# Properties

 Label 1150.2.b.f.599.2 Level $1150$ Weight $2$ Character 1150.599 Analytic conductor $9.183$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1150 = 2 \cdot 5^{2} \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1150.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$9.18279623245$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{13})$$ Defining polynomial: $$x^{4} + 7 x^{2} + 9$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 230) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 599.2 Root $$1.30278i$$ of defining polynomial Character $$\chi$$ $$=$$ 1150.599 Dual form 1150.2.b.f.599.3

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000i q^{2} +0.302776i q^{3} -1.00000 q^{4} +0.302776 q^{6} +3.30278i q^{7} +1.00000i q^{8} +2.90833 q^{9} +O(q^{10})$$ $$q-1.00000i q^{2} +0.302776i q^{3} -1.00000 q^{4} +0.302776 q^{6} +3.30278i q^{7} +1.00000i q^{8} +2.90833 q^{9} -1.69722 q^{11} -0.302776i q^{12} -3.30278i q^{13} +3.30278 q^{14} +1.00000 q^{16} +6.90833i q^{17} -2.90833i q^{18} -5.90833 q^{19} -1.00000 q^{21} +1.69722i q^{22} +1.00000i q^{23} -0.302776 q^{24} -3.30278 q^{26} +1.78890i q^{27} -3.30278i q^{28} +2.60555 q^{29} -7.90833 q^{31} -1.00000i q^{32} -0.513878i q^{33} +6.90833 q^{34} -2.90833 q^{36} +8.00000i q^{37} +5.90833i q^{38} +1.00000 q^{39} +0.908327 q^{41} +1.00000i q^{42} +9.21110i q^{43} +1.69722 q^{44} +1.00000 q^{46} -2.60555i q^{47} +0.302776i q^{48} -3.90833 q^{49} -2.09167 q^{51} +3.30278i q^{52} +11.2111i q^{53} +1.78890 q^{54} -3.30278 q^{56} -1.78890i q^{57} -2.60555i q^{58} +3.39445 q^{59} +11.5139 q^{61} +7.90833i q^{62} +9.60555i q^{63} -1.00000 q^{64} -0.513878 q^{66} -4.00000i q^{67} -6.90833i q^{68} -0.302776 q^{69} -16.3028 q^{71} +2.90833i q^{72} +5.81665i q^{73} +8.00000 q^{74} +5.90833 q^{76} -5.60555i q^{77} -1.00000i q^{78} +14.4222 q^{79} +8.18335 q^{81} -0.908327i q^{82} -11.2111i q^{83} +1.00000 q^{84} +9.21110 q^{86} +0.788897i q^{87} -1.69722i q^{88} +10.9083 q^{91} -1.00000i q^{92} -2.39445i q^{93} -2.60555 q^{94} +0.302776 q^{96} +6.30278i q^{97} +3.90833i q^{98} -4.93608 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{4} - 6q^{6} - 10q^{9} + O(q^{10})$$ $$4q - 4q^{4} - 6q^{6} - 10q^{9} - 14q^{11} + 6q^{14} + 4q^{16} - 2q^{19} - 4q^{21} + 6q^{24} - 6q^{26} - 4q^{29} - 10q^{31} + 6q^{34} + 10q^{36} + 4q^{39} - 18q^{41} + 14q^{44} + 4q^{46} + 6q^{49} - 30q^{51} + 36q^{54} - 6q^{56} + 28q^{59} + 10q^{61} - 4q^{64} + 34q^{66} + 6q^{69} - 58q^{71} + 32q^{74} + 2q^{76} + 76q^{81} + 4q^{84} + 8q^{86} + 22q^{91} + 4q^{94} - 6q^{96} + 74q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1150\mathbb{Z}\right)^\times$$.

 $$n$$ $$51$$ $$277$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ − 1.00000i − 0.707107i
$$3$$ 0.302776i 0.174808i 0.996173 + 0.0874038i $$0.0278570\pi$$
−0.996173 + 0.0874038i $$0.972143\pi$$
$$4$$ −1.00000 −0.500000
$$5$$ 0 0
$$6$$ 0.302776 0.123608
$$7$$ 3.30278i 1.24833i 0.781292 + 0.624166i $$0.214561\pi$$
−0.781292 + 0.624166i $$0.785439\pi$$
$$8$$ 1.00000i 0.353553i
$$9$$ 2.90833 0.969442
$$10$$ 0 0
$$11$$ −1.69722 −0.511732 −0.255866 0.966712i $$-0.582361\pi$$
−0.255866 + 0.966712i $$0.582361\pi$$
$$12$$ − 0.302776i − 0.0874038i
$$13$$ − 3.30278i − 0.916025i −0.888946 0.458013i $$-0.848561\pi$$
0.888946 0.458013i $$-0.151439\pi$$
$$14$$ 3.30278 0.882704
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 6.90833i 1.67552i 0.546042 + 0.837758i $$0.316134\pi$$
−0.546042 + 0.837758i $$0.683866\pi$$
$$18$$ − 2.90833i − 0.685499i
$$19$$ −5.90833 −1.35546 −0.677732 0.735309i $$-0.737037\pi$$
−0.677732 + 0.735309i $$0.737037\pi$$
$$20$$ 0 0
$$21$$ −1.00000 −0.218218
$$22$$ 1.69722i 0.361849i
$$23$$ 1.00000i 0.208514i
$$24$$ −0.302776 −0.0618038
$$25$$ 0 0
$$26$$ −3.30278 −0.647728
$$27$$ 1.78890i 0.344273i
$$28$$ − 3.30278i − 0.624166i
$$29$$ 2.60555 0.483839 0.241919 0.970296i $$-0.422223\pi$$
0.241919 + 0.970296i $$0.422223\pi$$
$$30$$ 0 0
$$31$$ −7.90833 −1.42038 −0.710189 0.704011i $$-0.751390\pi$$
−0.710189 + 0.704011i $$0.751390\pi$$
$$32$$ − 1.00000i − 0.176777i
$$33$$ − 0.513878i − 0.0894547i
$$34$$ 6.90833 1.18477
$$35$$ 0 0
$$36$$ −2.90833 −0.484721
$$37$$ 8.00000i 1.31519i 0.753371 + 0.657596i $$0.228427\pi$$
−0.753371 + 0.657596i $$0.771573\pi$$
$$38$$ 5.90833i 0.958457i
$$39$$ 1.00000 0.160128
$$40$$ 0 0
$$41$$ 0.908327 0.141857 0.0709284 0.997481i $$-0.477404\pi$$
0.0709284 + 0.997481i $$0.477404\pi$$
$$42$$ 1.00000i 0.154303i
$$43$$ 9.21110i 1.40468i 0.711842 + 0.702340i $$0.247861\pi$$
−0.711842 + 0.702340i $$0.752139\pi$$
$$44$$ 1.69722 0.255866
$$45$$ 0 0
$$46$$ 1.00000 0.147442
$$47$$ − 2.60555i − 0.380059i −0.981778 0.190029i $$-0.939142\pi$$
0.981778 0.190029i $$-0.0608583\pi$$
$$48$$ 0.302776i 0.0437019i
$$49$$ −3.90833 −0.558332
$$50$$ 0 0
$$51$$ −2.09167 −0.292893
$$52$$ 3.30278i 0.458013i
$$53$$ 11.2111i 1.53996i 0.638066 + 0.769982i $$0.279735\pi$$
−0.638066 + 0.769982i $$0.720265\pi$$
$$54$$ 1.78890 0.243438
$$55$$ 0 0
$$56$$ −3.30278 −0.441352
$$57$$ − 1.78890i − 0.236945i
$$58$$ − 2.60555i − 0.342126i
$$59$$ 3.39445 0.441920 0.220960 0.975283i $$-0.429081\pi$$
0.220960 + 0.975283i $$0.429081\pi$$
$$60$$ 0 0
$$61$$ 11.5139 1.47420 0.737101 0.675783i $$-0.236194\pi$$
0.737101 + 0.675783i $$0.236194\pi$$
$$62$$ 7.90833i 1.00436i
$$63$$ 9.60555i 1.21019i
$$64$$ −1.00000 −0.125000
$$65$$ 0 0
$$66$$ −0.513878 −0.0632540
$$67$$ − 4.00000i − 0.488678i −0.969690 0.244339i $$-0.921429\pi$$
0.969690 0.244339i $$-0.0785709\pi$$
$$68$$ − 6.90833i − 0.837758i
$$69$$ −0.302776 −0.0364499
$$70$$ 0 0
$$71$$ −16.3028 −1.93478 −0.967392 0.253285i $$-0.918489\pi$$
−0.967392 + 0.253285i $$0.918489\pi$$
$$72$$ 2.90833i 0.342750i
$$73$$ 5.81665i 0.680788i 0.940283 + 0.340394i $$0.110560\pi$$
−0.940283 + 0.340394i $$0.889440\pi$$
$$74$$ 8.00000 0.929981
$$75$$ 0 0
$$76$$ 5.90833 0.677732
$$77$$ − 5.60555i − 0.638812i
$$78$$ − 1.00000i − 0.113228i
$$79$$ 14.4222 1.62262 0.811312 0.584613i $$-0.198754\pi$$
0.811312 + 0.584613i $$0.198754\pi$$
$$80$$ 0 0
$$81$$ 8.18335 0.909261
$$82$$ − 0.908327i − 0.100308i
$$83$$ − 11.2111i − 1.23058i −0.788301 0.615289i $$-0.789039\pi$$
0.788301 0.615289i $$-0.210961\pi$$
$$84$$ 1.00000 0.109109
$$85$$ 0 0
$$86$$ 9.21110 0.993259
$$87$$ 0.788897i 0.0845787i
$$88$$ − 1.69722i − 0.180925i
$$89$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$90$$ 0 0
$$91$$ 10.9083 1.14350
$$92$$ − 1.00000i − 0.104257i
$$93$$ − 2.39445i − 0.248293i
$$94$$ −2.60555 −0.268742
$$95$$ 0 0
$$96$$ 0.302776 0.0309019
$$97$$ 6.30278i 0.639950i 0.947426 + 0.319975i $$0.103675\pi$$
−0.947426 + 0.319975i $$0.896325\pi$$
$$98$$ 3.90833i 0.394801i
$$99$$ −4.93608 −0.496095
$$100$$ 0 0
$$101$$ 2.60555 0.259262 0.129631 0.991562i $$-0.458621\pi$$
0.129631 + 0.991562i $$0.458621\pi$$
$$102$$ 2.09167i 0.207106i
$$103$$ − 8.11943i − 0.800031i −0.916508 0.400016i $$-0.869005\pi$$
0.916508 0.400016i $$-0.130995\pi$$
$$104$$ 3.30278 0.323864
$$105$$ 0 0
$$106$$ 11.2111 1.08892
$$107$$ − 2.60555i − 0.251888i −0.992037 0.125944i $$-0.959804\pi$$
0.992037 0.125944i $$-0.0401960\pi$$
$$108$$ − 1.78890i − 0.172137i
$$109$$ −1.48612 −0.142345 −0.0711723 0.997464i $$-0.522674\pi$$
−0.0711723 + 0.997464i $$0.522674\pi$$
$$110$$ 0 0
$$111$$ −2.42221 −0.229906
$$112$$ 3.30278i 0.312083i
$$113$$ 16.4222i 1.54487i 0.635093 + 0.772436i $$0.280962\pi$$
−0.635093 + 0.772436i $$0.719038\pi$$
$$114$$ −1.78890 −0.167546
$$115$$ 0 0
$$116$$ −2.60555 −0.241919
$$117$$ − 9.60555i − 0.888034i
$$118$$ − 3.39445i − 0.312484i
$$119$$ −22.8167 −2.09160
$$120$$ 0 0
$$121$$ −8.11943 −0.738130
$$122$$ − 11.5139i − 1.04242i
$$123$$ 0.275019i 0.0247977i
$$124$$ 7.90833 0.710189
$$125$$ 0 0
$$126$$ 9.60555 0.855731
$$127$$ 9.81665i 0.871087i 0.900168 + 0.435544i $$0.143444\pi$$
−0.900168 + 0.435544i $$0.856556\pi$$
$$128$$ 1.00000i 0.0883883i
$$129$$ −2.78890 −0.245549
$$130$$ 0 0
$$131$$ −11.2111 −0.979519 −0.489759 0.871858i $$-0.662915\pi$$
−0.489759 + 0.871858i $$0.662915\pi$$
$$132$$ 0.513878i 0.0447274i
$$133$$ − 19.5139i − 1.69207i
$$134$$ −4.00000 −0.345547
$$135$$ 0 0
$$136$$ −6.90833 −0.592384
$$137$$ 3.90833i 0.333911i 0.985964 + 0.166955i $$0.0533936\pi$$
−0.985964 + 0.166955i $$0.946606\pi$$
$$138$$ 0.302776i 0.0257740i
$$139$$ 12.6056 1.06919 0.534594 0.845109i $$-0.320464\pi$$
0.534594 + 0.845109i $$0.320464\pi$$
$$140$$ 0 0
$$141$$ 0.788897 0.0664372
$$142$$ 16.3028i 1.36810i
$$143$$ 5.60555i 0.468760i
$$144$$ 2.90833 0.242361
$$145$$ 0 0
$$146$$ 5.81665 0.481390
$$147$$ − 1.18335i − 0.0976007i
$$148$$ − 8.00000i − 0.657596i
$$149$$ −13.3028 −1.08981 −0.544903 0.838499i $$-0.683433\pi$$
−0.544903 + 0.838499i $$0.683433\pi$$
$$150$$ 0 0
$$151$$ 8.90833 0.724949 0.362475 0.931994i $$-0.381932\pi$$
0.362475 + 0.931994i $$0.381932\pi$$
$$152$$ − 5.90833i − 0.479229i
$$153$$ 20.0917i 1.62432i
$$154$$ −5.60555 −0.451708
$$155$$ 0 0
$$156$$ −1.00000 −0.0800641
$$157$$ − 18.6056i − 1.48488i −0.669910 0.742442i $$-0.733667\pi$$
0.669910 0.742442i $$-0.266333\pi$$
$$158$$ − 14.4222i − 1.14737i
$$159$$ −3.39445 −0.269197
$$160$$ 0 0
$$161$$ −3.30278 −0.260295
$$162$$ − 8.18335i − 0.642944i
$$163$$ − 9.30278i − 0.728650i −0.931272 0.364325i $$-0.881300\pi$$
0.931272 0.364325i $$-0.118700\pi$$
$$164$$ −0.908327 −0.0709284
$$165$$ 0 0
$$166$$ −11.2111 −0.870150
$$167$$ 6.78890i 0.525341i 0.964886 + 0.262670i $$0.0846032\pi$$
−0.964886 + 0.262670i $$0.915397\pi$$
$$168$$ − 1.00000i − 0.0771517i
$$169$$ 2.09167 0.160898
$$170$$ 0 0
$$171$$ −17.1833 −1.31404
$$172$$ − 9.21110i − 0.702340i
$$173$$ − 19.6972i − 1.49755i −0.662823 0.748776i $$-0.730642\pi$$
0.662823 0.748776i $$-0.269358\pi$$
$$174$$ 0.788897 0.0598062
$$175$$ 0 0
$$176$$ −1.69722 −0.127933
$$177$$ 1.02776i 0.0772509i
$$178$$ 0 0
$$179$$ −9.39445 −0.702174 −0.351087 0.936343i $$-0.614188\pi$$
−0.351087 + 0.936343i $$0.614188\pi$$
$$180$$ 0 0
$$181$$ 17.1194 1.27248 0.636239 0.771492i $$-0.280489\pi$$
0.636239 + 0.771492i $$0.280489\pi$$
$$182$$ − 10.9083i − 0.808579i
$$183$$ 3.48612i 0.257702i
$$184$$ −1.00000 −0.0737210
$$185$$ 0 0
$$186$$ −2.39445 −0.175569
$$187$$ − 11.7250i − 0.857416i
$$188$$ 2.60555i 0.190029i
$$189$$ −5.90833 −0.429768
$$190$$ 0 0
$$191$$ −8.60555 −0.622676 −0.311338 0.950299i $$-0.600777\pi$$
−0.311338 + 0.950299i $$0.600777\pi$$
$$192$$ − 0.302776i − 0.0218509i
$$193$$ 17.8167i 1.28247i 0.767344 + 0.641235i $$0.221578\pi$$
−0.767344 + 0.641235i $$0.778422\pi$$
$$194$$ 6.30278 0.452513
$$195$$ 0 0
$$196$$ 3.90833 0.279166
$$197$$ 4.30278i 0.306560i 0.988183 + 0.153280i $$0.0489837\pi$$
−0.988183 + 0.153280i $$0.951016\pi$$
$$198$$ 4.93608i 0.350792i
$$199$$ 20.4222 1.44769 0.723846 0.689962i $$-0.242373\pi$$
0.723846 + 0.689962i $$0.242373\pi$$
$$200$$ 0 0
$$201$$ 1.21110 0.0854246
$$202$$ − 2.60555i − 0.183326i
$$203$$ 8.60555i 0.603991i
$$204$$ 2.09167 0.146446
$$205$$ 0 0
$$206$$ −8.11943 −0.565707
$$207$$ 2.90833i 0.202143i
$$208$$ − 3.30278i − 0.229006i
$$209$$ 10.0278 0.693634
$$210$$ 0 0
$$211$$ 7.21110 0.496433 0.248216 0.968705i $$-0.420156\pi$$
0.248216 + 0.968705i $$0.420156\pi$$
$$212$$ − 11.2111i − 0.769982i
$$213$$ − 4.93608i − 0.338215i
$$214$$ −2.60555 −0.178112
$$215$$ 0 0
$$216$$ −1.78890 −0.121719
$$217$$ − 26.1194i − 1.77310i
$$218$$ 1.48612i 0.100653i
$$219$$ −1.76114 −0.119007
$$220$$ 0 0
$$221$$ 22.8167 1.53481
$$222$$ 2.42221i 0.162568i
$$223$$ 4.00000i 0.267860i 0.990991 + 0.133930i $$0.0427597\pi$$
−0.990991 + 0.133930i $$0.957240\pi$$
$$224$$ 3.30278 0.220676
$$225$$ 0 0
$$226$$ 16.4222 1.09239
$$227$$ − 14.6056i − 0.969404i −0.874679 0.484702i $$-0.838928\pi$$
0.874679 0.484702i $$-0.161072\pi$$
$$228$$ 1.78890i 0.118473i
$$229$$ −2.00000 −0.132164 −0.0660819 0.997814i $$-0.521050\pi$$
−0.0660819 + 0.997814i $$0.521050\pi$$
$$230$$ 0 0
$$231$$ 1.69722 0.111669
$$232$$ 2.60555i 0.171063i
$$233$$ − 25.8167i − 1.69131i −0.533734 0.845653i $$-0.679212\pi$$
0.533734 0.845653i $$-0.320788\pi$$
$$234$$ −9.60555 −0.627935
$$235$$ 0 0
$$236$$ −3.39445 −0.220960
$$237$$ 4.36669i 0.283647i
$$238$$ 22.8167i 1.47898i
$$239$$ −5.21110 −0.337078 −0.168539 0.985695i $$-0.553905\pi$$
−0.168539 + 0.985695i $$0.553905\pi$$
$$240$$ 0 0
$$241$$ −14.4222 −0.929016 −0.464508 0.885569i $$-0.653769\pi$$
−0.464508 + 0.885569i $$0.653769\pi$$
$$242$$ 8.11943i 0.521937i
$$243$$ 7.84441i 0.503219i
$$244$$ −11.5139 −0.737101
$$245$$ 0 0
$$246$$ 0.275019 0.0175346
$$247$$ 19.5139i 1.24164i
$$248$$ − 7.90833i − 0.502179i
$$249$$ 3.39445 0.215114
$$250$$ 0 0
$$251$$ 12.5139 0.789869 0.394934 0.918709i $$-0.370767\pi$$
0.394934 + 0.918709i $$0.370767\pi$$
$$252$$ − 9.60555i − 0.605093i
$$253$$ − 1.69722i − 0.106704i
$$254$$ 9.81665 0.615952
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ − 1.81665i − 0.113320i −0.998394 0.0566599i $$-0.981955\pi$$
0.998394 0.0566599i $$-0.0180451\pi$$
$$258$$ 2.78890i 0.173629i
$$259$$ −26.4222 −1.64180
$$260$$ 0 0
$$261$$ 7.57779 0.469054
$$262$$ 11.2111i 0.692624i
$$263$$ − 3.51388i − 0.216675i −0.994114 0.108338i $$-0.965447\pi$$
0.994114 0.108338i $$-0.0345527\pi$$
$$264$$ 0.513878 0.0316270
$$265$$ 0 0
$$266$$ −19.5139 −1.19647
$$267$$ 0 0
$$268$$ 4.00000i 0.244339i
$$269$$ −4.18335 −0.255063 −0.127532 0.991835i $$-0.540705\pi$$
−0.127532 + 0.991835i $$0.540705\pi$$
$$270$$ 0 0
$$271$$ −2.69722 −0.163845 −0.0819224 0.996639i $$-0.526106\pi$$
−0.0819224 + 0.996639i $$0.526106\pi$$
$$272$$ 6.90833i 0.418879i
$$273$$ 3.30278i 0.199893i
$$274$$ 3.90833 0.236111
$$275$$ 0 0
$$276$$ 0.302776 0.0182250
$$277$$ − 27.2111i − 1.63496i −0.575959 0.817478i $$-0.695371\pi$$
0.575959 0.817478i $$-0.304629\pi$$
$$278$$ − 12.6056i − 0.756031i
$$279$$ −23.0000 −1.37697
$$280$$ 0 0
$$281$$ −26.6056 −1.58715 −0.793577 0.608470i $$-0.791784\pi$$
−0.793577 + 0.608470i $$0.791784\pi$$
$$282$$ − 0.788897i − 0.0469782i
$$283$$ − 2.00000i − 0.118888i −0.998232 0.0594438i $$-0.981067\pi$$
0.998232 0.0594438i $$-0.0189327\pi$$
$$284$$ 16.3028 0.967392
$$285$$ 0 0
$$286$$ 5.60555 0.331463
$$287$$ 3.00000i 0.177084i
$$288$$ − 2.90833i − 0.171375i
$$289$$ −30.7250 −1.80735
$$290$$ 0 0
$$291$$ −1.90833 −0.111868
$$292$$ − 5.81665i − 0.340394i
$$293$$ − 23.2111i − 1.35601i −0.735059 0.678004i $$-0.762845\pi$$
0.735059 0.678004i $$-0.237155\pi$$
$$294$$ −1.18335 −0.0690142
$$295$$ 0 0
$$296$$ −8.00000 −0.464991
$$297$$ − 3.03616i − 0.176176i
$$298$$ 13.3028i 0.770609i
$$299$$ 3.30278 0.191004
$$300$$ 0 0
$$301$$ −30.4222 −1.75351
$$302$$ − 8.90833i − 0.512617i
$$303$$ 0.788897i 0.0453210i
$$304$$ −5.90833 −0.338866
$$305$$ 0 0
$$306$$ 20.0917 1.14856
$$307$$ − 11.6972i − 0.667596i −0.942645 0.333798i $$-0.891670\pi$$
0.942645 0.333798i $$-0.108330\pi$$
$$308$$ 5.60555i 0.319406i
$$309$$ 2.45837 0.139852
$$310$$ 0 0
$$311$$ 22.4222 1.27145 0.635723 0.771917i $$-0.280702\pi$$
0.635723 + 0.771917i $$0.280702\pi$$
$$312$$ 1.00000i 0.0566139i
$$313$$ − 19.7250i − 1.11492i −0.830203 0.557461i $$-0.811776\pi$$
0.830203 0.557461i $$-0.188224\pi$$
$$314$$ −18.6056 −1.04997
$$315$$ 0 0
$$316$$ −14.4222 −0.811312
$$317$$ 17.7250i 0.995534i 0.867311 + 0.497767i $$0.165847\pi$$
−0.867311 + 0.497767i $$0.834153\pi$$
$$318$$ 3.39445i 0.190351i
$$319$$ −4.42221 −0.247596
$$320$$ 0 0
$$321$$ 0.788897 0.0440320
$$322$$ 3.30278i 0.184056i
$$323$$ − 40.8167i − 2.27110i
$$324$$ −8.18335 −0.454630
$$325$$ 0 0
$$326$$ −9.30278 −0.515233
$$327$$ − 0.449961i − 0.0248829i
$$328$$ 0.908327i 0.0501540i
$$329$$ 8.60555 0.474439
$$330$$ 0 0
$$331$$ 16.6056 0.912724 0.456362 0.889794i $$-0.349152\pi$$
0.456362 + 0.889794i $$0.349152\pi$$
$$332$$ 11.2111i 0.615289i
$$333$$ 23.2666i 1.27500i
$$334$$ 6.78890 0.371472
$$335$$ 0 0
$$336$$ −1.00000 −0.0545545
$$337$$ − 22.5139i − 1.22641i −0.789924 0.613205i $$-0.789880\pi$$
0.789924 0.613205i $$-0.210120\pi$$
$$338$$ − 2.09167i − 0.113772i
$$339$$ −4.97224 −0.270055
$$340$$ 0 0
$$341$$ 13.4222 0.726853
$$342$$ 17.1833i 0.929169i
$$343$$ 10.2111i 0.551348i
$$344$$ −9.21110 −0.496629
$$345$$ 0 0
$$346$$ −19.6972 −1.05893
$$347$$ 28.5416i 1.53220i 0.642724 + 0.766098i $$0.277804\pi$$
−0.642724 + 0.766098i $$0.722196\pi$$
$$348$$ − 0.788897i − 0.0422893i
$$349$$ 27.2111 1.45658 0.728288 0.685271i $$-0.240316\pi$$
0.728288 + 0.685271i $$0.240316\pi$$
$$350$$ 0 0
$$351$$ 5.90833 0.315363
$$352$$ 1.69722i 0.0904624i
$$353$$ 10.4222i 0.554718i 0.960766 + 0.277359i $$0.0894591\pi$$
−0.960766 + 0.277359i $$0.910541\pi$$
$$354$$ 1.02776 0.0546246
$$355$$ 0 0
$$356$$ 0 0
$$357$$ − 6.90833i − 0.365627i
$$358$$ 9.39445i 0.496512i
$$359$$ −11.2111 −0.591699 −0.295850 0.955235i $$-0.595603\pi$$
−0.295850 + 0.955235i $$0.595603\pi$$
$$360$$ 0 0
$$361$$ 15.9083 0.837280
$$362$$ − 17.1194i − 0.899777i
$$363$$ − 2.45837i − 0.129031i
$$364$$ −10.9083 −0.571752
$$365$$ 0 0
$$366$$ 3.48612 0.182223
$$367$$ 14.7889i 0.771974i 0.922504 + 0.385987i $$0.126139\pi$$
−0.922504 + 0.385987i $$0.873861\pi$$
$$368$$ 1.00000i 0.0521286i
$$369$$ 2.64171 0.137522
$$370$$ 0 0
$$371$$ −37.0278 −1.92239
$$372$$ 2.39445i 0.124146i
$$373$$ − 4.60555i − 0.238466i −0.992866 0.119233i $$-0.961956\pi$$
0.992866 0.119233i $$-0.0380436\pi$$
$$374$$ −11.7250 −0.606284
$$375$$ 0 0
$$376$$ 2.60555 0.134371
$$377$$ − 8.60555i − 0.443208i
$$378$$ 5.90833i 0.303892i
$$379$$ −14.9083 −0.765789 −0.382895 0.923792i $$-0.625073\pi$$
−0.382895 + 0.923792i $$0.625073\pi$$
$$380$$ 0 0
$$381$$ −2.97224 −0.152273
$$382$$ 8.60555i 0.440298i
$$383$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$384$$ −0.302776 −0.0154510
$$385$$ 0 0
$$386$$ 17.8167 0.906844
$$387$$ 26.7889i 1.36176i
$$388$$ − 6.30278i − 0.319975i
$$389$$ 25.9361 1.31501 0.657506 0.753449i $$-0.271612\pi$$
0.657506 + 0.753449i $$0.271612\pi$$
$$390$$ 0 0
$$391$$ −6.90833 −0.349369
$$392$$ − 3.90833i − 0.197400i
$$393$$ − 3.39445i − 0.171227i
$$394$$ 4.30278 0.216771
$$395$$ 0 0
$$396$$ 4.93608 0.248048
$$397$$ 10.7250i 0.538271i 0.963102 + 0.269136i $$0.0867380\pi$$
−0.963102 + 0.269136i $$0.913262\pi$$
$$398$$ − 20.4222i − 1.02367i
$$399$$ 5.90833 0.295786
$$400$$ 0 0
$$401$$ −8.60555 −0.429741 −0.214870 0.976643i $$-0.568933\pi$$
−0.214870 + 0.976643i $$0.568933\pi$$
$$402$$ − 1.21110i − 0.0604043i
$$403$$ 26.1194i 1.30110i
$$404$$ −2.60555 −0.129631
$$405$$ 0 0
$$406$$ 8.60555 0.427086
$$407$$ − 13.5778i − 0.673026i
$$408$$ − 2.09167i − 0.103553i
$$409$$ 25.9083 1.28108 0.640542 0.767923i $$-0.278710\pi$$
0.640542 + 0.767923i $$0.278710\pi$$
$$410$$ 0 0
$$411$$ −1.18335 −0.0583702
$$412$$ 8.11943i 0.400016i
$$413$$ 11.2111i 0.551662i
$$414$$ 2.90833 0.142936
$$415$$ 0 0
$$416$$ −3.30278 −0.161932
$$417$$ 3.81665i 0.186902i
$$418$$ − 10.0278i − 0.490474i
$$419$$ −3.63331 −0.177499 −0.0887493 0.996054i $$-0.528287\pi$$
−0.0887493 + 0.996054i $$0.528287\pi$$
$$420$$ 0 0
$$421$$ 30.6972 1.49609 0.748046 0.663647i $$-0.230992\pi$$
0.748046 + 0.663647i $$0.230992\pi$$
$$422$$ − 7.21110i − 0.351031i
$$423$$ − 7.57779i − 0.368445i
$$424$$ −11.2111 −0.544459
$$425$$ 0 0
$$426$$ −4.93608 −0.239154
$$427$$ 38.0278i 1.84029i
$$428$$ 2.60555i 0.125944i
$$429$$ −1.69722 −0.0819428
$$430$$ 0 0
$$431$$ −30.2389 −1.45655 −0.728277 0.685283i $$-0.759679\pi$$
−0.728277 + 0.685283i $$0.759679\pi$$
$$432$$ 1.78890i 0.0860684i
$$433$$ 24.0917i 1.15777i 0.815409 + 0.578886i $$0.196512\pi$$
−0.815409 + 0.578886i $$0.803488\pi$$
$$434$$ −26.1194 −1.25377
$$435$$ 0 0
$$436$$ 1.48612 0.0711723
$$437$$ − 5.90833i − 0.282634i
$$438$$ 1.76114i 0.0841506i
$$439$$ 14.6972 0.701460 0.350730 0.936477i $$-0.385933\pi$$
0.350730 + 0.936477i $$0.385933\pi$$
$$440$$ 0 0
$$441$$ −11.3667 −0.541271
$$442$$ − 22.8167i − 1.08528i
$$443$$ − 17.4861i − 0.830791i −0.909641 0.415395i $$-0.863643\pi$$
0.909641 0.415395i $$-0.136357\pi$$
$$444$$ 2.42221 0.114953
$$445$$ 0 0
$$446$$ 4.00000 0.189405
$$447$$ − 4.02776i − 0.190506i
$$448$$ − 3.30278i − 0.156041i
$$449$$ 2.09167 0.0987122 0.0493561 0.998781i $$-0.484283\pi$$
0.0493561 + 0.998781i $$0.484283\pi$$
$$450$$ 0 0
$$451$$ −1.54163 −0.0725927
$$452$$ − 16.4222i − 0.772436i
$$453$$ 2.69722i 0.126727i
$$454$$ −14.6056 −0.685472
$$455$$ 0 0
$$456$$ 1.78890 0.0837728
$$457$$ − 32.4222i − 1.51665i −0.651879 0.758323i $$-0.726019\pi$$
0.651879 0.758323i $$-0.273981\pi$$
$$458$$ 2.00000i 0.0934539i
$$459$$ −12.3583 −0.576836
$$460$$ 0 0
$$461$$ 10.1833 0.474286 0.237143 0.971475i $$-0.423789\pi$$
0.237143 + 0.971475i $$0.423789\pi$$
$$462$$ − 1.69722i − 0.0789620i
$$463$$ − 17.6333i − 0.819489i −0.912200 0.409745i $$-0.865618\pi$$
0.912200 0.409745i $$-0.134382\pi$$
$$464$$ 2.60555 0.120960
$$465$$ 0 0
$$466$$ −25.8167 −1.19593
$$467$$ − 1.81665i − 0.0840647i −0.999116 0.0420324i $$-0.986617\pi$$
0.999116 0.0420324i $$-0.0133833\pi$$
$$468$$ 9.60555i 0.444017i
$$469$$ 13.2111 0.610032
$$470$$ 0 0
$$471$$ 5.63331 0.259569
$$472$$ 3.39445i 0.156242i
$$473$$ − 15.6333i − 0.718820i
$$474$$ 4.36669 0.200569
$$475$$ 0 0
$$476$$ 22.8167 1.04580
$$477$$ 32.6056i 1.49291i
$$478$$ 5.21110i 0.238350i
$$479$$ 30.0000 1.37073 0.685367 0.728197i $$-0.259642\pi$$
0.685367 + 0.728197i $$0.259642\pi$$
$$480$$ 0 0
$$481$$ 26.4222 1.20475
$$482$$ 14.4222i 0.656913i
$$483$$ − 1.00000i − 0.0455016i
$$484$$ 8.11943 0.369065
$$485$$ 0 0
$$486$$ 7.84441 0.355830
$$487$$ 9.81665i 0.444835i 0.974952 + 0.222418i $$0.0713948\pi$$
−0.974952 + 0.222418i $$0.928605\pi$$
$$488$$ 11.5139i 0.521209i
$$489$$ 2.81665 0.127373
$$490$$ 0 0
$$491$$ −4.18335 −0.188792 −0.0943959 0.995535i $$-0.530092\pi$$
−0.0943959 + 0.995535i $$0.530092\pi$$
$$492$$ − 0.275019i − 0.0123988i
$$493$$ 18.0000i 0.810679i
$$494$$ 19.5139 0.877971
$$495$$ 0 0
$$496$$ −7.90833 −0.355094
$$497$$ − 53.8444i − 2.41525i
$$498$$ − 3.39445i − 0.152109i
$$499$$ 31.6333 1.41610 0.708051 0.706162i $$-0.249575\pi$$
0.708051 + 0.706162i $$0.249575\pi$$
$$500$$ 0 0
$$501$$ −2.05551 −0.0918335
$$502$$ − 12.5139i − 0.558522i
$$503$$ − 29.7250i − 1.32537i −0.748898 0.662686i $$-0.769417\pi$$
0.748898 0.662686i $$-0.230583\pi$$
$$504$$ −9.60555 −0.427865
$$505$$ 0 0
$$506$$ −1.69722 −0.0754508
$$507$$ 0.633308i 0.0281262i
$$508$$ − 9.81665i − 0.435544i
$$509$$ 35.4500 1.57129 0.785646 0.618676i $$-0.212331\pi$$
0.785646 + 0.618676i $$0.212331\pi$$
$$510$$ 0 0
$$511$$ −19.2111 −0.849849
$$512$$ − 1.00000i − 0.0441942i
$$513$$ − 10.5694i − 0.466650i
$$514$$ −1.81665 −0.0801292
$$515$$ 0 0
$$516$$ 2.78890 0.122774
$$517$$ 4.42221i 0.194488i
$$518$$ 26.4222i 1.16093i
$$519$$ 5.96384 0.261784
$$520$$ 0 0
$$521$$ 6.00000 0.262865 0.131432 0.991325i $$-0.458042\pi$$
0.131432 + 0.991325i $$0.458042\pi$$
$$522$$ − 7.57779i − 0.331671i
$$523$$ 20.4222i 0.893001i 0.894783 + 0.446500i $$0.147330\pi$$
−0.894783 + 0.446500i $$0.852670\pi$$
$$524$$ 11.2111 0.489759
$$525$$ 0 0
$$526$$ −3.51388 −0.153212
$$527$$ − 54.6333i − 2.37986i
$$528$$ − 0.513878i − 0.0223637i
$$529$$ −1.00000 −0.0434783
$$530$$ 0 0
$$531$$ 9.87217 0.428416
$$532$$ 19.5139i 0.846034i
$$533$$ − 3.00000i − 0.129944i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 4.00000 0.172774
$$537$$ − 2.84441i − 0.122745i
$$538$$ 4.18335i 0.180357i
$$539$$ 6.63331 0.285717
$$540$$ 0 0
$$541$$ 28.8444 1.24012 0.620059 0.784555i $$-0.287109\pi$$
0.620059 + 0.784555i $$0.287109\pi$$
$$542$$ 2.69722i 0.115856i
$$543$$ 5.18335i 0.222439i
$$544$$ 6.90833 0.296192
$$545$$ 0 0
$$546$$ 3.30278 0.141346
$$547$$ − 10.5139i − 0.449541i −0.974412 0.224770i $$-0.927837\pi$$
0.974412 0.224770i $$-0.0721632\pi$$
$$548$$ − 3.90833i − 0.166955i
$$549$$ 33.4861 1.42915
$$550$$ 0 0
$$551$$ −15.3944 −0.655826
$$552$$ − 0.302776i − 0.0128870i
$$553$$ 47.6333i 2.02557i
$$554$$ −27.2111 −1.15609
$$555$$ 0 0
$$556$$ −12.6056 −0.534594
$$557$$ 22.4222i 0.950059i 0.879970 + 0.475030i $$0.157563\pi$$
−0.879970 + 0.475030i $$0.842437\pi$$
$$558$$ 23.0000i 0.973668i
$$559$$ 30.4222 1.28672
$$560$$ 0 0
$$561$$ 3.55004 0.149883
$$562$$ 26.6056i 1.12229i
$$563$$ 3.63331i 0.153126i 0.997065 + 0.0765628i $$0.0243946\pi$$
−0.997065 + 0.0765628i $$0.975605\pi$$
$$564$$ −0.788897 −0.0332186
$$565$$ 0 0
$$566$$ −2.00000 −0.0840663
$$567$$ 27.0278i 1.13506i
$$568$$ − 16.3028i − 0.684049i
$$569$$ −28.4222 −1.19152 −0.595760 0.803162i $$-0.703149\pi$$
−0.595760 + 0.803162i $$0.703149\pi$$
$$570$$ 0 0
$$571$$ −16.1194 −0.674577 −0.337289 0.941401i $$-0.609510\pi$$
−0.337289 + 0.941401i $$0.609510\pi$$
$$572$$ − 5.60555i − 0.234380i
$$573$$ − 2.60555i − 0.108848i
$$574$$ 3.00000 0.125218
$$575$$ 0 0
$$576$$ −2.90833 −0.121180
$$577$$ 2.00000i 0.0832611i 0.999133 + 0.0416305i $$0.0132552\pi$$
−0.999133 + 0.0416305i $$0.986745\pi$$
$$578$$ 30.7250i 1.27799i
$$579$$ −5.39445 −0.224186
$$580$$ 0 0
$$581$$ 37.0278 1.53617
$$582$$ 1.90833i 0.0791027i
$$583$$ − 19.0278i − 0.788049i
$$584$$ −5.81665 −0.240695
$$585$$ 0 0
$$586$$ −23.2111 −0.958842
$$587$$ 16.5416i 0.682746i 0.939928 + 0.341373i $$0.110892\pi$$
−0.939928 + 0.341373i $$0.889108\pi$$
$$588$$ 1.18335i 0.0488004i
$$589$$ 46.7250 1.92527
$$590$$ 0 0
$$591$$ −1.30278 −0.0535890
$$592$$ 8.00000i 0.328798i
$$593$$ 1.81665i 0.0746010i 0.999304 + 0.0373005i $$0.0118759\pi$$
−0.999304 + 0.0373005i $$0.988124\pi$$
$$594$$ −3.03616 −0.124575
$$595$$ 0 0
$$596$$ 13.3028 0.544903
$$597$$ 6.18335i 0.253068i
$$598$$ − 3.30278i − 0.135061i
$$599$$ 35.3305 1.44357 0.721783 0.692119i $$-0.243323\pi$$
0.721783 + 0.692119i $$0.243323\pi$$
$$600$$ 0 0
$$601$$ 42.9361 1.75140 0.875700 0.482856i $$-0.160401\pi$$
0.875700 + 0.482856i $$0.160401\pi$$
$$602$$ 30.4222i 1.23992i
$$603$$ − 11.6333i − 0.473745i
$$604$$ −8.90833 −0.362475
$$605$$ 0 0
$$606$$ 0.788897 0.0320468
$$607$$ 46.0555i 1.86934i 0.355522 + 0.934668i $$0.384303\pi$$
−0.355522 + 0.934668i $$0.615697\pi$$
$$608$$ 5.90833i 0.239614i
$$609$$ −2.60555 −0.105582
$$610$$ 0 0
$$611$$ −8.60555 −0.348143
$$612$$ − 20.0917i − 0.812158i
$$613$$ − 3.57779i − 0.144506i −0.997386 0.0722529i $$-0.976981\pi$$
0.997386 0.0722529i $$-0.0230189\pi$$
$$614$$ −11.6972 −0.472062
$$615$$ 0 0
$$616$$ 5.60555 0.225854
$$617$$ 18.9083i 0.761221i 0.924736 + 0.380610i $$0.124286\pi$$
−0.924736 + 0.380610i $$0.875714\pi$$
$$618$$ − 2.45837i − 0.0988900i
$$619$$ 12.3305 0.495606 0.247803 0.968810i $$-0.420291\pi$$
0.247803 + 0.968810i $$0.420291\pi$$
$$620$$ 0 0
$$621$$ −1.78890 −0.0717860
$$622$$ − 22.4222i − 0.899049i
$$623$$ 0 0
$$624$$ 1.00000 0.0400320
$$625$$ 0 0
$$626$$ −19.7250 −0.788369
$$627$$ 3.03616i 0.121253i
$$628$$ 18.6056i 0.742442i
$$629$$ −55.2666 −2.20362
$$630$$ 0 0
$$631$$ 23.3944 0.931318 0.465659 0.884964i $$-0.345817\pi$$
0.465659 + 0.884964i $$0.345817\pi$$
$$632$$ 14.4222i 0.573685i
$$633$$ 2.18335i 0.0867802i
$$634$$ 17.7250 0.703949
$$635$$ 0 0
$$636$$ 3.39445 0.134599
$$637$$ 12.9083i 0.511447i
$$638$$ 4.42221i 0.175077i
$$639$$ −47.4138 −1.87566
$$640$$ 0 0
$$641$$ −36.0000 −1.42191 −0.710957 0.703235i $$-0.751738\pi$$
−0.710957 + 0.703235i $$0.751738\pi$$
$$642$$ − 0.788897i − 0.0311353i
$$643$$ 34.2389i 1.35025i 0.737704 + 0.675124i $$0.235910\pi$$
−0.737704 + 0.675124i $$0.764090\pi$$
$$644$$ 3.30278 0.130148
$$645$$ 0 0
$$646$$ −40.8167 −1.60591
$$647$$ 26.8444i 1.05536i 0.849442 + 0.527681i $$0.176938\pi$$
−0.849442 + 0.527681i $$0.823062\pi$$
$$648$$ 8.18335i 0.321472i
$$649$$ −5.76114 −0.226145
$$650$$ 0 0
$$651$$ 7.90833 0.309952
$$652$$ 9.30278i 0.364325i
$$653$$ − 41.7250i − 1.63282i −0.577469 0.816412i $$-0.695960\pi$$
0.577469 0.816412i $$-0.304040\pi$$
$$654$$ −0.449961 −0.0175949
$$655$$ 0 0
$$656$$ 0.908327 0.0354642
$$657$$ 16.9167i 0.659985i
$$658$$ − 8.60555i − 0.335479i
$$659$$ −15.6333 −0.608987 −0.304494 0.952514i $$-0.598487\pi$$
−0.304494 + 0.952514i $$0.598487\pi$$
$$660$$ 0 0
$$661$$ −34.9083 −1.35778 −0.678888 0.734242i $$-0.737538\pi$$
−0.678888 + 0.734242i $$0.737538\pi$$
$$662$$ − 16.6056i − 0.645393i
$$663$$ 6.90833i 0.268297i
$$664$$ 11.2111 0.435075
$$665$$ 0 0
$$666$$ 23.2666 0.901563
$$667$$ 2.60555i 0.100887i
$$668$$ − 6.78890i − 0.262670i
$$669$$ −1.21110 −0.0468239
$$670$$ 0 0
$$671$$ −19.5416 −0.754396
$$672$$ 1.00000i 0.0385758i
$$673$$ 37.6333i 1.45066i 0.688403 + 0.725329i $$0.258312\pi$$
−0.688403 + 0.725329i $$0.741688\pi$$
$$674$$ −22.5139 −0.867202
$$675$$ 0 0
$$676$$ −2.09167 −0.0804490
$$677$$ 16.4222i 0.631157i 0.948900 + 0.315578i $$0.102198\pi$$
−0.948900 + 0.315578i $$0.897802\pi$$
$$678$$ 4.97224i 0.190958i
$$679$$ −20.8167 −0.798870
$$680$$ 0 0
$$681$$ 4.42221 0.169459
$$682$$ − 13.4222i − 0.513963i
$$683$$ 0.275019i 0.0105233i 0.999986 + 0.00526166i $$0.00167485\pi$$
−0.999986 + 0.00526166i $$0.998325\pi$$
$$684$$ 17.1833 0.657022
$$685$$ 0 0
$$686$$ 10.2111 0.389862
$$687$$ − 0.605551i − 0.0231032i
$$688$$ 9.21110i 0.351170i
$$689$$ 37.0278 1.41065
$$690$$ 0 0
$$691$$ 51.8167 1.97120 0.985599 0.169098i $$-0.0540855\pi$$
0.985599 + 0.169098i $$0.0540855\pi$$
$$692$$ 19.6972i 0.748776i
$$693$$ − 16.3028i − 0.619291i
$$694$$ 28.5416 1.08343
$$695$$ 0 0
$$696$$ −0.788897 −0.0299031
$$697$$ 6.27502i 0.237683i
$$698$$ − 27.2111i − 1.02996i
$$699$$ 7.81665 0.295653
$$700$$ 0 0
$$701$$ 32.0917 1.21209 0.606043 0.795432i $$-0.292756\pi$$
0.606043 + 0.795432i $$0.292756\pi$$
$$702$$ − 5.90833i − 0.222995i
$$703$$ − 47.2666i − 1.78269i
$$704$$ 1.69722 0.0639665
$$705$$ 0 0
$$706$$ 10.4222 0.392245
$$707$$ 8.60555i 0.323645i
$$708$$ − 1.02776i − 0.0386254i
$$709$$ 15.8806 0.596407 0.298204 0.954502i $$-0.403613\pi$$
0.298204 + 0.954502i $$0.403613\pi$$
$$710$$ 0 0
$$711$$ 41.9445 1.57304
$$712$$ 0 0
$$713$$ − 7.90833i − 0.296169i
$$714$$ −6.90833 −0.258538
$$715$$ 0 0
$$716$$ 9.39445 0.351087
$$717$$ − 1.57779i − 0.0589238i
$$718$$ 11.2111i 0.418395i
$$719$$ −10.6972 −0.398939 −0.199470 0.979904i $$-0.563922\pi$$
−0.199470 + 0.979904i $$0.563922\pi$$
$$720$$ 0 0
$$721$$ 26.8167 0.998704
$$722$$ − 15.9083i − 0.592047i
$$723$$ − 4.36669i − 0.162399i
$$724$$ −17.1194 −0.636239
$$725$$ 0 0
$$726$$ −2.45837 −0.0912385
$$727$$ 2.90833i 0.107864i 0.998545 + 0.0539319i $$0.0171754\pi$$
−0.998545 + 0.0539319i $$0.982825\pi$$
$$728$$ 10.9083i 0.404289i
$$729$$ 22.1749 0.821294
$$730$$ 0 0
$$731$$ −63.6333 −2.35356
$$732$$ − 3.48612i − 0.128851i
$$733$$ − 29.6333i − 1.09453i −0.836959 0.547266i $$-0.815669\pi$$
0.836959 0.547266i $$-0.184331\pi$$
$$734$$ 14.7889 0.545868
$$735$$ 0 0
$$736$$ 1.00000 0.0368605
$$737$$ 6.78890i 0.250072i
$$738$$ − 2.64171i − 0.0972427i
$$739$$ −35.6333 −1.31079 −0.655396 0.755285i $$-0.727498\pi$$
−0.655396 + 0.755285i $$0.727498\pi$$
$$740$$ 0 0
$$741$$ −5.90833 −0.217048
$$742$$ 37.0278i 1.35933i
$$743$$ 32.3305i 1.18609i 0.805169 + 0.593046i $$0.202075\pi$$
−0.805169 + 0.593046i $$0.797925\pi$$
$$744$$ 2.39445 0.0877847
$$745$$ 0 0
$$746$$ −4.60555 −0.168621
$$747$$ − 32.6056i − 1.19297i
$$748$$ 11.7250i 0.428708i
$$749$$ 8.60555 0.314440
$$750$$ 0 0
$$751$$ 21.8167 0.796101 0.398051 0.917364i $$-0.369687\pi$$
0.398051 + 0.917364i $$0.369687\pi$$
$$752$$ − 2.60555i − 0.0950147i
$$753$$ 3.78890i 0.138075i
$$754$$ −8.60555 −0.313396
$$755$$ 0 0
$$756$$ 5.90833 0.214884
$$757$$ 13.2111i 0.480166i 0.970752 + 0.240083i $$0.0771746\pi$$
−0.970752 + 0.240083i $$0.922825\pi$$
$$758$$ 14.9083i 0.541495i
$$759$$ 0.513878 0.0186526
$$760$$ 0 0
$$761$$ −49.5416 −1.79588 −0.897941 0.440115i $$-0.854938\pi$$
−0.897941 + 0.440115i $$0.854938\pi$$
$$762$$ 2.97224i 0.107673i
$$763$$ − 4.90833i − 0.177693i
$$764$$ 8.60555 0.311338
$$765$$ 0 0
$$766$$ 0 0
$$767$$ − 11.2111i − 0.404809i
$$768$$ 0.302776i 0.0109255i
$$769$$ −45.2666 −1.63236 −0.816178 0.577801i $$-0.803911\pi$$
−0.816178 + 0.577801i $$0.803911\pi$$
$$770$$ 0 0
$$771$$ 0.550039 0.0198092
$$772$$ − 17.8167i − 0.641235i
$$773$$ − 12.0000i − 0.431610i −0.976436 0.215805i $$-0.930762\pi$$
0.976436 0.215805i $$-0.0692376\pi$$
$$774$$ 26.7889 0.962907
$$775$$ 0 0
$$776$$ −6.30278 −0.226256
$$777$$ − 8.00000i − 0.286998i
$$778$$ − 25.9361i − 0.929854i
$$779$$ −5.36669 −0.192282
$$780$$ 0 0
$$781$$ 27.6695 0.990091
$$782$$ 6.90833i 0.247041i
$$783$$ 4.66106i 0.166573i
$$784$$ −3.90833 −0.139583
$$785$$ 0 0
$$786$$ −3.39445 −0.121076
$$787$$ 37.4500i 1.33495i 0.744634 + 0.667473i $$0.232624\pi$$
−0.744634 + 0.667473i $$0.767376\pi$$
$$788$$ − 4.30278i − 0.153280i
$$789$$ 1.06392 0.0378764
$$790$$ 0 0
$$791$$ −54.2389 −1.92851
$$792$$ − 4.93608i − 0.175396i
$$793$$ − 38.0278i − 1.35041i
$$794$$ 10.7250 0.380615
$$795$$ 0 0
$$796$$ −20.4222 −0.723846
$$797$$ 1.81665i 0.0643492i 0.999482 + 0.0321746i $$0.0102433\pi$$
−0.999482 + 0.0321746i $$0.989757\pi$$
$$798$$ − 5.90833i − 0.209153i
$$799$$ 18.0000 0.636794
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 8.60555i 0.303873i
$$803$$ − 9.87217i − 0.348381i
$$804$$ −1.21110 −0.0427123
$$805$$ 0 0
$$806$$ 26.1194 0.920018
$$807$$ − 1.26662i − 0.0445870i
$$808$$ 2.60555i 0.0916630i
$$809$$ −50.7250 −1.78340 −0.891698 0.452631i $$-0.850485\pi$$
−0.891698 + 0.452631i $$0.850485\pi$$
$$810$$ 0 0
$$811$$ −41.0278 −1.44068 −0.720340 0.693621i $$-0.756014\pi$$
−0.720340 + 0.693621i $$0.756014\pi$$
$$812$$ − 8.60555i − 0.301996i
$$813$$ − 0.816654i − 0.0286413i
$$814$$ −13.5778 −0.475901
$$815$$ 0 0
$$816$$ −2.09167 −0.0732232
$$817$$ − 54.4222i − 1.90399i
$$818$$ − 25.9083i − 0.905863i
$$819$$ 31.7250 1.10856
$$820$$ 0 0
$$821$$ −30.0000 −1.04701 −0.523504 0.852023i $$-0.675375\pi$$
−0.523504 + 0.852023i $$0.675375\pi$$
$$822$$ 1.18335i 0.0412739i
$$823$$ 15.2111i 0.530226i 0.964217 + 0.265113i $$0.0854092\pi$$
−0.964217 + 0.265113i $$0.914591\pi$$
$$824$$ 8.11943 0.282854
$$825$$ 0 0
$$826$$ 11.2111 0.390084
$$827$$ − 29.4500i − 1.02408i −0.858963 0.512038i $$-0.828891\pi$$
0.858963 0.512038i $$-0.171109\pi$$
$$828$$ − 2.90833i − 0.101071i
$$829$$ −31.2111 −1.08401 −0.542003 0.840376i $$-0.682334\pi$$
−0.542003 + 0.840376i $$0.682334\pi$$
$$830$$ 0 0
$$831$$ 8.23886 0.285803
$$832$$ 3.30278i 0.114503i
$$833$$ − 27.0000i − 0.935495i
$$834$$ 3.81665 0.132160
$$835$$ 0 0
$$836$$ −10.0278 −0.346817
$$837$$ − 14.1472i − 0.488998i
$$838$$ 3.63331i 0.125511i
$$839$$ 43.8167 1.51272 0.756359 0.654156i $$-0.226976\pi$$
0.756359 + 0.654156i $$0.226976\pi$$
$$840$$ 0 0
$$841$$ −22.2111 −0.765900
$$842$$ − 30.6972i − 1.05790i
$$843$$ − 8.05551i − 0.277447i
$$844$$ −7.21110 −0.248216
$$845$$ 0 0
$$846$$ −7.57779 −0.260530
$$847$$ − 26.8167i − 0.921431i
$$848$$ 11.2111i 0.384991i
$$849$$ 0.605551 0.0207825
$$850$$ 0 0
$$851$$ −8.00000 −0.274236
$$852$$ 4.93608i 0.169107i
$$853$$ 21.7250i 0.743849i 0.928263 + 0.371925i $$0.121302\pi$$
−0.928263 + 0.371925i $$0.878698\pi$$
$$854$$ 38.0278 1.30128
$$855$$ 0 0
$$856$$ 2.60555 0.0890559
$$857$$ 9.63331i 0.329068i 0.986371 + 0.164534i $$0.0526119\pi$$
−0.986371 + 0.164534i $$0.947388\pi$$
$$858$$ 1.69722i 0.0579423i
$$859$$ 35.8167 1.22205 0.611024 0.791612i $$-0.290758\pi$$
0.611024 + 0.791612i $$0.290758\pi$$
$$860$$ 0 0
$$861$$ −0.908327 −0.0309557
$$862$$ 30.2389i 1.02994i
$$863$$ 41.4500i 1.41097i 0.708723 + 0.705487i $$0.249271\pi$$
−0.708723 + 0.705487i $$0.750729\pi$$
$$864$$ 1.78890 0.0608595
$$865$$ 0 0
$$866$$ 24.0917 0.818668
$$867$$ − 9.30278i − 0.315939i
$$868$$ 26.1194i 0.886551i
$$869$$ −24.4777 −0.830350
$$870$$ 0 0
$$871$$ −13.2111 −0.447641
$$872$$ − 1.48612i − 0.0503264i
$$873$$ 18.3305i 0.620395i
$$874$$ −5.90833 −0.199852
$$875$$ 0 0
$$876$$ 1.76114 0.0595034
$$877$$ − 48.1749i − 1.62675i −0.581738 0.813376i $$-0.697627\pi$$
0.581738 0.813376i $$-0.302373\pi$$
$$878$$ − 14.6972i − 0.496007i
$$879$$ 7.02776 0.237040
$$880$$ 0 0
$$881$$ 55.2666 1.86198 0.930990 0.365045i $$-0.118946\pi$$
0.930990 + 0.365045i $$0.118946\pi$$
$$882$$ 11.3667i 0.382736i
$$883$$ − 8.27502i − 0.278477i −0.990259 0.139238i $$-0.955535\pi$$
0.990259 0.139238i $$-0.0444654\pi$$
$$884$$ −22.8167 −0.767407
$$885$$ 0 0
$$886$$ −17.4861 −0.587458
$$887$$ 27.6333i 0.927836i 0.885878 + 0.463918i $$0.153557\pi$$
−0.885878 + 0.463918i $$0.846443\pi$$
$$888$$ − 2.42221i − 0.0812839i
$$889$$ −32.4222 −1.08741
$$890$$ 0 0
$$891$$ −13.8890 −0.465298
$$892$$ − 4.00000i − 0.133930i
$$893$$ 15.3944i 0.515156i
$$894$$ −4.02776 −0.134708
$$895$$ 0 0
$$896$$ −3.30278 −0.110338
$$897$$ 1.00000i 0.0333890i
$$898$$ − 2.09167i − 0.0698000i
$$899$$ −20.6056 −0.687234
$$900$$ 0 0
$$901$$ −77.4500 −2.58023
$$902$$ 1.54163i 0.0513308i
$$903$$ − 9.21110i − 0.306526i
$$904$$ −16.4222 −0.546194
$$905$$ 0 0
$$906$$ 2.69722 0.0896093
$$907$$ 48.6611i 1.61576i 0.589344 + 0.807882i $$0.299386\pi$$
−0.589344 + 0.807882i $$0.700614\pi$$
$$908$$ 14.6056i 0.484702i
$$909$$ 7.57779 0.251340
$$910$$ 0 0
$$911$$ −4.18335 −0.138600 −0.0693002 0.997596i $$-0.522077\pi$$
−0.0693002 + 0.997596i $$0.522077\pi$$
$$912$$ − 1.78890i − 0.0592363i
$$913$$ 19.0278i 0.629727i
$$914$$ −32.4222 −1.07243
$$915$$ 0 0
$$916$$ 2.00000 0.0660819
$$917$$ − 37.0278i − 1.22276i
$$918$$ 12.3583i 0.407884i
$$919$$ −44.0000 −1.45143 −0.725713 0.687998i $$-0.758490\pi$$
−0.725713 + 0.687998i $$0.758490\pi$$
$$920$$ 0 0
$$921$$ 3.54163 0.116701
$$922$$ − 10.1833i − 0.335371i
$$923$$ 53.8444i 1.77231i
$$924$$ −1.69722 −0.0558346
$$925$$ 0 0
$$926$$ −17.6333 −0.579466
$$927$$ − 23.6140i − 0.775584i
$$928$$ − 2.60555i − 0.0855314i
$$929$$ 14.3667 0.471356 0.235678 0.971831i $$-0.424269\pi$$
0.235678 + 0.971831i $$0.424269\pi$$
$$930$$ 0 0
$$931$$ 23.0917 0.756799
$$932$$ 25.8167i 0.845653i
$$933$$ 6.78890i 0.222259i
$$934$$ −1.81665 −0.0594427
$$935$$ 0 0
$$936$$ 9.60555 0.313967
$$937$$ 37.9638i 1.24022i 0.784513 + 0.620112i $$0.212913\pi$$
−0.784513 + 0.620112i $$0.787087\pi$$
$$938$$ − 13.2111i − 0.431358i
$$939$$ 5.97224 0.194897
$$940$$ 0 0
$$941$$ 25.9361 0.845492 0.422746 0.906248i $$-0.361066\pi$$
0.422746 + 0.906248i $$0.361066\pi$$
$$942$$ − 5.63331i − 0.183543i
$$943$$ 0.908327i 0.0295792i
$$944$$ 3.39445 0.110480
$$945$$ 0 0
$$946$$ −15.6333 −0.508283
$$947$$ − 4.93608i − 0.160401i −0.996779 0.0802006i $$-0.974444\pi$$
0.996779 0.0802006i $$-0.0255561\pi$$
$$948$$ − 4.36669i − 0.141824i
$$949$$ 19.2111 0.623619
$$950$$ 0 0
$$951$$ −5.36669 −0.174027
$$952$$ − 22.8167i − 0.739492i
$$953$$ 41.3305i 1.33883i 0.742890 + 0.669414i $$0.233455\pi$$
−0.742890 + 0.669414i $$0.766545\pi$$
$$954$$ 32.6056 1.05564
$$955$$ 0 0
$$956$$ 5.21110 0.168539
$$957$$ − 1.33894i − 0.0432817i
$$958$$ − 30.0000i − 0.969256i
$$959$$ −12.9083 −0.416832
$$960$$ 0 0
$$961$$ 31.5416 1.01747
$$962$$ − 26.4222i − 0.851886i
$$963$$ − 7.57779i − 0.244191i
$$964$$ 14.4222 0.464508
$$965$$ 0 0
$$966$$ −1.00000 −0.0321745
$$967$$ − 12.6056i − 0.405367i −0.979244 0.202684i $$-0.935034\pi$$
0.979244 0.202684i $$-0.0649663\pi$$
$$968$$ − 8.11943i − 0.260968i
$$969$$ 12.3583 0.397005
$$970$$ 0 0
$$971$$ −17.0917 −0.548498 −0.274249 0.961659i $$-0.588429\pi$$
−0.274249 + 0.961659i $$0.588429\pi$$
$$972$$ − 7.84441i − 0.251610i
$$973$$ 41.6333i 1.33470i
$$974$$ 9.81665 0.314546
$$975$$ 0 0
$$976$$ 11.5139 0.368550
$$977$$ 6.51388i 0.208397i 0.994556 + 0.104199i $$0.0332278\pi$$
−0.994556 + 0.104199i $$0.966772\pi$$
$$978$$ − 2.81665i − 0.0900667i
$$979$$ 0 0
$$980$$ 0 0
$$981$$ −4.32213 −0.137995
$$982$$ 4.18335i 0.133496i
$$983$$ − 34.5416i − 1.10171i −0.834602 0.550854i $$-0.814302\pi$$
0.834602 0.550854i $$-0.185698\pi$$
$$984$$ −0.275019 −0.00876729
$$985$$ 0 0
$$986$$ 18.0000 0.573237
$$987$$ 2.60555i 0.0829356i
$$988$$ − 19.5139i − 0.620819i
$$989$$ −9.21110 −0.292896
$$990$$ 0 0
$$991$$ −15.3305 −0.486990 −0.243495 0.969902i $$-0.578294\pi$$
−0.243495 + 0.969902i $$0.578294\pi$$
$$992$$ 7.90833i 0.251090i
$$993$$ 5.02776i 0.159551i
$$994$$ −53.8444 −1.70784
$$995$$ 0 0
$$996$$ −3.39445 −0.107557
$$997$$ − 16.7889i − 0.531710i −0.964013 0.265855i $$-0.914346\pi$$
0.964013 0.265855i $$-0.0856542\pi$$
$$998$$ − 31.6333i − 1.00133i
$$999$$ −14.3112 −0.452786
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 1150.2.b.f.599.2 4
5.2 odd 4 1150.2.a.m.1.2 2
5.3 odd 4 230.2.a.b.1.1 2
5.4 even 2 inner 1150.2.b.f.599.3 4
15.8 even 4 2070.2.a.w.1.2 2
20.3 even 4 1840.2.a.j.1.2 2
20.7 even 4 9200.2.a.ca.1.1 2
40.3 even 4 7360.2.a.bu.1.1 2
40.13 odd 4 7360.2.a.bc.1.2 2
115.68 even 4 5290.2.a.j.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
230.2.a.b.1.1 2 5.3 odd 4
1150.2.a.m.1.2 2 5.2 odd 4
1150.2.b.f.599.2 4 1.1 even 1 trivial
1150.2.b.f.599.3 4 5.4 even 2 inner
1840.2.a.j.1.2 2 20.3 even 4
2070.2.a.w.1.2 2 15.8 even 4
5290.2.a.j.1.1 2 115.68 even 4
7360.2.a.bc.1.2 2 40.13 odd 4
7360.2.a.bu.1.1 2 40.3 even 4
9200.2.a.ca.1.1 2 20.7 even 4