8541i 0.539940i
$$483$$ 10.7639i 0.489776i
$$484$$ 1.23607 0.0561849
$$485$$ 0 0
$$486$$ 1.61803 0.0733955
$$487$$ 16.6180i 0.753035i 0.926410 + 0.376518i $$0.122879\pi$$
−0.926410 + 0.376518i $$0.877121\pi$$
$$488$$ 19.4721i 0.881462i
$$489$$ −18.2705 −0.826221
$$490$$ 0 0
$$491$$ 22.3262 1.00757 0.503785 0.863829i $$-0.331941\pi$$
0.503785 + 0.863829i $$0.331941\pi$$
$$492$$ − 5.79837i − 0.261411i
$$493$$ − 15.3262i − 0.690259i
$$494$$ −10.8541 −0.488349
$$495$$ 0 0
$$496$$ −42.2705 −1.89800
$$497$$ − 20.1803i − 0.905212i
$$498$$ − 14.5623i − 0.652553i
$$499$$ 15.0000 0.671492 0.335746 0.941953i $$-0.391012\pi$$
0.335746 + 0.941953i $$0.391012\pi$$
$$500$$ 0 0
$$501$$ 17.7984 0.795173
$$502$$ − 30.7082i − 1.37057i
$$503$$ 3.96556i 0.176815i 0.996084 + 0.0884077i $$0.0281778\pi$$
−0.996084 + 0.0884077i $$0.971822\pi$$
$$504$$ −4.47214 −0.199205
$$505$$ 0 0
$$506$$ −26.1246 −1.16138
$$507$$ − 12.0000i − 0.532939i
$$508$$ − 8.43769i − 0.374362i
$$509$$ −32.8885 −1.45776 −0.728880 0.684642i $$-0.759959\pi$$
−0.728880 + 0.684642i $$0.759959\pi$$
$$510$$ 0 0
$$511$$ −31.4164 −1.38978
$$512$$ − 5.29180i − 0.233867i
$$513$$ 6.70820i 0.296174i
$$514$$ −50.5066 −2.22775
$$515$$ 0 0
$$516$$ −4.56231 −0.200844
$$517$$ 14.2918i 0.628552i
$$518$$ − 6.47214i − 0.284369i
$$519$$ −0.909830 −0.0399371
$$520$$ 0 0
$$521$$ 40.0902 1.75638 0.878191 0.478310i $$-0.158750\pi$$
0.878191 + 0.478310i $$0.158750\pi$$
$$522$$ − 5.85410i − 0.256227i
$$523$$ − 1.56231i − 0.0683149i −0.999416 0.0341574i $$-0.989125\pi$$
0.999416 0.0341574i $$-0.0108748\pi$$
$$524$$ 8.76393 0.382854
$$525$$ 0 0
$$526$$ −20.2361 −0.882334
$$527$$ − 36.8885i − 1.60689i
$$528$$ − 14.5623i − 0.633743i
$$529$$ −5.96556 −0.259372
$$530$$ 0 0
$$531$$ 3.94427 0.171167
$$532$$ 8.29180i 0.359495i
$$533$$ − 9.38197i − 0.406378i
$$534$$ 18.0902 0.782838
$$535$$ 0 0
$$536$$ 29.4721 1.27300
$$537$$ − 15.6525i − 0.675454i
$$538$$ 33.2148i 1.43199i
$$539$$ −9.00000 −0.387657
$$540$$ 0 0
$$541$$ −26.2918 −1.13037 −0.565186 0.824963i $$-0.691196\pi$$
−0.565186 + 0.824963i $$0.691196\pi$$
$$542$$ − 18.4721i − 0.793446i
$$543$$ 12.4721i 0.535231i
$$544$$ −14.3262 −0.614232
$$545$$ 0 0
$$546$$ 3.23607 0.138491
$$547$$ 24.7082i 1.05645i 0.849106 + 0.528223i $$0.177142\pi$$
−0.849106 + 0.528223i $$0.822858\pi$$
$$548$$ − 0.270510i − 0.0115556i
$$549$$ −8.70820 −0.371657
$$550$$ 0 0
$$551$$ 24.2705 1.03396
$$552$$ − 12.0344i − 0.512220i
$$553$$ − 18.2918i − 0.777846i
$$554$$ 21.1246 0.897499
$$555$$ 0 0
$$556$$ 8.29180 0.351650
$$557$$ − 37.6525i − 1.59539i −0.603063 0.797693i $$-0.706053\pi$$
0.603063 0.797693i $$-0.293947\pi$$
$$558$$ − 14.0902i − 0.596484i
$$559$$ −7.38197 −0.312224
$$560$$ 0 0
$$561$$ 12.7082 0.536541
$$562$$ − 22.9443i − 0.967846i
$$563$$ − 9.00000i − 0.379305i −0.981851 0.189652i $$-0.939264\pi$$
0.981851 0.189652i $$-0.0607361\pi$$
$$564$$ 2.94427 0.123976
$$565$$ 0 0
$$566$$ 3.70820 0.155867
$$567$$ − 2.00000i − 0.0839921i
$$568$$ 22.5623i 0.946693i
$$569$$ 10.8541 0.455028 0.227514 0.973775i $$-0.426940\pi$$
0.227514 + 0.973775i $$0.426940\pi$$
$$570$$ 0 0
$$571$$ −38.1246 −1.59547 −0.797733 0.603011i $$-0.793967\pi$$
−0.797733 + 0.603011i $$0.793967\pi$$
$$572$$ 1.85410i 0.0775239i
$$573$$ − 17.3262i − 0.723814i
$$574$$ −30.3607 −1.26723
$$575$$ 0 0
$$576$$ 4.23607 0.176503
$$577$$ 3.72949i 0.155261i 0.996982 + 0.0776304i $$0.0247354\pi$$
−0.996982 + 0.0776304i $$0.975265\pi$$
$$578$$ − 1.52786i − 0.0635508i
$$579$$ 11.0000 0.457144
$$580$$ 0 0
$$581$$ −18.0000 −0.746766
$$582$$ 6.23607i 0.258493i
$$583$$ 33.7082i 1.39605i
$$584$$ 35.1246 1.45347
$$585$$ 0 0
$$586$$ −10.2361 −0.422848
$$587$$ 39.3050i 1.62229i 0.584846 + 0.811144i $$0.301155\pi$$
−0.584846 + 0.811144i $$0.698845\pi$$
$$588$$ 1.85410i 0.0764619i
$$589$$ 58.4164 2.40701
$$590$$ 0 0
$$591$$ −0.0901699 −0.00370910
$$592$$ 9.70820i 0.399005i
$$593$$ − 17.6180i − 0.723486i −0.932278 0.361743i $$-0.882182\pi$$
0.932278 0.361743i $$-0.117818\pi$$
$$594$$ 4.85410 0.199166
$$595$$ 0 0
$$596$$ −8.09017 −0.331386
$$597$$ 11.7082i 0.479185i
$$598$$ 8.70820i 0.356105i
$$599$$ −39.2705 −1.60455 −0.802275 0.596955i $$-0.796377\pi$$
−0.802275 + 0.596955i $$0.796377\pi$$
$$600$$ 0 0
$$601$$ −24.7082 −1.00787 −0.503934 0.863742i $$-0.668115\pi$$
−0.503934 + 0.863742i $$0.668115\pi$$
$$602$$ 23.8885i 0.973624i
$$603$$ 13.1803i 0.536745i
$$604$$ 4.09017 0.166427
$$605$$ 0 0
$$606$$ −15.1803 −0.616659
$$607$$ − 22.8541i − 0.927619i −0.885935 0.463810i $$-0.846482\pi$$
0.885935 0.463810i $$-0.153518\pi$$
$$608$$ − 22.6869i − 0.920076i
$$609$$ −7.23607 −0.293220
$$610$$ 0 0
$$611$$ 4.76393 0.192728
$$612$$ − 2.61803i − 0.105828i
$$613$$ 5.87539i 0.237305i 0.992936 + 0.118652i $$0.0378574\pi$$
−0.992936 + 0.118652i $$0.962143\pi$$
$$614$$ −14.3262 −0.578160
$$615$$ 0 0
$$616$$ −13.4164 −0.540562
$$617$$ 25.2361i 1.01597i 0.861367 + 0.507983i $$0.169609\pi$$
−0.861367 + 0.507983i $$0.830391\pi$$
$$618$$ 23.3262i 0.938319i
$$619$$ −34.2705 −1.37745 −0.688724 0.725024i $$-0.741829\pi$$
−0.688724 + 0.725024i $$0.741829\pi$$
$$620$$ 0 0
$$621$$ 5.38197 0.215971
$$622$$ − 21.8885i − 0.877651i
$$623$$ − 22.3607i − 0.895862i
$$624$$ −4.85410 −0.194320
$$625$$ 0 0
$$626$$ 3.70820 0.148210
$$627$$ 20.1246i 0.803700i
$$628$$ 1.76393i 0.0703886i
$$629$$ −8.47214 −0.337806
$$630$$ 0 0
$$631$$ −10.7639 −0.428505 −0.214253 0.976778i $$-0.568732\pi$$
−0.214253 + 0.976778i $$0.568732\pi$$
$$632$$ 20.4508i 0.813491i
$$633$$ 3.00000i 0.119239i
$$634$$ −33.2705 −1.32134
$$635$$ 0 0
$$636$$ 6.94427 0.275358
$$637$$ 3.00000i 0.118864i
$$638$$ − 17.5623i − 0.695298i
$$639$$ −10.0902 −0.399161
$$640$$ 0 0
$$641$$ −23.3262 −0.921331 −0.460666 0.887574i $$-0.652389\pi$$
−0.460666 + 0.887574i $$0.652389\pi$$
$$642$$ − 1.85410i − 0.0731756i
$$643$$ − 5.90983i − 0.233061i −0.993187 0.116530i $$-0.962823\pi$$
0.993187 0.116530i $$-0.0371773\pi$$
$$644$$ 6.65248 0.262144
$$645$$ 0 0
$$646$$ 45.9787 1.80901
$$647$$ − 19.0344i − 0.748321i −0.927364 0.374161i $$-0.877931\pi$$
0.927364 0.374161i $$-0.122069\pi$$
$$648$$ 2.23607i 0.0878410i
$$649$$ 11.8328 0.464479
$$650$$ 0 0
$$651$$ −17.4164 −0.682603
$$652$$ 11.2918i 0.442221i
$$653$$ − 29.6525i − 1.16039i −0.814477 0.580196i $$-0.802976\pi$$
0.814477 0.580196i $$-0.197024\pi$$
$$654$$ 6.70820 0.262312
$$655$$ 0 0
$$656$$ 45.5410 1.77808
$$657$$ 15.7082i 0.612835i
$$658$$ − 15.4164i − 0.600994i
$$659$$ 2.23607 0.0871048 0.0435524 0.999051i $$-0.486132\pi$$
0.0435524 + 0.999051i $$0.486132\pi$$
$$660$$ 0 0
$$661$$ 4.88854 0.190142 0.0950712 0.995470i $$-0.469692\pi$$
0.0950712 + 0.995470i $$0.469692\pi$$
$$662$$ − 49.6525i − 1.92980i
$$663$$ − 4.23607i − 0.164515i
$$664$$ 20.1246 0.780986
$$665$$ 0 0
$$666$$ −3.23607 −0.125395
$$667$$ − 19.4721i − 0.753964i
$$668$$ − 11.0000i − 0.425603i
$$669$$ 10.1459 0.392263
$$670$$ 0 0
$$671$$ −26.1246 −1.00853
$$672$$ 6.76393i 0.260924i
$$673$$ 46.7771i 1.80312i 0.432650 + 0.901562i $$0.357579\pi$$
−0.432650 + 0.901562i $$0.642421\pi$$
$$674$$ 53.6869 2.06794
$$675$$ 0 0
$$676$$ −7.41641 −0.285246
$$677$$ − 44.8885i − 1.72521i −0.505881 0.862603i $$-0.668832\pi$$
0.505881 0.862603i $$-0.331168\pi$$
$$678$$ 6.09017i 0.233892i
$$679$$ 7.70820 0.295814
$$680$$ 0 0
$$681$$ 5.76393 0.220874
$$682$$ − 42.2705i − 1.61862i
$$683$$ 0.596748i 0.0228339i 0.999935 + 0.0114170i $$0.00363421\pi$$
−0.999935 + 0.0114170i $$0.996366\pi$$
$$684$$ 4.14590 0.158522
$$685$$ 0 0
$$686$$ 32.3607 1.23554
$$687$$ − 16.1803i − 0.617318i
$$688$$ − 35.8328i − 1.36611i
$$689$$ 11.2361 0.428060
$$690$$ 0 0
$$691$$ 15.0902 0.574057 0.287029 0.957922i $$-0.407333\pi$$
0.287029 + 0.957922i $$0.407333\pi$$
$$692$$ 0.562306i 0.0213757i
$$693$$ − 6.00000i − 0.227921i
$$694$$ −19.8541 −0.753651
$$695$$ 0 0
$$696$$ 8.09017 0.306657
$$697$$ 39.7426i 1.50536i
$$698$$ − 11.7082i − 0.443162i
$$699$$ −10.1803 −0.385056
$$700$$ 0 0
$$701$$ −48.6525 −1.83758 −0.918789 0.394748i $$-0.870832\pi$$
−0.918789 + 0.394748i $$0.870832\pi$$
$$702$$ − 1.61803i − 0.0610688i
$$703$$ − 13.4164i − 0.506009i
$$704$$ 12.7082 0.478958
$$705$$ 0 0
$$706$$ −20.0344 −0.754006
$$707$$ 18.7639i 0.705690i
$$708$$ − 2.43769i − 0.0916142i
$$709$$ 5.20163 0.195351 0.0976756 0.995218i $$-0.468859\pi$$
0.0976756 + 0.995218i $$0.468859\pi$$
$$710$$ 0 0
$$711$$ −9.14590 −0.342998
$$712$$ 25.0000i 0.936915i
$$713$$ − 46.8673i − 1.75519i
$$714$$ −13.7082 −0.513017
$$715$$ 0 0
$$716$$ −9.67376 −0.361525
$$717$$ 21.3820i 0.798524i
$$718$$ − 38.7426i − 1.44586i
$$719$$ 35.1246 1.30993 0.654963 0.755661i $$-0.272684\pi$$
0.654963 + 0.755661i $$0.272684\pi$$
$$720$$ 0 0
$$721$$ 28.8328 1.07379
$$722$$ 42.0689i 1.56564i
$$723$$ − 7.32624i − 0.272466i
$$724$$ 7.70820 0.286473
$$725$$ 0 0
$$726$$ −3.23607 −0.120102
$$727$$ − 14.4377i − 0.535464i −0.963493 0.267732i $$-0.913726\pi$$
0.963493 0.267732i $$-0.0862743\pi$$
$$728$$ 4.47214i 0.165748i
$$729$$ −1.00000 −0.0370370
$$730$$ 0 0
$$731$$ 31.2705 1.15658
$$732$$ 5.38197i 0.198923i
$$733$$ 10.1459i 0.374747i 0.982289 + 0.187374i $$0.0599975\pi$$
−0.982289 + 0.187374i $$0.940002\pi$$
$$734$$ 20.2705 0.748198
$$735$$ 0 0
$$736$$ −18.2016 −0.670921
$$737$$ 39.5410i 1.45651i
$$738$$ 15.1803i 0.558796i
$$739$$ 1.70820 0.0628373 0.0314186 0.999506i $$-0.489997\pi$$
0.0314186 + 0.999506i $$0.489997\pi$$
$$740$$ 0 0
$$741$$ 6.70820 0.246432
$$742$$ − 36.3607i − 1.33484i
$$743$$ 16.5279i 0.606349i 0.952935 + 0.303174i $$0.0980464\pi$$
−0.952935 + 0.303174i $$0.901954\pi$$
$$744$$ 19.4721 0.713883
$$745$$ 0 0
$$746$$ 28.1803 1.03176
$$747$$ 9.00000i 0.329293i
$$748$$ − 7.85410i − 0.287174i
$$749$$ −2.29180 −0.0837404
$$750$$ 0 0
$$751$$ 15.2918 0.558006 0.279003 0.960290i $$-0.409996\pi$$
0.279003 + 0.960290i $$0.409996\pi$$
$$752$$ 23.1246i 0.843268i
$$753$$ 18.9787i 0.691623i
$$754$$ −5.85410 −0.213194
$$755$$ 0 0
$$756$$ −1.23607 −0.0449554
$$757$$ 32.2705i 1.17289i 0.809988 + 0.586446i $$0.199473\pi$$
−0.809988 + 0.586446i $$0.800527\pi$$
$$758$$ 22.0344i 0.800327i
$$759$$ 16.1459 0.586059
$$760$$ 0 0
$$761$$ 3.18034 0.115287 0.0576436 0.998337i $$-0.481641\pi$$
0.0576436 + 0.998337i $$0.481641\pi$$
$$762$$ 22.0902i 0.800242i
$$763$$ − 8.29180i − 0.300183i
$$764$$ −10.7082 −0.387409
$$765$$ 0 0
$$766$$ 8.18034 0.295568
$$767$$ − 3.94427i − 0.142419i
$$768$$ − 13.5623i − 0.489388i
$$769$$ 36.3050 1.30919 0.654595 0.755980i $$-0.272839\pi$$
0.654595 + 0.755980i $$0.272839\pi$$
$$770$$ 0 0
$$771$$ 31.2148 1.12417
$$772$$ − 6.79837i − 0.244679i
$$773$$ 41.7771i 1.50262i 0.659951 + 0.751309i $$0.270577\pi$$
−0.659951 + 0.751309i $$0.729423\pi$$
$$774$$ 11.9443 0.429328
$$775$$ 0 0
$$776$$ −8.61803 −0.309369
$$777$$ 4.00000i 0.143499i
$$778$$ − 1.05573i − 0.0378497i
$$779$$ −62.9361 −2.25492
$$780$$ 0 0
$$781$$ −30.2705 −1.08316
$$782$$ − 36.8885i − 1.31913i
$$783$$ 3.61803i 0.129298i
$$784$$ −14.5623 −0.520082
$$785$$ 0 0
$$786$$ −22.9443 −0.818395
$$787$$ 17.1459i 0.611185i 0.952162 + 0.305593i $$0.0988546\pi$$
−0.952162 + 0.305593i $$0.901145\pi$$
$$788$$ 0.0557281i 0.00198523i
$$789$$ 12.5066 0.445246
$$790$$ 0 0
$$791$$ 7.52786 0.267660
$$792$$ 6.70820i 0.238366i
$$793$$ 8.70820i 0.309237i
$$794$$ 4.09017 0.145155
$$795$$ 0 0
$$796$$ 7.23607 0.256476
$$797$$ − 30.0132i − 1.06312i −0.847020 0.531560i $$-0.821606\pi$$
0.847020 0.531560i $$-0.178394\pi$$
$$798$$ − 21.7082i − 0.768462i
$$799$$ −20.1803 −0.713929
$$800$$ 0 0
$$801$$ −11.1803 −0.395038
$$802$$ 58.6869i 2.07231i
$$803$$ 47.1246i 1.66299i
$$804$$ 8.14590 0.287284
$$805$$ 0 0
$$806$$ −14.0902 −0.496305
$$807$$ − 20.5279i − 0.722615i
$$808$$ − 20.9787i − 0.738029i
$$809$$ 39.9230 1.40362 0.701809 0.712365i $$-0.252376\pi$$
0.701809 + 0.712365i $$0.252376\pi$$
$$810$$ 0 0
$$811$$ −33.7771 −1.18607 −0.593037 0.805175i $$-0.702071\pi$$
−0.593037 + 0.805175i $$0.702071\pi$$
$$812$$ 4.47214i 0.156941i
$$813$$ 11.4164i 0.400391i
$$814$$ −9.70820 −0.340272
$$815$$ 0 0
$$816$$ 20.5623 0.719825
$$817$$ 49.5197i 1.73248i
$$818$$ 8.29180i 0.289916i
$$819$$ −2.00000 −0.0698857
$$820$$ 0 0
$$821$$ 5.94427 0.207457 0.103728 0.994606i $$-0.466923\pi$$
0.103728 + 0.994606i $$0.466923\pi$$
$$822$$ 0.708204i 0.0247014i
$$823$$ 28.5623i 0.995619i 0.867286 + 0.497810i $$0.165862\pi$$
−0.867286 + 0.497810i $$0.834138\pi$$
$$824$$ −32.2361 −1.12300
$$825$$ 0 0
$$826$$ −12.7639 −0.444114
$$827$$ 48.9787i 1.70316i 0.524227 + 0.851578i $$0.324354\pi$$
−0.524227 + 0.851578i $$0.675646\pi$$
$$828$$ − 3.32624i − 0.115595i
$$829$$ 50.1246 1.74090 0.870450 0.492257i $$-0.163828\pi$$
0.870450 + 0.492257i $$0.163828\pi$$
$$830$$ 0 0
$$831$$ −13.0557 −0.452898
$$832$$ − 4.23607i − 0.146859i
$$833$$ − 12.7082i − 0.440313i
$$834$$ −21.7082 −0.751694
$$835$$ 0 0
$$836$$ 12.4377 0.430167
$$837$$ 8.70820i 0.301000i
$$838$$ − 0.527864i − 0.0182348i
$$839$$ 3.21478 0.110987 0.0554933 0.998459i $$-0.482327\pi$$
0.0554933 + 0.998459i $$0.482327\pi$$
$$840$$ 0 0
$$841$$ −15.9098 −0.548615
$$842$$ − 49.1246i − 1.69295i
$$843$$ 14.1803i 0.488397i
$$844$$ 1.85410 0.0638208
$$845$$ 0 0
$$846$$ −7.70820 −0.265014
$$847$$ 4.00000i 0.137442i
$$848$$ 54.5410i 1.87295i
$$849$$ −2.29180 −0.0786542
$$850$$ 0 0
$$851$$ −10.7639 −0.368983
$$852$$ 6.23607i 0.213644i
$$853$$ 20.3951i 0.698316i 0.937064 + 0.349158i $$0.113532\pi$$
−0.937064 + 0.349158i $$0.886468\pi$$
$$854$$ 28.1803 0.964311
$$855$$ 0 0
$$856$$ 2.56231 0.0875778
$$857$$ 9.05573i 0.309338i 0.987966 + 0.154669i $$0.0494311\pi$$
−0.987966 + 0.154669i $$0.950569\pi$$
$$858$$ − 4.85410i − 0.165716i
$$859$$ −15.1246 −0.516045 −0.258023 0.966139i $$-0.583071\pi$$
−0.258023 + 0.966139i $$0.583071\pi$$
$$860$$ 0 0
$$861$$ 18.7639 0.639473
$$862$$ 48.1591i 1.64030i
$$863$$ − 13.0689i − 0.444870i −0.974948 0.222435i $$-0.928599\pi$$
0.974948 0.222435i $$-0.0714005\pi$$
$$864$$ 3.38197 0.115057
$$865$$ 0 0
$$866$$ 5.61803 0.190909
$$867$$ 0.944272i 0.0320692i
$$868$$ 10.7639i 0.365352i
$$869$$ −27.4377 −0.930760
$$870$$ 0 0
$$871$$ 13.1803 0.446599
$$872$$ 9.27051i 0.313939i
$$873$$ − 3.85410i − 0.130442i
$$874$$ 58.4164 1.97596
$$875$$ 0 0
$$876$$ 9.70820 0.328010
$$877$$ 43.1246i 1.45621i 0.685463 + 0.728107i $$0.259600\pi$$
−0.685463 + 0.728107i $$0.740400\pi$$
$$878$$ 51.8328i 1.74927i
$$879$$ 6.32624 0.213379
$$880$$ 0 0
$$881$$ 15.0902 0.508401 0.254200 0.967152i $$-0.418188\pi$$
0.254200 + 0.967152i $$0.418188\pi$$
$$882$$ − 4.85410i − 0.163446i
$$883$$ − 29.2016i − 0.982713i −0.870959 0.491356i $$-0.836501\pi$$
0.870959 0.491356i $$-0.163499\pi$$
$$884$$ −2.61803 −0.0880540
$$885$$ 0 0
$$886$$ −31.4164 −1.05545
$$887$$ 42.9230i 1.44121i 0.693344 + 0.720606i $$0.256136\pi$$
−0.693344 + 0.720606i $$0.743864\pi$$
$$888$$ − 4.47214i − 0.150075i
$$889$$ 27.3050 0.915779
$$890$$ 0 0
$$891$$ −3.00000 −0.100504
$$892$$ − 6.27051i − 0.209952i
$$893$$ − 31.9574i − 1.06941i
$$894$$ 21.1803 0.708377
$$895$$ 0 0
$$896$$ −27.2361 −0.909893
$$897$$ − 5.38197i − 0.179699i
$$898$$ − 26.7082i − 0.891264i
$$899$$ 31.5066 1.05080
$$900$$ 0 0
$$901$$ −47.5967 −1.58568
$$902$$ 45.5410i 1.51635i
$$903$$ − 14.7639i − 0.491313i
$$904$$ −8.41641 −0.279926
$$905$$ 0 0
$$906$$ −10.7082 −0.355756
$$907$$ − 17.0000i − 0.564476i −0.959344 0.282238i $$-0.908923\pi$$
0.959344 0.282238i $$-0.0910767\pi$$
$$908$$ − 3.56231i − 0.118219i
$$909$$ 9.38197 0.311180
$$910$$ 0 0
$$911$$ 14.8885 0.493279 0.246640 0.969107i $$-0.420674\pi$$
0.246640 + 0.969107i $$0.420674\pi$$
$$912$$ 32.5623i 1.07825i
$$913$$ 27.0000i 0.893570i
$$914$$ 16.0000 0.529233
$$915$$ 0 0
$$916$$ −10.0000 −0.330409
$$917$$ 28.3607i 0.936552i
$$918$$ 6.85410i 0.226219i
$$919$$ 5.00000 0.164935 0.0824674 0.996594i $$-0.473720\pi$$
0.0824674 + 0.996594i $$0.473720\pi$$
$$920$$ 0 0
$$921$$ 8.85410 0.291753
$$922$$ − 31.0344i − 1.02206i
$$923$$ 10.0902i 0.332122i
$$924$$ −3.70820 −0.121991
$$925$$ 0 0
$$926$$ −54.5066 −1.79120
$$927$$ − 14.4164i − 0.473497i
$$928$$ − 12.2361i − 0.401669i
$$929$$ 29.5967 0.971038 0.485519 0.874226i $$-0.338631\pi$$
0.485519 + 0.874226i $$0.338631\pi$$
$$930$$ 0 0
$$931$$ 20.1246 0.659558
$$932$$ 6.29180i 0.206095i
$$933$$ 13.5279i 0.442882i
$$934$$ 16.8541 0.551483
$$935$$ 0 0
$$936$$ 2.23607 0.0730882
$$937$$ − 10.4164i − 0.340289i −0.985419 0.170145i $$-0.945577\pi$$
0.985419 0.170145i $$-0.0544235\pi$$
$$938$$ − 42.6525i − 1.39265i
$$939$$ −2.29180 −0.0747899
$$940$$ 0 0
$$941$$ 57.9787 1.89005 0.945026 0.326995i $$-0.106036\pi$$
0.945026 + 0.326995i $$0.106036\pi$$
$$942$$ − 4.61803i − 0.150464i
$$943$$ 50.4934i 1.64429i
$$944$$ 19.1459 0.623146
$$945$$ 0 0
$$946$$ 35.8328 1.16503
$$947$$ − 23.8328i − 0.774462i −0.921983 0.387231i $$-0.873432\pi$$
0.921983 0.387231i $$-0.126568\pi$$
$$948$$ 5.65248i 0.183584i
$$949$$ 15.7082 0.509910
$$950$$ 0 0
$$951$$ 20.5623 0.666778
$$952$$ − 18.9443i − 0.613987i
$$953$$ 42.0557i 1.36232i 0.732135 + 0.681159i $$0.238524\pi$$
−0.732135 + 0.681159i $$0.761476\pi$$
$$954$$ −18.1803 −0.588610
$$955$$ 0 0
$$956$$ 13.2148 0.427397
$$957$$ 10.8541i 0.350863i
$$958$$ − 7.76393i − 0.250841i
$$959$$ 0.875388 0.0282678
$$960$$ 0 0
$$961$$ 44.8328 1.44622
$$962$$ 3.23607i 0.104335i
$$963$$ 1.14590i 0.0369260i
$$964$$ −4.52786 −0.145833
$$965$$ 0 0
$$966$$ −17.4164 −0.560364
$$967$$ − 35.4164i − 1.13891i −0.822021 0.569457i $$-0.807153\pi$$
0.822021 0.569457i $$-0.192847\pi$$
$$968$$ − 4.47214i − 0.143740i
$$969$$ −28.4164 −0.912867
$$970$$ 0 0
$$971$$ 29.8885 0.959169 0.479585 0.877496i $$-0.340787\pi$$
0.479585 + 0.877496i $$0.340787\pi$$
$$972$$ 0.618034i 0.0198234i
$$973$$ 26.8328i 0.860221i
$$974$$ −26.8885 −0.861565
$$975$$ 0 0
$$976$$ −42.2705 −1.35305
$$977$$ − 37.6525i − 1.20461i −0.798266 0.602305i $$-0.794249\pi$$
0.798266 0.602305i $$-0.205751\pi$$
$$978$$ − 29.5623i − 0.945298i
$$979$$ −33.5410 −1.07198
$$980$$ 0 0
$$981$$ −4.14590 −0.132368
$$982$$ 36.1246i 1.15278i
$$983$$ 24.6180i 0.785193i 0.919711 + 0.392597i $$0.128423\pi$$
−0.919711 + 0.392597i $$0.871577\pi$$
$$984$$ −20.9787 −0.668777
$$985$$ 0 0
$$986$$ 24.7984 0.789741
$$987$$ 9.52786i 0.303275i
$$988$$ − 4.14590i − 0.131899i
$$989$$ 39.7295 1.26332
$$990$$ 0 0
$$991$$ 32.0000 1.01651 0.508257 0.861206i $$-0.330290\pi$$
0.508257 + 0.861206i $$0.330290\pi$$
$$992$$ − 29.4508i − 0.935065i
$$993$$ 30.6869i 0.973820i
$$994$$ 32.6525 1.03567
$$995$$ 0 0
$$996$$ 5.56231 0.176248
$$997$$ − 2.93112i − 0.0928294i −0.998922 0.0464147i $$-0.985220\pi$$
0.998922 0.0464147i $$-0.0147796\pi$$
$$998$$ 24.2705i 0.768270i
$$999$$ 2.00000 0.0632772
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1875.2.b.a.1249.4 4
5.2 odd 4 1875.2.a.b.1.1 2
5.3 odd 4 1875.2.a.c.1.2 yes 2
5.4 even 2 inner 1875.2.b.a.1249.1 4
15.2 even 4 5625.2.a.g.1.2 2
15.8 even 4 5625.2.a.b.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
1875.2.a.b.1.1 2 5.2 odd 4
1875.2.a.c.1.2 yes 2 5.3 odd 4
1875.2.b.a.1249.1 4 5.4 even 2 inner
1875.2.b.a.1249.4 4 1.1 even 1 trivial
5625.2.a.b.1.1 2 15.8 even 4
5625.2.a.g.1.2 2 15.2 even 4