# Properties

 Label 1840.2.a.k.1.2 Level $1840$ Weight $2$ Character 1840.1 Self dual yes Analytic conductor $14.692$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1840 = 2^{4} \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1840.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$14.6924739719$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{17})$$ Defining polynomial: $$x^{2} - x - 4$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 920) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$-1.56155$$ of defining polynomial Character $$\chi$$ $$=$$ 1840.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.56155 q^{3} +1.00000 q^{5} +3.12311 q^{7} -0.561553 q^{9} +O(q^{10})$$ $$q+1.56155 q^{3} +1.00000 q^{5} +3.12311 q^{7} -0.561553 q^{9} +4.00000 q^{11} +3.56155 q^{13} +1.56155 q^{15} +5.12311 q^{17} -4.00000 q^{19} +4.87689 q^{21} -1.00000 q^{23} +1.00000 q^{25} -5.56155 q^{27} -4.43845 q^{29} -5.56155 q^{31} +6.24621 q^{33} +3.12311 q^{35} +1.12311 q^{37} +5.56155 q^{39} -3.56155 q^{41} +0.876894 q^{43} -0.561553 q^{45} -8.68466 q^{47} +2.75379 q^{49} +8.00000 q^{51} +12.2462 q^{53} +4.00000 q^{55} -6.24621 q^{57} -10.2462 q^{59} +2.87689 q^{61} -1.75379 q^{63} +3.56155 q^{65} +10.2462 q^{67} -1.56155 q^{69} +8.68466 q^{71} +12.4384 q^{73} +1.56155 q^{75} +12.4924 q^{77} -6.24621 q^{79} -7.00000 q^{81} -12.0000 q^{83} +5.12311 q^{85} -6.93087 q^{87} +10.0000 q^{89} +11.1231 q^{91} -8.68466 q^{93} -4.00000 q^{95} +0.246211 q^{97} -2.24621 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{3} + 2 q^{5} - 2 q^{7} + 3 q^{9} + O(q^{10})$$ $$2 q - q^{3} + 2 q^{5} - 2 q^{7} + 3 q^{9} + 8 q^{11} + 3 q^{13} - q^{15} + 2 q^{17} - 8 q^{19} + 18 q^{21} - 2 q^{23} + 2 q^{25} - 7 q^{27} - 13 q^{29} - 7 q^{31} - 4 q^{33} - 2 q^{35} - 6 q^{37} + 7 q^{39} - 3 q^{41} + 10 q^{43} + 3 q^{45} - 5 q^{47} + 22 q^{49} + 16 q^{51} + 8 q^{53} + 8 q^{55} + 4 q^{57} - 4 q^{59} + 14 q^{61} - 20 q^{63} + 3 q^{65} + 4 q^{67} + q^{69} + 5 q^{71} + 29 q^{73} - q^{75} - 8 q^{77} + 4 q^{79} - 14 q^{81} - 24 q^{83} + 2 q^{85} + 15 q^{87} + 20 q^{89} + 14 q^{91} - 5 q^{93} - 8 q^{95} - 16 q^{97} + 12 q^{99} + O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 1.56155 0.901563 0.450781 0.892634i $$-0.351145\pi$$
0.450781 + 0.892634i $$0.351145\pi$$
$$4$$ 0 0
$$5$$ 1.00000 0.447214
$$6$$ 0 0
$$7$$ 3.12311 1.18042 0.590211 0.807249i $$-0.299044\pi$$
0.590211 + 0.807249i $$0.299044\pi$$
$$8$$ 0 0
$$9$$ −0.561553 −0.187184
$$10$$ 0 0
$$11$$ 4.00000 1.20605 0.603023 0.797724i $$-0.293963\pi$$
0.603023 + 0.797724i $$0.293963\pi$$
$$12$$ 0 0
$$13$$ 3.56155 0.987797 0.493899 0.869520i $$-0.335571\pi$$
0.493899 + 0.869520i $$0.335571\pi$$
$$14$$ 0 0
$$15$$ 1.56155 0.403191
$$16$$ 0 0
$$17$$ 5.12311 1.24254 0.621268 0.783598i $$-0.286618\pi$$
0.621268 + 0.783598i $$0.286618\pi$$
$$18$$ 0 0
$$19$$ −4.00000 −0.917663 −0.458831 0.888523i $$-0.651732\pi$$
−0.458831 + 0.888523i $$0.651732\pi$$
$$20$$ 0 0
$$21$$ 4.87689 1.06423
$$22$$ 0 0
$$23$$ −1.00000 −0.208514
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ −5.56155 −1.07032
$$28$$ 0 0
$$29$$ −4.43845 −0.824199 −0.412099 0.911139i $$-0.635204\pi$$
−0.412099 + 0.911139i $$0.635204\pi$$
$$30$$ 0 0
$$31$$ −5.56155 −0.998884 −0.499442 0.866347i $$-0.666462\pi$$
−0.499442 + 0.866347i $$0.666462\pi$$
$$32$$ 0 0
$$33$$ 6.24621 1.08733
$$34$$ 0 0
$$35$$ 3.12311 0.527901
$$36$$ 0 0
$$37$$ 1.12311 0.184637 0.0923187 0.995730i $$-0.470572\pi$$
0.0923187 + 0.995730i $$0.470572\pi$$
$$38$$ 0 0
$$39$$ 5.56155 0.890561
$$40$$ 0 0
$$41$$ −3.56155 −0.556221 −0.278111 0.960549i $$-0.589708\pi$$
−0.278111 + 0.960549i $$0.589708\pi$$
$$42$$ 0 0
$$43$$ 0.876894 0.133725 0.0668626 0.997762i $$-0.478701\pi$$
0.0668626 + 0.997762i $$0.478701\pi$$
$$44$$ 0 0
$$45$$ −0.561553 −0.0837114
$$46$$ 0 0
$$47$$ −8.68466 −1.26679 −0.633394 0.773830i $$-0.718339\pi$$
−0.633394 + 0.773830i $$0.718339\pi$$
$$48$$ 0 0
$$49$$ 2.75379 0.393398
$$50$$ 0 0
$$51$$ 8.00000 1.12022
$$52$$ 0 0
$$53$$ 12.2462 1.68215 0.841073 0.540921i $$-0.181924\pi$$
0.841073 + 0.540921i $$0.181924\pi$$
$$54$$ 0 0
$$55$$ 4.00000 0.539360
$$56$$ 0 0
$$57$$ −6.24621 −0.827331
$$58$$ 0 0
$$59$$ −10.2462 −1.33394 −0.666972 0.745083i $$-0.732410\pi$$
−0.666972 + 0.745083i $$0.732410\pi$$
$$60$$ 0 0
$$61$$ 2.87689 0.368349 0.184174 0.982894i $$-0.441039\pi$$
0.184174 + 0.982894i $$0.441039\pi$$
$$62$$ 0 0
$$63$$ −1.75379 −0.220957
$$64$$ 0 0
$$65$$ 3.56155 0.441756
$$66$$ 0 0
$$67$$ 10.2462 1.25177 0.625887 0.779914i $$-0.284737\pi$$
0.625887 + 0.779914i $$0.284737\pi$$
$$68$$ 0 0
$$69$$ −1.56155 −0.187989
$$70$$ 0 0
$$71$$ 8.68466 1.03068 0.515340 0.856986i $$-0.327666\pi$$
0.515340 + 0.856986i $$0.327666\pi$$
$$72$$ 0 0
$$73$$ 12.4384 1.45581 0.727905 0.685678i $$-0.240494\pi$$
0.727905 + 0.685678i $$0.240494\pi$$
$$74$$ 0 0
$$75$$ 1.56155 0.180313
$$76$$ 0 0
$$77$$ 12.4924 1.42364
$$78$$ 0 0
$$79$$ −6.24621 −0.702754 −0.351377 0.936234i $$-0.614286\pi$$
−0.351377 + 0.936234i $$0.614286\pi$$
$$80$$ 0 0
$$81$$ −7.00000 −0.777778
$$82$$ 0 0
$$83$$ −12.0000 −1.31717 −0.658586 0.752506i $$-0.728845\pi$$
−0.658586 + 0.752506i $$0.728845\pi$$
$$84$$ 0 0
$$85$$ 5.12311 0.555679
$$86$$ 0 0
$$87$$ −6.93087 −0.743067
$$88$$ 0 0
$$89$$ 10.0000 1.06000 0.529999 0.847998i $$-0.322192\pi$$
0.529999 + 0.847998i $$0.322192\pi$$
$$90$$ 0 0
$$91$$ 11.1231 1.16602
$$92$$ 0 0
$$93$$ −8.68466 −0.900557
$$94$$ 0 0
$$95$$ −4.00000 −0.410391
$$96$$ 0 0
$$97$$ 0.246211 0.0249990 0.0124995 0.999922i $$-0.496021\pi$$
0.0124995 + 0.999922i $$0.496021\pi$$
$$98$$ 0 0
$$99$$ −2.24621 −0.225753
$$100$$ 0 0
$$101$$ −16.2462 −1.61656 −0.808279 0.588799i $$-0.799601\pi$$
−0.808279 + 0.588799i $$0.799601\pi$$
$$102$$ 0 0
$$103$$ −14.2462 −1.40372 −0.701860 0.712314i $$-0.747647\pi$$
−0.701860 + 0.712314i $$0.747647\pi$$
$$104$$ 0 0
$$105$$ 4.87689 0.475936
$$106$$ 0 0
$$107$$ 12.0000 1.16008 0.580042 0.814587i $$-0.303036\pi$$
0.580042 + 0.814587i $$0.303036\pi$$
$$108$$ 0 0
$$109$$ −8.24621 −0.789844 −0.394922 0.918715i $$-0.629228\pi$$
−0.394922 + 0.918715i $$0.629228\pi$$
$$110$$ 0 0
$$111$$ 1.75379 0.166462
$$112$$ 0 0
$$113$$ 3.75379 0.353127 0.176563 0.984289i $$-0.443502\pi$$
0.176563 + 0.984289i $$0.443502\pi$$
$$114$$ 0 0
$$115$$ −1.00000 −0.0932505
$$116$$ 0 0
$$117$$ −2.00000 −0.184900
$$118$$ 0 0
$$119$$ 16.0000 1.46672
$$120$$ 0 0
$$121$$ 5.00000 0.454545
$$122$$ 0 0
$$123$$ −5.56155 −0.501468
$$124$$ 0 0
$$125$$ 1.00000 0.0894427
$$126$$ 0 0
$$127$$ −3.80776 −0.337884 −0.168942 0.985626i $$-0.554035\pi$$
−0.168942 + 0.985626i $$0.554035\pi$$
$$128$$ 0 0
$$129$$ 1.36932 0.120562
$$130$$ 0 0
$$131$$ 18.9309 1.65400 0.826999 0.562204i $$-0.190046\pi$$
0.826999 + 0.562204i $$0.190046\pi$$
$$132$$ 0 0
$$133$$ −12.4924 −1.08323
$$134$$ 0 0
$$135$$ −5.56155 −0.478662
$$136$$ 0 0
$$137$$ 6.87689 0.587533 0.293766 0.955877i $$-0.405091\pi$$
0.293766 + 0.955877i $$0.405091\pi$$
$$138$$ 0 0
$$139$$ −12.6847 −1.07590 −0.537949 0.842977i $$-0.680801\pi$$
−0.537949 + 0.842977i $$0.680801\pi$$
$$140$$ 0 0
$$141$$ −13.5616 −1.14209
$$142$$ 0 0
$$143$$ 14.2462 1.19133
$$144$$ 0 0
$$145$$ −4.43845 −0.368593
$$146$$ 0 0
$$147$$ 4.30019 0.354673
$$148$$ 0 0
$$149$$ 12.2462 1.00325 0.501624 0.865086i $$-0.332736\pi$$
0.501624 + 0.865086i $$0.332736\pi$$
$$150$$ 0 0
$$151$$ −3.80776 −0.309871 −0.154936 0.987925i $$-0.549517\pi$$
−0.154936 + 0.987925i $$0.549517\pi$$
$$152$$ 0 0
$$153$$ −2.87689 −0.232583
$$154$$ 0 0
$$155$$ −5.56155 −0.446715
$$156$$ 0 0
$$157$$ −2.00000 −0.159617 −0.0798087 0.996810i $$-0.525431\pi$$
−0.0798087 + 0.996810i $$0.525431\pi$$
$$158$$ 0 0
$$159$$ 19.1231 1.51656
$$160$$ 0 0
$$161$$ −3.12311 −0.246135
$$162$$ 0 0
$$163$$ 12.6847 0.993539 0.496770 0.867882i $$-0.334519\pi$$
0.496770 + 0.867882i $$0.334519\pi$$
$$164$$ 0 0
$$165$$ 6.24621 0.486267
$$166$$ 0 0
$$167$$ 8.00000 0.619059 0.309529 0.950890i $$-0.399829\pi$$
0.309529 + 0.950890i $$0.399829\pi$$
$$168$$ 0 0
$$169$$ −0.315342 −0.0242570
$$170$$ 0 0
$$171$$ 2.24621 0.171772
$$172$$ 0 0
$$173$$ −22.4924 −1.71007 −0.855034 0.518573i $$-0.826464\pi$$
−0.855034 + 0.518573i $$0.826464\pi$$
$$174$$ 0 0
$$175$$ 3.12311 0.236085
$$176$$ 0 0
$$177$$ −16.0000 −1.20263
$$178$$ 0 0
$$179$$ −4.68466 −0.350148 −0.175074 0.984555i $$-0.556016\pi$$
−0.175074 + 0.984555i $$0.556016\pi$$
$$180$$ 0 0
$$181$$ −5.12311 −0.380797 −0.190399 0.981707i $$-0.560978\pi$$
−0.190399 + 0.981707i $$0.560978\pi$$
$$182$$ 0 0
$$183$$ 4.49242 0.332089
$$184$$ 0 0
$$185$$ 1.12311 0.0825724
$$186$$ 0 0
$$187$$ 20.4924 1.49855
$$188$$ 0 0
$$189$$ −17.3693 −1.26343
$$190$$ 0 0
$$191$$ 11.1231 0.804840 0.402420 0.915455i $$-0.368169\pi$$
0.402420 + 0.915455i $$0.368169\pi$$
$$192$$ 0 0
$$193$$ 20.4384 1.47119 0.735596 0.677421i $$-0.236902\pi$$
0.735596 + 0.677421i $$0.236902\pi$$
$$194$$ 0 0
$$195$$ 5.56155 0.398271
$$196$$ 0 0
$$197$$ −15.5616 −1.10871 −0.554357 0.832279i $$-0.687036\pi$$
−0.554357 + 0.832279i $$0.687036\pi$$
$$198$$ 0 0
$$199$$ −24.0000 −1.70131 −0.850657 0.525720i $$-0.823796\pi$$
−0.850657 + 0.525720i $$0.823796\pi$$
$$200$$ 0 0
$$201$$ 16.0000 1.12855
$$202$$ 0 0
$$203$$ −13.8617 −0.972903
$$204$$ 0 0
$$205$$ −3.56155 −0.248750
$$206$$ 0 0
$$207$$ 0.561553 0.0390306
$$208$$ 0 0
$$209$$ −16.0000 −1.10674
$$210$$ 0 0
$$211$$ −2.24621 −0.154636 −0.0773178 0.997006i $$-0.524636\pi$$
−0.0773178 + 0.997006i $$0.524636\pi$$
$$212$$ 0 0
$$213$$ 13.5616 0.929222
$$214$$ 0 0
$$215$$ 0.876894 0.0598037
$$216$$ 0 0
$$217$$ −17.3693 −1.17911
$$218$$ 0 0
$$219$$ 19.4233 1.31250
$$220$$ 0 0
$$221$$ 18.2462 1.22737
$$222$$ 0 0
$$223$$ −12.4924 −0.836554 −0.418277 0.908319i $$-0.637366\pi$$
−0.418277 + 0.908319i $$0.637366\pi$$
$$224$$ 0 0
$$225$$ −0.561553 −0.0374369
$$226$$ 0 0
$$227$$ −0.876894 −0.0582015 −0.0291008 0.999576i $$-0.509264\pi$$
−0.0291008 + 0.999576i $$0.509264\pi$$
$$228$$ 0 0
$$229$$ −16.2462 −1.07358 −0.536790 0.843716i $$-0.680363\pi$$
−0.536790 + 0.843716i $$0.680363\pi$$
$$230$$ 0 0
$$231$$ 19.5076 1.28350
$$232$$ 0 0
$$233$$ −8.43845 −0.552821 −0.276411 0.961040i $$-0.589145\pi$$
−0.276411 + 0.961040i $$0.589145\pi$$
$$234$$ 0 0
$$235$$ −8.68466 −0.566525
$$236$$ 0 0
$$237$$ −9.75379 −0.633577
$$238$$ 0 0
$$239$$ −11.8078 −0.763781 −0.381890 0.924208i $$-0.624727\pi$$
−0.381890 + 0.924208i $$0.624727\pi$$
$$240$$ 0 0
$$241$$ 14.4924 0.933539 0.466769 0.884379i $$-0.345418\pi$$
0.466769 + 0.884379i $$0.345418\pi$$
$$242$$ 0 0
$$243$$ 5.75379 0.369106
$$244$$ 0 0
$$245$$ 2.75379 0.175933
$$246$$ 0 0
$$247$$ −14.2462 −0.906465
$$248$$ 0 0
$$249$$ −18.7386 −1.18751
$$250$$ 0 0
$$251$$ 10.2462 0.646735 0.323368 0.946273i $$-0.395185\pi$$
0.323368 + 0.946273i $$0.395185\pi$$
$$252$$ 0 0
$$253$$ −4.00000 −0.251478
$$254$$ 0 0
$$255$$ 8.00000 0.500979
$$256$$ 0 0
$$257$$ 28.0540 1.74996 0.874979 0.484160i $$-0.160875\pi$$
0.874979 + 0.484160i $$0.160875\pi$$
$$258$$ 0 0
$$259$$ 3.50758 0.217950
$$260$$ 0 0
$$261$$ 2.49242 0.154277
$$262$$ 0 0
$$263$$ 8.00000 0.493301 0.246651 0.969104i $$-0.420670\pi$$
0.246651 + 0.969104i $$0.420670\pi$$
$$264$$ 0 0
$$265$$ 12.2462 0.752279
$$266$$ 0 0
$$267$$ 15.6155 0.955655
$$268$$ 0 0
$$269$$ 30.3002 1.84743 0.923717 0.383074i $$-0.125135\pi$$
0.923717 + 0.383074i $$0.125135\pi$$
$$270$$ 0 0
$$271$$ −14.2462 −0.865396 −0.432698 0.901539i $$-0.642438\pi$$
−0.432698 + 0.901539i $$0.642438\pi$$
$$272$$ 0 0
$$273$$ 17.3693 1.05124
$$274$$ 0 0
$$275$$ 4.00000 0.241209
$$276$$ 0 0
$$277$$ −15.5616 −0.935003 −0.467502 0.883992i $$-0.654846\pi$$
−0.467502 + 0.883992i $$0.654846\pi$$
$$278$$ 0 0
$$279$$ 3.12311 0.186975
$$280$$ 0 0
$$281$$ −2.49242 −0.148685 −0.0743427 0.997233i $$-0.523686\pi$$
−0.0743427 + 0.997233i $$0.523686\pi$$
$$282$$ 0 0
$$283$$ −31.1231 −1.85008 −0.925038 0.379874i $$-0.875967\pi$$
−0.925038 + 0.379874i $$0.875967\pi$$
$$284$$ 0 0
$$285$$ −6.24621 −0.369994
$$286$$ 0 0
$$287$$ −11.1231 −0.656576
$$288$$ 0 0
$$289$$ 9.24621 0.543895
$$290$$ 0 0
$$291$$ 0.384472 0.0225381
$$292$$ 0 0
$$293$$ −21.1231 −1.23403 −0.617013 0.786953i $$-0.711657\pi$$
−0.617013 + 0.786953i $$0.711657\pi$$
$$294$$ 0 0
$$295$$ −10.2462 −0.596557
$$296$$ 0 0
$$297$$ −22.2462 −1.29086
$$298$$ 0 0
$$299$$ −3.56155 −0.205970
$$300$$ 0 0
$$301$$ 2.73863 0.157852
$$302$$ 0 0
$$303$$ −25.3693 −1.45743
$$304$$ 0 0
$$305$$ 2.87689 0.164730
$$306$$ 0 0
$$307$$ −12.0000 −0.684876 −0.342438 0.939540i $$-0.611253\pi$$
−0.342438 + 0.939540i $$0.611253\pi$$
$$308$$ 0 0
$$309$$ −22.2462 −1.26554
$$310$$ 0 0
$$311$$ 14.9309 0.846652 0.423326 0.905977i $$-0.360863\pi$$
0.423326 + 0.905977i $$0.360863\pi$$
$$312$$ 0 0
$$313$$ 6.87689 0.388705 0.194353 0.980932i $$-0.437739\pi$$
0.194353 + 0.980932i $$0.437739\pi$$
$$314$$ 0 0
$$315$$ −1.75379 −0.0988148
$$316$$ 0 0
$$317$$ −28.7386 −1.61412 −0.807061 0.590468i $$-0.798943\pi$$
−0.807061 + 0.590468i $$0.798943\pi$$
$$318$$ 0 0
$$319$$ −17.7538 −0.994021
$$320$$ 0 0
$$321$$ 18.7386 1.04589
$$322$$ 0 0
$$323$$ −20.4924 −1.14023
$$324$$ 0 0
$$325$$ 3.56155 0.197559
$$326$$ 0 0
$$327$$ −12.8769 −0.712094
$$328$$ 0 0
$$329$$ −27.1231 −1.49535
$$330$$ 0 0
$$331$$ −6.43845 −0.353889 −0.176945 0.984221i $$-0.556621\pi$$
−0.176945 + 0.984221i $$0.556621\pi$$
$$332$$ 0 0
$$333$$ −0.630683 −0.0345612
$$334$$ 0 0
$$335$$ 10.2462 0.559810
$$336$$ 0 0
$$337$$ 32.2462 1.75656 0.878282 0.478144i $$-0.158690\pi$$
0.878282 + 0.478144i $$0.158690\pi$$
$$338$$ 0 0
$$339$$ 5.86174 0.318366
$$340$$ 0 0
$$341$$ −22.2462 −1.20470
$$342$$ 0 0
$$343$$ −13.2614 −0.716046
$$344$$ 0 0
$$345$$ −1.56155 −0.0840712
$$346$$ 0 0
$$347$$ −16.4924 −0.885360 −0.442680 0.896680i $$-0.645972\pi$$
−0.442680 + 0.896680i $$0.645972\pi$$
$$348$$ 0 0
$$349$$ 17.8078 0.953228 0.476614 0.879113i $$-0.341864\pi$$
0.476614 + 0.879113i $$0.341864\pi$$
$$350$$ 0 0
$$351$$ −19.8078 −1.05726
$$352$$ 0 0
$$353$$ 14.1922 0.755377 0.377688 0.925933i $$-0.376719\pi$$
0.377688 + 0.925933i $$0.376719\pi$$
$$354$$ 0 0
$$355$$ 8.68466 0.460934
$$356$$ 0 0
$$357$$ 24.9848 1.32234
$$358$$ 0 0
$$359$$ 30.2462 1.59633 0.798167 0.602436i $$-0.205803\pi$$
0.798167 + 0.602436i $$0.205803\pi$$
$$360$$ 0 0
$$361$$ −3.00000 −0.157895
$$362$$ 0 0
$$363$$ 7.80776 0.409801
$$364$$ 0 0
$$365$$ 12.4384 0.651058
$$366$$ 0 0
$$367$$ 12.4924 0.652099 0.326050 0.945353i $$-0.394282\pi$$
0.326050 + 0.945353i $$0.394282\pi$$
$$368$$ 0 0
$$369$$ 2.00000 0.104116
$$370$$ 0 0
$$371$$ 38.2462 1.98564
$$372$$ 0 0
$$373$$ 24.7386 1.28092 0.640459 0.767992i $$-0.278744\pi$$
0.640459 + 0.767992i $$0.278744\pi$$
$$374$$ 0 0
$$375$$ 1.56155 0.0806382
$$376$$ 0 0
$$377$$ −15.8078 −0.814141
$$378$$ 0 0
$$379$$ −2.24621 −0.115380 −0.0576901 0.998335i $$-0.518374\pi$$
−0.0576901 + 0.998335i $$0.518374\pi$$
$$380$$ 0 0
$$381$$ −5.94602 −0.304624
$$382$$ 0 0
$$383$$ 7.61553 0.389135 0.194568 0.980889i $$-0.437670\pi$$
0.194568 + 0.980889i $$0.437670\pi$$
$$384$$ 0 0
$$385$$ 12.4924 0.636673
$$386$$ 0 0
$$387$$ −0.492423 −0.0250312
$$388$$ 0 0
$$389$$ −33.6155 −1.70437 −0.852187 0.523237i $$-0.824724\pi$$
−0.852187 + 0.523237i $$0.824724\pi$$
$$390$$ 0 0
$$391$$ −5.12311 −0.259087
$$392$$ 0 0
$$393$$ 29.5616 1.49118
$$394$$ 0 0
$$395$$ −6.24621 −0.314281
$$396$$ 0 0
$$397$$ −24.9309 −1.25124 −0.625622 0.780126i $$-0.715155\pi$$
−0.625622 + 0.780126i $$0.715155\pi$$
$$398$$ 0 0
$$399$$ −19.5076 −0.976600
$$400$$ 0 0
$$401$$ 20.7386 1.03564 0.517819 0.855490i $$-0.326744\pi$$
0.517819 + 0.855490i $$0.326744\pi$$
$$402$$ 0 0
$$403$$ −19.8078 −0.986695
$$404$$ 0 0
$$405$$ −7.00000 −0.347833
$$406$$ 0 0
$$407$$ 4.49242 0.222681
$$408$$ 0 0
$$409$$ −14.6847 −0.726110 −0.363055 0.931768i $$-0.618266\pi$$
−0.363055 + 0.931768i $$0.618266\pi$$
$$410$$ 0 0
$$411$$ 10.7386 0.529698
$$412$$ 0 0
$$413$$ −32.0000 −1.57462
$$414$$ 0 0
$$415$$ −12.0000 −0.589057
$$416$$ 0 0
$$417$$ −19.8078 −0.969990
$$418$$ 0 0
$$419$$ 0.876894 0.0428391 0.0214195 0.999771i $$-0.493181\pi$$
0.0214195 + 0.999771i $$0.493181\pi$$
$$420$$ 0 0
$$421$$ −6.49242 −0.316421 −0.158211 0.987405i $$-0.550573\pi$$
−0.158211 + 0.987405i $$0.550573\pi$$
$$422$$ 0 0
$$423$$ 4.87689 0.237123
$$424$$ 0 0
$$425$$ 5.12311 0.248507
$$426$$ 0 0
$$427$$ 8.98485 0.434807
$$428$$ 0 0
$$429$$ 22.2462 1.07406
$$430$$ 0 0
$$431$$ −7.61553 −0.366827 −0.183414 0.983036i $$-0.558715\pi$$
−0.183414 + 0.983036i $$0.558715\pi$$
$$432$$ 0 0
$$433$$ −24.7386 −1.18886 −0.594431 0.804146i $$-0.702623\pi$$
−0.594431 + 0.804146i $$0.702623\pi$$
$$434$$ 0 0
$$435$$ −6.93087 −0.332310
$$436$$ 0 0
$$437$$ 4.00000 0.191346
$$438$$ 0 0
$$439$$ 32.3002 1.54160 0.770802 0.637075i $$-0.219856\pi$$
0.770802 + 0.637075i $$0.219856\pi$$
$$440$$ 0 0
$$441$$ −1.54640 −0.0736380
$$442$$ 0 0
$$443$$ 3.31534 0.157517 0.0787583 0.996894i $$-0.474904\pi$$
0.0787583 + 0.996894i $$0.474904\pi$$
$$444$$ 0 0
$$445$$ 10.0000 0.474045
$$446$$ 0 0
$$447$$ 19.1231 0.904492
$$448$$ 0 0
$$449$$ −32.7386 −1.54503 −0.772516 0.634996i $$-0.781002\pi$$
−0.772516 + 0.634996i $$0.781002\pi$$
$$450$$ 0 0
$$451$$ −14.2462 −0.670828
$$452$$ 0 0
$$453$$ −5.94602 −0.279369
$$454$$ 0 0
$$455$$ 11.1231 0.521459
$$456$$ 0 0
$$457$$ −9.12311 −0.426761 −0.213380 0.976969i $$-0.568447\pi$$
−0.213380 + 0.976969i $$0.568447\pi$$
$$458$$ 0 0
$$459$$ −28.4924 −1.32991
$$460$$ 0 0
$$461$$ 27.5616 1.28367 0.641835 0.766843i $$-0.278174\pi$$
0.641835 + 0.766843i $$0.278174\pi$$
$$462$$ 0 0
$$463$$ 6.24621 0.290286 0.145143 0.989411i $$-0.453636\pi$$
0.145143 + 0.989411i $$0.453636\pi$$
$$464$$ 0 0
$$465$$ −8.68466 −0.402741
$$466$$ 0 0
$$467$$ −10.6307 −0.491929 −0.245965 0.969279i $$-0.579105\pi$$
−0.245965 + 0.969279i $$0.579105\pi$$
$$468$$ 0 0
$$469$$ 32.0000 1.47762
$$470$$ 0 0
$$471$$ −3.12311 −0.143905
$$472$$ 0 0
$$473$$ 3.50758 0.161279
$$474$$ 0 0
$$475$$ −4.00000 −0.183533
$$476$$ 0 0
$$477$$ −6.87689 −0.314871
$$478$$ 0 0
$$479$$ −16.0000 −0.731059 −0.365529 0.930800i $$-0.619112\pi$$
−0.365529 + 0.930800i $$0.619112\pi$$
$$480$$ 0 0
$$481$$ 4.00000 0.182384
$$482$$ 0 0
$$483$$ −4.87689 −0.221906
$$484$$ 0 0
$$485$$ 0.246211 0.0111799
$$486$$ 0 0
$$487$$ −34.0540 −1.54313 −0.771566 0.636149i $$-0.780526\pi$$
−0.771566 + 0.636149i $$0.780526\pi$$
$$488$$ 0 0
$$489$$ 19.8078 0.895738
$$490$$ 0 0
$$491$$ −6.43845 −0.290563 −0.145282 0.989390i $$-0.546409\pi$$
−0.145282 + 0.989390i $$0.546409\pi$$
$$492$$ 0 0
$$493$$ −22.7386 −1.02410
$$494$$ 0 0
$$495$$ −2.24621 −0.100960
$$496$$ 0 0
$$497$$ 27.1231 1.21664
$$498$$ 0 0
$$499$$ −38.0540 −1.70353 −0.851765 0.523924i $$-0.824468\pi$$
−0.851765 + 0.523924i $$0.824468\pi$$
$$500$$ 0 0
$$501$$ 12.4924 0.558120
$$502$$ 0 0
$$503$$ 15.6155 0.696262 0.348131 0.937446i $$-0.386816\pi$$
0.348131 + 0.937446i $$0.386816\pi$$
$$504$$ 0 0
$$505$$ −16.2462 −0.722947
$$506$$ 0 0
$$507$$ −0.492423 −0.0218693
$$508$$ 0 0
$$509$$ −4.43845 −0.196731 −0.0983654 0.995150i $$-0.531361\pi$$
−0.0983654 + 0.995150i $$0.531361\pi$$
$$510$$ 0 0
$$511$$ 38.8466 1.71847
$$512$$ 0 0
$$513$$ 22.2462 0.982194
$$514$$ 0 0
$$515$$ −14.2462 −0.627763
$$516$$ 0 0
$$517$$ −34.7386 −1.52780
$$518$$ 0 0
$$519$$ −35.1231 −1.54173
$$520$$ 0 0
$$521$$ 39.8617 1.74637 0.873187 0.487385i $$-0.162049\pi$$
0.873187 + 0.487385i $$0.162049\pi$$
$$522$$ 0 0
$$523$$ −33.8617 −1.48067 −0.740335 0.672238i $$-0.765333\pi$$
−0.740335 + 0.672238i $$0.765333\pi$$
$$524$$ 0 0
$$525$$ 4.87689 0.212845
$$526$$ 0 0
$$527$$ −28.4924 −1.24115
$$528$$ 0 0
$$529$$ 1.00000 0.0434783
$$530$$ 0 0
$$531$$ 5.75379 0.249693
$$532$$ 0 0
$$533$$ −12.6847 −0.549434
$$534$$ 0 0
$$535$$ 12.0000 0.518805
$$536$$ 0 0
$$537$$ −7.31534 −0.315680
$$538$$ 0 0
$$539$$ 11.0152 0.474456
$$540$$ 0 0
$$541$$ 21.3153 0.916418 0.458209 0.888844i $$-0.348491\pi$$
0.458209 + 0.888844i $$0.348491\pi$$
$$542$$ 0 0
$$543$$ −8.00000 −0.343313
$$544$$ 0 0
$$545$$ −8.24621 −0.353229
$$546$$ 0 0
$$547$$ −38.0540 −1.62707 −0.813535 0.581516i $$-0.802460\pi$$
−0.813535 + 0.581516i $$0.802460\pi$$
$$548$$ 0 0
$$549$$ −1.61553 −0.0689491
$$550$$ 0 0
$$551$$ 17.7538 0.756337
$$552$$ 0 0
$$553$$ −19.5076 −0.829547
$$554$$ 0 0
$$555$$ 1.75379 0.0744442
$$556$$ 0 0
$$557$$ −24.2462 −1.02734 −0.513672 0.857986i $$-0.671715\pi$$
−0.513672 + 0.857986i $$0.671715\pi$$
$$558$$ 0 0
$$559$$ 3.12311 0.132093
$$560$$ 0 0
$$561$$ 32.0000 1.35104
$$562$$ 0 0
$$563$$ −43.2311 −1.82197 −0.910986 0.412438i $$-0.864678\pi$$
−0.910986 + 0.412438i $$0.864678\pi$$
$$564$$ 0 0
$$565$$ 3.75379 0.157923
$$566$$ 0 0
$$567$$ −21.8617 −0.918107
$$568$$ 0 0
$$569$$ 28.7386 1.20479 0.602393 0.798200i $$-0.294214\pi$$
0.602393 + 0.798200i $$0.294214\pi$$
$$570$$ 0 0
$$571$$ −12.0000 −0.502184 −0.251092 0.967963i $$-0.580790\pi$$
−0.251092 + 0.967963i $$0.580790\pi$$
$$572$$ 0 0
$$573$$ 17.3693 0.725614
$$574$$ 0 0
$$575$$ −1.00000 −0.0417029
$$576$$ 0 0
$$577$$ 29.4233 1.22491 0.612454 0.790506i $$-0.290183\pi$$
0.612454 + 0.790506i $$0.290183\pi$$
$$578$$ 0 0
$$579$$ 31.9157 1.32637
$$580$$ 0 0
$$581$$ −37.4773 −1.55482
$$582$$ 0 0
$$583$$ 48.9848 2.02874
$$584$$ 0 0
$$585$$ −2.00000 −0.0826898
$$586$$ 0 0
$$587$$ −9.17708 −0.378779 −0.189389 0.981902i $$-0.560651\pi$$
−0.189389 + 0.981902i $$0.560651\pi$$
$$588$$ 0 0
$$589$$ 22.2462 0.916639
$$590$$ 0 0
$$591$$ −24.3002 −0.999576
$$592$$ 0 0
$$593$$ −38.9848 −1.60092 −0.800458 0.599389i $$-0.795410\pi$$
−0.800458 + 0.599389i $$0.795410\pi$$
$$594$$ 0 0
$$595$$ 16.0000 0.655936
$$596$$ 0 0
$$597$$ −37.4773 −1.53384
$$598$$ 0 0
$$599$$ 32.0000 1.30748 0.653742 0.756717i $$-0.273198\pi$$
0.653742 + 0.756717i $$0.273198\pi$$
$$600$$ 0 0
$$601$$ −27.1771 −1.10858 −0.554288 0.832325i $$-0.687009\pi$$
−0.554288 + 0.832325i $$0.687009\pi$$
$$602$$ 0 0
$$603$$ −5.75379 −0.234312
$$604$$ 0 0
$$605$$ 5.00000 0.203279
$$606$$ 0 0
$$607$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$608$$ 0 0
$$609$$ −21.6458 −0.877134
$$610$$ 0 0
$$611$$ −30.9309 −1.25133
$$612$$ 0 0
$$613$$ −5.12311 −0.206920 −0.103460 0.994634i $$-0.532991\pi$$
−0.103460 + 0.994634i $$0.532991\pi$$
$$614$$ 0 0
$$615$$ −5.56155 −0.224263
$$616$$ 0 0
$$617$$ −5.61553 −0.226073 −0.113036 0.993591i $$-0.536058\pi$$
−0.113036 + 0.993591i $$0.536058\pi$$
$$618$$ 0 0
$$619$$ −36.9848 −1.48655 −0.743273 0.668988i $$-0.766728\pi$$
−0.743273 + 0.668988i $$0.766728\pi$$
$$620$$ 0 0
$$621$$ 5.56155 0.223177
$$622$$ 0 0
$$623$$ 31.2311 1.25125
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ −24.9848 −0.997799
$$628$$ 0 0
$$629$$ 5.75379 0.229419
$$630$$ 0 0
$$631$$ 6.63068 0.263963 0.131982 0.991252i $$-0.457866\pi$$
0.131982 + 0.991252i $$0.457866\pi$$
$$632$$ 0 0
$$633$$ −3.50758 −0.139414
$$634$$ 0 0
$$635$$ −3.80776 −0.151107
$$636$$ 0 0
$$637$$ 9.80776 0.388598
$$638$$ 0 0
$$639$$ −4.87689 −0.192927
$$640$$ 0 0
$$641$$ 40.2462 1.58963 0.794815 0.606852i $$-0.207568\pi$$
0.794815 + 0.606852i $$0.207568\pi$$
$$642$$ 0 0
$$643$$ 46.3542 1.82803 0.914015 0.405681i $$-0.132966\pi$$
0.914015 + 0.405681i $$0.132966\pi$$
$$644$$ 0 0
$$645$$ 1.36932 0.0539168
$$646$$ 0 0
$$647$$ 5.56155 0.218647 0.109324 0.994006i $$-0.465132\pi$$
0.109324 + 0.994006i $$0.465132\pi$$
$$648$$ 0 0
$$649$$ −40.9848 −1.60880
$$650$$ 0 0
$$651$$ −27.1231 −1.06304
$$652$$ 0 0
$$653$$ 50.7926 1.98767 0.993834 0.110876i $$-0.0353658\pi$$
0.993834 + 0.110876i $$0.0353658\pi$$
$$654$$ 0 0
$$655$$ 18.9309 0.739690
$$656$$ 0 0
$$657$$ −6.98485 −0.272505
$$658$$ 0 0
$$659$$ 2.24621 0.0875000 0.0437500 0.999043i $$-0.486070\pi$$
0.0437500 + 0.999043i $$0.486070\pi$$
$$660$$ 0 0
$$661$$ −22.4924 −0.874854 −0.437427 0.899254i $$-0.644110\pi$$
−0.437427 + 0.899254i $$0.644110\pi$$
$$662$$ 0 0
$$663$$ 28.4924 1.10655
$$664$$ 0 0
$$665$$ −12.4924 −0.484435
$$666$$ 0 0
$$667$$ 4.43845 0.171857
$$668$$ 0 0
$$669$$ −19.5076 −0.754207
$$670$$ 0 0
$$671$$ 11.5076 0.444245
$$672$$ 0 0
$$673$$ −0.438447 −0.0169009 −0.00845045 0.999964i $$-0.502690\pi$$
−0.00845045 + 0.999964i $$0.502690\pi$$
$$674$$ 0 0
$$675$$ −5.56155 −0.214064
$$676$$ 0 0
$$677$$ −28.7386 −1.10452 −0.552258 0.833673i $$-0.686234\pi$$
−0.552258 + 0.833673i $$0.686234\pi$$
$$678$$ 0 0
$$679$$ 0.768944 0.0295094
$$680$$ 0 0
$$681$$ −1.36932 −0.0524723
$$682$$ 0 0
$$683$$ 14.4384 0.552472 0.276236 0.961090i $$-0.410913\pi$$
0.276236 + 0.961090i $$0.410913\pi$$
$$684$$ 0 0
$$685$$ 6.87689 0.262753
$$686$$ 0 0
$$687$$ −25.3693 −0.967900
$$688$$ 0 0
$$689$$ 43.6155 1.66162
$$690$$ 0 0
$$691$$ 10.2462 0.389784 0.194892 0.980825i $$-0.437564\pi$$
0.194892 + 0.980825i $$0.437564\pi$$
$$692$$ 0 0
$$693$$ −7.01515 −0.266484
$$694$$ 0 0
$$695$$ −12.6847 −0.481157
$$696$$ 0 0
$$697$$ −18.2462 −0.691125
$$698$$ 0 0
$$699$$ −13.1771 −0.498403
$$700$$ 0 0
$$701$$ −26.9848 −1.01920 −0.509602 0.860410i $$-0.670207\pi$$
−0.509602 + 0.860410i $$0.670207\pi$$
$$702$$ 0 0
$$703$$ −4.49242 −0.169435
$$704$$ 0 0
$$705$$ −13.5616 −0.510758
$$706$$ 0 0
$$707$$ −50.7386 −1.90822
$$708$$ 0 0
$$709$$ 8.73863 0.328186 0.164093 0.986445i $$-0.447530\pi$$
0.164093 + 0.986445i $$0.447530\pi$$
$$710$$ 0 0
$$711$$ 3.50758 0.131544
$$712$$ 0 0
$$713$$ 5.56155 0.208282
$$714$$ 0 0
$$715$$ 14.2462 0.532778
$$716$$ 0 0
$$717$$ −18.4384 −0.688596
$$718$$ 0 0
$$719$$ 10.7386 0.400483 0.200242 0.979747i $$-0.435827\pi$$
0.200242 + 0.979747i $$0.435827\pi$$
$$720$$ 0 0
$$721$$ −44.4924 −1.65698
$$722$$ 0 0
$$723$$ 22.6307 0.841644
$$724$$ 0 0
$$725$$ −4.43845 −0.164840
$$726$$ 0 0
$$727$$ −28.8769 −1.07098 −0.535492 0.844540i $$-0.679874\pi$$
−0.535492 + 0.844540i $$0.679874\pi$$
$$728$$ 0 0
$$729$$ 29.9848 1.11055
$$730$$ 0 0
$$731$$ 4.49242 0.166158
$$732$$ 0 0
$$733$$ 2.87689 0.106261 0.0531303 0.998588i $$-0.483080\pi$$
0.0531303 + 0.998588i $$0.483080\pi$$
$$734$$ 0 0
$$735$$ 4.30019 0.158615
$$736$$ 0 0
$$737$$ 40.9848 1.50970
$$738$$ 0 0
$$739$$ 42.5464 1.56509 0.782547 0.622591i $$-0.213920\pi$$
0.782547 + 0.622591i $$0.213920\pi$$
$$740$$ 0 0
$$741$$ −22.2462 −0.817235
$$742$$ 0 0
$$743$$ 30.2462 1.10963 0.554813 0.831975i $$-0.312790\pi$$
0.554813 + 0.831975i $$0.312790\pi$$
$$744$$ 0 0
$$745$$ 12.2462 0.448666
$$746$$ 0 0
$$747$$ 6.73863 0.246554
$$748$$ 0 0
$$749$$ 37.4773 1.36939
$$750$$ 0 0
$$751$$ −8.38447 −0.305954 −0.152977 0.988230i $$-0.548886\pi$$
−0.152977 + 0.988230i $$0.548886\pi$$
$$752$$ 0 0
$$753$$ 16.0000 0.583072
$$754$$ 0 0
$$755$$ −3.80776 −0.138579
$$756$$ 0 0
$$757$$ 33.1231 1.20388 0.601940 0.798541i $$-0.294395\pi$$
0.601940 + 0.798541i $$0.294395\pi$$
$$758$$ 0 0
$$759$$ −6.24621 −0.226723
$$760$$ 0 0
$$761$$ 42.3002 1.53338 0.766690 0.642017i $$-0.221902\pi$$
0.766690 + 0.642017i $$0.221902\pi$$
$$762$$ 0 0
$$763$$ −25.7538 −0.932350
$$764$$ 0 0
$$765$$ −2.87689 −0.104014
$$766$$ 0 0
$$767$$ −36.4924 −1.31767
$$768$$ 0 0
$$769$$ 5.50758 0.198608 0.0993042 0.995057i $$-0.468338\pi$$
0.0993042 + 0.995057i $$0.468338\pi$$
$$770$$ 0 0
$$771$$ 43.8078 1.57770
$$772$$ 0 0
$$773$$ 12.2462 0.440466 0.220233 0.975447i $$-0.429318\pi$$
0.220233 + 0.975447i $$0.429318\pi$$
$$774$$ 0 0
$$775$$ −5.56155 −0.199777
$$776$$ 0 0
$$777$$ 5.47727 0.196496
$$778$$ 0 0
$$779$$ 14.2462 0.510423
$$780$$ 0 0
$$781$$ 34.7386 1.24305
$$782$$ 0 0
$$783$$ 24.6847 0.882158
$$784$$ 0 0
$$785$$ −2.00000 −0.0713831
$$786$$ 0 0
$$787$$ 29.7538 1.06061 0.530304 0.847808i $$-0.322078\pi$$
0.530304 + 0.847808i $$0.322078\pi$$
$$788$$ 0 0
$$789$$ 12.4924 0.444742
$$790$$ 0 0
$$791$$ 11.7235 0.416839
$$792$$ 0 0
$$793$$ 10.2462 0.363854
$$794$$ 0 0
$$795$$ 19.1231 0.678227
$$796$$ 0 0
$$797$$ 18.8769 0.668654 0.334327 0.942457i $$-0.391491\pi$$
0.334327 + 0.942457i $$0.391491\pi$$
$$798$$ 0 0
$$799$$ −44.4924 −1.57403
$$800$$ 0 0
$$801$$ −5.61553 −0.198415
$$802$$ 0 0
$$803$$ 49.7538 1.75577
$$804$$ 0 0
$$805$$ −3.12311 −0.110075
$$806$$ 0 0
$$807$$ 47.3153 1.66558
$$808$$ 0 0
$$809$$ 22.4924 0.790791 0.395396 0.918511i $$-0.370607\pi$$
0.395396 + 0.918511i $$0.370607\pi$$
$$810$$ 0 0
$$811$$ 20.6847 0.726337 0.363168 0.931724i $$-0.381695\pi$$
0.363168 + 0.931724i $$0.381695\pi$$
$$812$$ 0 0
$$813$$ −22.2462 −0.780209
$$814$$ 0 0
$$815$$ 12.6847 0.444324
$$816$$ 0 0
$$817$$ −3.50758 −0.122715
$$818$$ 0 0
$$819$$ −6.24621 −0.218260
$$820$$ 0 0
$$821$$ −48.2462 −1.68380 −0.841902 0.539630i $$-0.818564\pi$$
−0.841902 + 0.539630i $$0.818564\pi$$
$$822$$ 0 0
$$823$$ 21.5616 0.751588 0.375794 0.926703i $$-0.377370\pi$$
0.375794 + 0.926703i $$0.377370\pi$$
$$824$$ 0 0
$$825$$ 6.24621 0.217465
$$826$$ 0 0
$$827$$ 12.0000 0.417281 0.208640 0.977992i $$-0.433096\pi$$
0.208640 + 0.977992i $$0.433096\pi$$
$$828$$ 0 0
$$829$$ 17.5076 0.608063 0.304032 0.952662i $$-0.401667\pi$$
0.304032 + 0.952662i $$0.401667\pi$$
$$830$$ 0 0
$$831$$ −24.3002 −0.842964
$$832$$ 0 0
$$833$$ 14.1080 0.488812
$$834$$ 0 0
$$835$$ 8.00000 0.276851
$$836$$ 0 0
$$837$$ 30.9309 1.06913
$$838$$ 0 0
$$839$$ 31.6155 1.09149 0.545745 0.837952i $$-0.316247\pi$$
0.545745 + 0.837952i $$0.316247\pi$$
$$840$$ 0 0
$$841$$ −9.30019 −0.320696
$$842$$ 0 0
$$843$$ −3.89205 −0.134049
$$844$$ 0 0
$$845$$ −0.315342 −0.0108481
$$846$$ 0 0
$$847$$ 15.6155 0.536556
$$848$$ 0 0
$$849$$ −48.6004 −1.66796
$$850$$ 0 0
$$851$$ −1.12311 −0.0384996
$$852$$ 0 0
$$853$$ −42.9848 −1.47177 −0.735887 0.677105i $$-0.763234\pi$$
−0.735887 + 0.677105i $$0.763234\pi$$
$$854$$ 0 0
$$855$$ 2.24621 0.0768188
$$856$$ 0 0
$$857$$ −11.1771 −0.381802 −0.190901 0.981609i $$-0.561141\pi$$
−0.190901 + 0.981609i $$0.561141\pi$$
$$858$$ 0 0
$$859$$ 23.4233 0.799192 0.399596 0.916691i $$-0.369150\pi$$
0.399596 + 0.916691i $$0.369150\pi$$
$$860$$ 0 0
$$861$$ −17.3693 −0.591945
$$862$$ 0 0
$$863$$ 43.4233 1.47815 0.739073 0.673625i $$-0.235264\pi$$
0.739073 + 0.673625i $$0.235264\pi$$
$$864$$ 0 0
$$865$$ −22.4924 −0.764765
$$866$$ 0 0
$$867$$ 14.4384 0.490355
$$868$$ 0 0
$$869$$ −24.9848 −0.847553
$$870$$ 0 0
$$871$$ 36.4924 1.23650
$$872$$ 0 0
$$873$$ −0.138261 −0.00467941
$$874$$ 0 0
$$875$$ 3.12311 0.105580
$$876$$ 0 0
$$877$$ 8.73863 0.295083 0.147541 0.989056i $$-0.452864\pi$$
0.147541 + 0.989056i $$0.452864\pi$$
$$878$$ 0 0
$$879$$ −32.9848 −1.11255
$$880$$ 0 0
$$881$$ −50.1080 −1.68818 −0.844090 0.536202i $$-0.819859\pi$$
−0.844090 + 0.536202i $$0.819859\pi$$
$$882$$ 0 0
$$883$$ 7.50758 0.252650 0.126325 0.991989i $$-0.459682\pi$$
0.126325 + 0.991989i $$0.459682\pi$$
$$884$$ 0 0
$$885$$ −16.0000 −0.537834
$$886$$ 0 0
$$887$$ 37.5616 1.26119 0.630597 0.776111i $$-0.282810\pi$$
0.630597 + 0.776111i $$0.282810\pi$$
$$888$$ 0 0
$$889$$ −11.8920 −0.398847
$$890$$ 0 0
$$891$$ −28.0000 −0.938035
$$892$$ 0 0
$$893$$ 34.7386 1.16248
$$894$$ 0 0
$$895$$ −4.68466 −0.156591
$$896$$ 0 0
$$897$$ −5.56155 −0.185695
$$898$$ 0 0
$$899$$ 24.6847 0.823279
$$900$$ 0 0
$$901$$ 62.7386 2.09013
$$902$$ 0 0
$$903$$ 4.27652 0.142314
$$904$$ 0 0
$$905$$ −5.12311 −0.170298
$$906$$ 0 0
$$907$$ −20.0000 −0.664089 −0.332045 0.943264i $$-0.607738\pi$$
−0.332045 + 0.943264i $$0.607738\pi$$
$$908$$ 0 0
$$909$$ 9.12311 0.302594
$$910$$ 0 0
$$911$$ −12.4924 −0.413892 −0.206946 0.978352i $$-0.566353\pi$$
−0.206946 + 0.978352i $$0.566353\pi$$
$$912$$ 0 0
$$913$$ −48.0000 −1.58857
$$914$$ 0 0
$$915$$ 4.49242 0.148515
$$916$$ 0 0
$$917$$ 59.1231 1.95242
$$918$$ 0 0
$$919$$ −21.8617 −0.721152 −0.360576 0.932730i $$-0.617420\pi$$
−0.360576 + 0.932730i $$0.617420\pi$$
$$920$$ 0 0
$$921$$ −18.7386 −0.617459
$$922$$ 0 0
$$923$$ 30.9309 1.01810
$$924$$ 0 0
$$925$$ 1.12311 0.0369275
$$926$$ 0 0
$$927$$ 8.00000 0.262754
$$928$$ 0 0
$$929$$ −24.0540 −0.789185 −0.394593 0.918856i $$-0.629114\pi$$
−0.394593 + 0.918856i $$0.629114\pi$$
$$930$$ 0 0
$$931$$ −11.0152 −0.361007
$$932$$ 0 0
$$933$$ 23.3153 0.763310
$$934$$ 0 0
$$935$$ 20.4924 0.670174
$$936$$ 0 0
$$937$$ 16.6307 0.543301 0.271650 0.962396i $$-0.412431\pi$$
0.271650 + 0.962396i $$0.412431\pi$$
$$938$$ 0 0
$$939$$ 10.7386 0.350442
$$940$$ 0 0
$$941$$ 46.6004 1.51913 0.759564 0.650432i $$-0.225412\pi$$
0.759564 + 0.650432i $$0.225412\pi$$
$$942$$ 0 0
$$943$$ 3.56155 0.115980
$$944$$ 0 0
$$945$$ −17.3693 −0.565024
$$946$$ 0 0
$$947$$ 55.8078 1.81351 0.906754 0.421659i $$-0.138552\pi$$
0.906754 + 0.421659i $$0.138552\pi$$
$$948$$ 0 0
$$949$$ 44.3002 1.43804
$$950$$ 0 0
$$951$$ −44.8769 −1.45523
$$952$$ 0 0
$$953$$ 46.4924 1.50604 0.753019 0.657999i $$-0.228597\pi$$
0.753019 + 0.657999i $$0.228597\pi$$
$$954$$ 0 0
$$955$$ 11.1231 0.359935
$$956$$ 0 0
$$957$$ −27.7235 −0.896173
$$958$$ 0 0
$$959$$ 21.4773 0.693537
$$960$$ 0 0
$$961$$ −0.0691303 −0.00223001
$$962$$ 0 0
$$963$$ −6.73863 −0.217149
$$964$$ 0 0
$$965$$ 20.4384 0.657937
$$966$$ 0 0
$$967$$ 45.1771 1.45280 0.726398 0.687274i $$-0.241193\pi$$
0.726398 + 0.687274i $$0.241193\pi$$
$$968$$ 0 0
$$969$$ −32.0000 −1.02799
$$970$$ 0 0
$$971$$ 21.3693 0.685774 0.342887 0.939377i $$-0.388595\pi$$
0.342887 + 0.939377i $$0.388595\pi$$
$$972$$ 0 0
$$973$$ −39.6155 −1.27002
$$974$$ 0 0
$$975$$ 5.56155 0.178112
$$976$$ 0 0
$$977$$ −26.1080 −0.835267 −0.417634 0.908615i $$-0.637140\pi$$
−0.417634 + 0.908615i $$0.637140\pi$$
$$978$$ 0 0
$$979$$ 40.0000 1.27841
$$980$$ 0 0
$$981$$ 4.63068 0.147846
$$982$$ 0 0
$$983$$ −28.8769 −0.921030 −0.460515 0.887652i $$-0.652335\pi$$
−0.460515 + 0.887652i $$0.652335\pi$$
$$984$$ 0 0
$$985$$ −15.5616 −0.495832
$$986$$ 0 0
$$987$$ −42.3542 −1.34815
$$988$$ 0 0
$$989$$ −0.876894 −0.0278836
$$990$$ 0 0
$$991$$ −20.4924 −0.650963 −0.325482 0.945548i $$-0.605526\pi$$
−0.325482 + 0.945548i $$0.605526\pi$$
$$992$$ 0 0
$$993$$ −10.0540 −0.319053
$$994$$ 0 0
$$995$$ −24.0000 −0.760851
$$996$$ 0 0
$$997$$ 0.738634 0.0233928 0.0116964 0.999932i $$-0.496277\pi$$
0.0116964 + 0.999932i $$0.496277\pi$$
$$998$$ 0 0
$$999$$ −6.24621 −0.197621
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 1840.2.a.k.1.2 2
4.3 odd 2 920.2.a.f.1.1 2
5.4 even 2 9200.2.a.bx.1.1 2
8.3 odd 2 7360.2.a.bj.1.2 2
8.5 even 2 7360.2.a.bm.1.1 2
12.11 even 2 8280.2.a.bb.1.1 2
20.3 even 4 4600.2.e.m.4049.2 4
20.7 even 4 4600.2.e.m.4049.3 4
20.19 odd 2 4600.2.a.r.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
920.2.a.f.1.1 2 4.3 odd 2
1840.2.a.k.1.2 2 1.1 even 1 trivial
4600.2.a.r.1.2 2 20.19 odd 2
4600.2.e.m.4049.2 4 20.3 even 4
4600.2.e.m.4049.3 4 20.7 even 4
7360.2.a.bj.1.2 2 8.3 odd 2
7360.2.a.bm.1.1 2 8.5 even 2
8280.2.a.bb.1.1 2 12.11 even 2
9200.2.a.bx.1.1 2 5.4 even 2