# Properties

 Label 2100.2.k.f.1849.1 Level 2100 Weight 2 Character 2100.1849 Analytic conductor 16.769 Analytic rank 0 Dimension 2 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2100 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2100.k (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$16.7685844245$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 1849.1 Root $$1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 2100.1849 Dual form 2100.2.k.f.1849.2

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000i q^{3} -1.00000i q^{7} -1.00000 q^{9} +O(q^{10})$$ $$q-1.00000i q^{3} -1.00000i q^{7} -1.00000 q^{9} +1.00000 q^{11} +4.00000i q^{13} -2.00000i q^{17} +4.00000 q^{19} -1.00000 q^{21} +7.00000i q^{23} +1.00000i q^{27} +9.00000 q^{29} -2.00000 q^{31} -1.00000i q^{33} -1.00000i q^{37} +4.00000 q^{39} +8.00000 q^{41} -9.00000i q^{43} -4.00000i q^{47} -1.00000 q^{49} -2.00000 q^{51} +6.00000i q^{53} -4.00000i q^{57} -4.00000 q^{59} +4.00000 q^{61} +1.00000i q^{63} -9.00000i q^{67} +7.00000 q^{69} +5.00000 q^{71} +10.0000i q^{73} -1.00000i q^{77} +15.0000 q^{79} +1.00000 q^{81} -6.00000i q^{83} -9.00000i q^{87} -8.00000 q^{89} +4.00000 q^{91} +2.00000i q^{93} -10.0000i q^{97} -1.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{9} + O(q^{10})$$ $$2q - 2q^{9} + 2q^{11} + 8q^{19} - 2q^{21} + 18q^{29} - 4q^{31} + 8q^{39} + 16q^{41} - 2q^{49} - 4q^{51} - 8q^{59} + 8q^{61} + 14q^{69} + 10q^{71} + 30q^{79} + 2q^{81} - 16q^{89} + 8q^{91} - 2q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2100\mathbb{Z}\right)^\times$$.

 $$n$$ $$701$$ $$1051$$ $$1177$$ $$1501$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$ $$1$$

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ − 1.00000i − 0.577350i
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ − 1.00000i − 0.377964i
$$8$$ 0 0
$$9$$ −1.00000 −0.333333
$$10$$ 0 0
$$11$$ 1.00000 0.301511 0.150756 0.988571i $$-0.451829\pi$$
0.150756 + 0.988571i $$0.451829\pi$$
$$12$$ 0 0
$$13$$ 4.00000i 1.10940i 0.832050 + 0.554700i $$0.187167\pi$$
−0.832050 + 0.554700i $$0.812833\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ − 2.00000i − 0.485071i −0.970143 0.242536i $$-0.922021\pi$$
0.970143 0.242536i $$-0.0779791\pi$$
$$18$$ 0 0
$$19$$ 4.00000 0.917663 0.458831 0.888523i $$-0.348268\pi$$
0.458831 + 0.888523i $$0.348268\pi$$
$$20$$ 0 0
$$21$$ −1.00000 −0.218218
$$22$$ 0 0
$$23$$ 7.00000i 1.45960i 0.683660 + 0.729800i $$0.260387\pi$$
−0.683660 + 0.729800i $$0.739613\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ 1.00000i 0.192450i
$$28$$ 0 0
$$29$$ 9.00000 1.67126 0.835629 0.549294i $$-0.185103\pi$$
0.835629 + 0.549294i $$0.185103\pi$$
$$30$$ 0 0
$$31$$ −2.00000 −0.359211 −0.179605 0.983739i $$-0.557482\pi$$
−0.179605 + 0.983739i $$0.557482\pi$$
$$32$$ 0 0
$$33$$ − 1.00000i − 0.174078i
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ − 1.00000i − 0.164399i −0.996616 0.0821995i $$-0.973806\pi$$
0.996616 0.0821995i $$-0.0261945\pi$$
$$38$$ 0 0
$$39$$ 4.00000 0.640513
$$40$$ 0 0
$$41$$ 8.00000 1.24939 0.624695 0.780869i $$-0.285223\pi$$
0.624695 + 0.780869i $$0.285223\pi$$
$$42$$ 0 0
$$43$$ − 9.00000i − 1.37249i −0.727372 0.686244i $$-0.759258\pi$$
0.727372 0.686244i $$-0.240742\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ − 4.00000i − 0.583460i −0.956501 0.291730i $$-0.905769\pi$$
0.956501 0.291730i $$-0.0942309\pi$$
$$48$$ 0 0
$$49$$ −1.00000 −0.142857
$$50$$ 0 0
$$51$$ −2.00000 −0.280056
$$52$$ 0 0
$$53$$ 6.00000i 0.824163i 0.911147 + 0.412082i $$0.135198\pi$$
−0.911147 + 0.412082i $$0.864802\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ − 4.00000i − 0.529813i
$$58$$ 0 0
$$59$$ −4.00000 −0.520756 −0.260378 0.965507i $$-0.583847\pi$$
−0.260378 + 0.965507i $$0.583847\pi$$
$$60$$ 0 0
$$61$$ 4.00000 0.512148 0.256074 0.966657i $$-0.417571\pi$$
0.256074 + 0.966657i $$0.417571\pi$$
$$62$$ 0 0
$$63$$ 1.00000i 0.125988i
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ − 9.00000i − 1.09952i −0.835321 0.549762i $$-0.814718\pi$$
0.835321 0.549762i $$-0.185282\pi$$
$$68$$ 0 0
$$69$$ 7.00000 0.842701
$$70$$ 0 0
$$71$$ 5.00000 0.593391 0.296695 0.954972i $$-0.404115\pi$$
0.296695 + 0.954972i $$0.404115\pi$$
$$72$$ 0 0
$$73$$ 10.0000i 1.17041i 0.810885 + 0.585206i $$0.198986\pi$$
−0.810885 + 0.585206i $$0.801014\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ − 1.00000i − 0.113961i
$$78$$ 0 0
$$79$$ 15.0000 1.68763 0.843816 0.536633i $$-0.180304\pi$$
0.843816 + 0.536633i $$0.180304\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ − 6.00000i − 0.658586i −0.944228 0.329293i $$-0.893190\pi$$
0.944228 0.329293i $$-0.106810\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ − 9.00000i − 0.964901i
$$88$$ 0 0
$$89$$ −8.00000 −0.847998 −0.423999 0.905663i $$-0.639374\pi$$
−0.423999 + 0.905663i $$0.639374\pi$$
$$90$$ 0 0
$$91$$ 4.00000 0.419314
$$92$$ 0 0
$$93$$ 2.00000i 0.207390i
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ − 10.0000i − 1.01535i −0.861550 0.507673i $$-0.830506\pi$$
0.861550 0.507673i $$-0.169494\pi$$
$$98$$ 0 0
$$99$$ −1.00000 −0.100504
$$100$$ 0 0
$$101$$ 6.00000 0.597022 0.298511 0.954406i $$-0.403510\pi$$
0.298511 + 0.954406i $$0.403510\pi$$
$$102$$ 0 0
$$103$$ − 4.00000i − 0.394132i −0.980390 0.197066i $$-0.936859\pi$$
0.980390 0.197066i $$-0.0631413\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ − 4.00000i − 0.386695i −0.981130 0.193347i $$-0.938066\pi$$
0.981130 0.193347i $$-0.0619344\pi$$
$$108$$ 0 0
$$109$$ −1.00000 −0.0957826 −0.0478913 0.998853i $$-0.515250\pi$$
−0.0478913 + 0.998853i $$0.515250\pi$$
$$110$$ 0 0
$$111$$ −1.00000 −0.0949158
$$112$$ 0 0
$$113$$ 13.0000i 1.22294i 0.791269 + 0.611469i $$0.209421\pi$$
−0.791269 + 0.611469i $$0.790579\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ − 4.00000i − 0.369800i
$$118$$ 0 0
$$119$$ −2.00000 −0.183340
$$120$$ 0 0
$$121$$ −10.0000 −0.909091
$$122$$ 0 0
$$123$$ − 8.00000i − 0.721336i
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ − 9.00000i − 0.798621i −0.916816 0.399310i $$-0.869250\pi$$
0.916816 0.399310i $$-0.130750\pi$$
$$128$$ 0 0
$$129$$ −9.00000 −0.792406
$$130$$ 0 0
$$131$$ 18.0000 1.57267 0.786334 0.617802i $$-0.211977\pi$$
0.786334 + 0.617802i $$0.211977\pi$$
$$132$$ 0 0
$$133$$ − 4.00000i − 0.346844i
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 2.00000i 0.170872i 0.996344 + 0.0854358i $$0.0272282\pi$$
−0.996344 + 0.0854358i $$0.972772\pi$$
$$138$$ 0 0
$$139$$ 10.0000 0.848189 0.424094 0.905618i $$-0.360592\pi$$
0.424094 + 0.905618i $$0.360592\pi$$
$$140$$ 0 0
$$141$$ −4.00000 −0.336861
$$142$$ 0 0
$$143$$ 4.00000i 0.334497i
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 1.00000i 0.0824786i
$$148$$ 0 0
$$149$$ 3.00000 0.245770 0.122885 0.992421i $$-0.460785\pi$$
0.122885 + 0.992421i $$0.460785\pi$$
$$150$$ 0 0
$$151$$ 11.0000 0.895167 0.447584 0.894242i $$-0.352285\pi$$
0.447584 + 0.894242i $$0.352285\pi$$
$$152$$ 0 0
$$153$$ 2.00000i 0.161690i
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ − 20.0000i − 1.59617i −0.602542 0.798087i $$-0.705846\pi$$
0.602542 0.798087i $$-0.294154\pi$$
$$158$$ 0 0
$$159$$ 6.00000 0.475831
$$160$$ 0 0
$$161$$ 7.00000 0.551677
$$162$$ 0 0
$$163$$ 12.0000i 0.939913i 0.882690 + 0.469956i $$0.155730\pi$$
−0.882690 + 0.469956i $$0.844270\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 16.0000i 1.23812i 0.785345 + 0.619059i $$0.212486\pi$$
−0.785345 + 0.619059i $$0.787514\pi$$
$$168$$ 0 0
$$169$$ −3.00000 −0.230769
$$170$$ 0 0
$$171$$ −4.00000 −0.305888
$$172$$ 0 0
$$173$$ 8.00000i 0.608229i 0.952636 + 0.304114i $$0.0983605\pi$$
−0.952636 + 0.304114i $$0.901639\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 4.00000i 0.300658i
$$178$$ 0 0
$$179$$ −4.00000 −0.298974 −0.149487 0.988764i $$-0.547762\pi$$
−0.149487 + 0.988764i $$0.547762\pi$$
$$180$$ 0 0
$$181$$ −10.0000 −0.743294 −0.371647 0.928374i $$-0.621207\pi$$
−0.371647 + 0.928374i $$0.621207\pi$$
$$182$$ 0 0
$$183$$ − 4.00000i − 0.295689i
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ − 2.00000i − 0.146254i
$$188$$ 0 0
$$189$$ 1.00000 0.0727393
$$190$$ 0 0
$$191$$ 20.0000 1.44715 0.723575 0.690246i $$-0.242498\pi$$
0.723575 + 0.690246i $$0.242498\pi$$
$$192$$ 0 0
$$193$$ − 17.0000i − 1.22369i −0.790979 0.611843i $$-0.790428\pi$$
0.790979 0.611843i $$-0.209572\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 15.0000i 1.06871i 0.845262 + 0.534353i $$0.179445\pi$$
−0.845262 + 0.534353i $$0.820555\pi$$
$$198$$ 0 0
$$199$$ −8.00000 −0.567105 −0.283552 0.958957i $$-0.591513\pi$$
−0.283552 + 0.958957i $$0.591513\pi$$
$$200$$ 0 0
$$201$$ −9.00000 −0.634811
$$202$$ 0 0
$$203$$ − 9.00000i − 0.631676i
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ − 7.00000i − 0.486534i
$$208$$ 0 0
$$209$$ 4.00000 0.276686
$$210$$ 0 0
$$211$$ −4.00000 −0.275371 −0.137686 0.990476i $$-0.543966\pi$$
−0.137686 + 0.990476i $$0.543966\pi$$
$$212$$ 0 0
$$213$$ − 5.00000i − 0.342594i
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 2.00000i 0.135769i
$$218$$ 0 0
$$219$$ 10.0000 0.675737
$$220$$ 0 0
$$221$$ 8.00000 0.538138
$$222$$ 0 0
$$223$$ − 6.00000i − 0.401790i −0.979613 0.200895i $$-0.935615\pi$$
0.979613 0.200895i $$-0.0643850\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$228$$ 0 0
$$229$$ −22.0000 −1.45380 −0.726900 0.686743i $$-0.759040\pi$$
−0.726900 + 0.686743i $$0.759040\pi$$
$$230$$ 0 0
$$231$$ −1.00000 −0.0657952
$$232$$ 0 0
$$233$$ 19.0000i 1.24473i 0.782727 + 0.622366i $$0.213828\pi$$
−0.782727 + 0.622366i $$0.786172\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ − 15.0000i − 0.974355i
$$238$$ 0 0
$$239$$ 16.0000 1.03495 0.517477 0.855697i $$-0.326871\pi$$
0.517477 + 0.855697i $$0.326871\pi$$
$$240$$ 0 0
$$241$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$242$$ 0 0
$$243$$ − 1.00000i − 0.0641500i
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 16.0000i 1.01806i
$$248$$ 0 0
$$249$$ −6.00000 −0.380235
$$250$$ 0 0
$$251$$ −2.00000 −0.126239 −0.0631194 0.998006i $$-0.520105\pi$$
−0.0631194 + 0.998006i $$0.520105\pi$$
$$252$$ 0 0
$$253$$ 7.00000i 0.440086i
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ − 26.0000i − 1.62184i −0.585160 0.810918i $$-0.698968\pi$$
0.585160 0.810918i $$-0.301032\pi$$
$$258$$ 0 0
$$259$$ −1.00000 −0.0621370
$$260$$ 0 0
$$261$$ −9.00000 −0.557086
$$262$$ 0 0
$$263$$ − 15.0000i − 0.924940i −0.886635 0.462470i $$-0.846963\pi$$
0.886635 0.462470i $$-0.153037\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 8.00000i 0.489592i
$$268$$ 0 0
$$269$$ −18.0000 −1.09748 −0.548740 0.835993i $$-0.684892\pi$$
−0.548740 + 0.835993i $$0.684892\pi$$
$$270$$ 0 0
$$271$$ −24.0000 −1.45790 −0.728948 0.684569i $$-0.759990\pi$$
−0.728948 + 0.684569i $$0.759990\pi$$
$$272$$ 0 0
$$273$$ − 4.00000i − 0.242091i
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ − 2.00000i − 0.120168i −0.998193 0.0600842i $$-0.980863\pi$$
0.998193 0.0600842i $$-0.0191369\pi$$
$$278$$ 0 0
$$279$$ 2.00000 0.119737
$$280$$ 0 0
$$281$$ 19.0000 1.13344 0.566722 0.823909i $$-0.308211\pi$$
0.566722 + 0.823909i $$0.308211\pi$$
$$282$$ 0 0
$$283$$ − 22.0000i − 1.30776i −0.756596 0.653882i $$-0.773139\pi$$
0.756596 0.653882i $$-0.226861\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ − 8.00000i − 0.472225i
$$288$$ 0 0
$$289$$ 13.0000 0.764706
$$290$$ 0 0
$$291$$ −10.0000 −0.586210
$$292$$ 0 0
$$293$$ 24.0000i 1.40209i 0.713115 + 0.701047i $$0.247284\pi$$
−0.713115 + 0.701047i $$0.752716\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 1.00000i 0.0580259i
$$298$$ 0 0
$$299$$ −28.0000 −1.61928
$$300$$ 0 0
$$301$$ −9.00000 −0.518751
$$302$$ 0 0
$$303$$ − 6.00000i − 0.344691i
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 22.0000i 1.25561i 0.778372 + 0.627803i $$0.216046\pi$$
−0.778372 + 0.627803i $$0.783954\pi$$
$$308$$ 0 0
$$309$$ −4.00000 −0.227552
$$310$$ 0 0
$$311$$ 2.00000 0.113410 0.0567048 0.998391i $$-0.481941\pi$$
0.0567048 + 0.998391i $$0.481941\pi$$
$$312$$ 0 0
$$313$$ 4.00000i 0.226093i 0.993590 + 0.113047i $$0.0360610\pi$$
−0.993590 + 0.113047i $$0.963939\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 31.0000i 1.74113i 0.492050 + 0.870567i $$0.336248\pi$$
−0.492050 + 0.870567i $$0.663752\pi$$
$$318$$ 0 0
$$319$$ 9.00000 0.503903
$$320$$ 0 0
$$321$$ −4.00000 −0.223258
$$322$$ 0 0
$$323$$ − 8.00000i − 0.445132i
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 1.00000i 0.0553001i
$$328$$ 0 0
$$329$$ −4.00000 −0.220527
$$330$$ 0 0
$$331$$ −23.0000 −1.26419 −0.632097 0.774889i $$-0.717806\pi$$
−0.632097 + 0.774889i $$0.717806\pi$$
$$332$$ 0 0
$$333$$ 1.00000i 0.0547997i
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 22.0000i 1.19842i 0.800593 + 0.599208i $$0.204518\pi$$
−0.800593 + 0.599208i $$0.795482\pi$$
$$338$$ 0 0
$$339$$ 13.0000 0.706063
$$340$$ 0 0
$$341$$ −2.00000 −0.108306
$$342$$ 0 0
$$343$$ 1.00000i 0.0539949i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 27.0000i 1.44944i 0.689046 + 0.724718i $$0.258030\pi$$
−0.689046 + 0.724718i $$0.741970\pi$$
$$348$$ 0 0
$$349$$ 28.0000 1.49881 0.749403 0.662114i $$-0.230341\pi$$
0.749403 + 0.662114i $$0.230341\pi$$
$$350$$ 0 0
$$351$$ −4.00000 −0.213504
$$352$$ 0 0
$$353$$ 18.0000i 0.958043i 0.877803 + 0.479022i $$0.159008\pi$$
−0.877803 + 0.479022i $$0.840992\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 2.00000i 0.105851i
$$358$$ 0 0
$$359$$ 35.0000 1.84723 0.923615 0.383322i $$-0.125220\pi$$
0.923615 + 0.383322i $$0.125220\pi$$
$$360$$ 0 0
$$361$$ −3.00000 −0.157895
$$362$$ 0 0
$$363$$ 10.0000i 0.524864i
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ − 4.00000i − 0.208798i −0.994535 0.104399i $$-0.966708\pi$$
0.994535 0.104399i $$-0.0332919\pi$$
$$368$$ 0 0
$$369$$ −8.00000 −0.416463
$$370$$ 0 0
$$371$$ 6.00000 0.311504
$$372$$ 0 0
$$373$$ 11.0000i 0.569558i 0.958593 + 0.284779i $$0.0919203\pi$$
−0.958593 + 0.284779i $$0.908080\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 36.0000i 1.85409i
$$378$$ 0 0
$$379$$ 23.0000 1.18143 0.590715 0.806880i $$-0.298846\pi$$
0.590715 + 0.806880i $$0.298846\pi$$
$$380$$ 0 0
$$381$$ −9.00000 −0.461084
$$382$$ 0 0
$$383$$ − 12.0000i − 0.613171i −0.951843 0.306586i $$-0.900813\pi$$
0.951843 0.306586i $$-0.0991866\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 9.00000i 0.457496i
$$388$$ 0 0
$$389$$ −27.0000 −1.36895 −0.684477 0.729034i $$-0.739969\pi$$
−0.684477 + 0.729034i $$0.739969\pi$$
$$390$$ 0 0
$$391$$ 14.0000 0.708010
$$392$$ 0 0
$$393$$ − 18.0000i − 0.907980i
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ − 10.0000i − 0.501886i −0.968002 0.250943i $$-0.919259\pi$$
0.968002 0.250943i $$-0.0807406\pi$$
$$398$$ 0 0
$$399$$ −4.00000 −0.200250
$$400$$ 0 0
$$401$$ −33.0000 −1.64794 −0.823971 0.566632i $$-0.808246\pi$$
−0.823971 + 0.566632i $$0.808246\pi$$
$$402$$ 0 0
$$403$$ − 8.00000i − 0.398508i
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ − 1.00000i − 0.0495682i
$$408$$ 0 0
$$409$$ −40.0000 −1.97787 −0.988936 0.148340i $$-0.952607\pi$$
−0.988936 + 0.148340i $$0.952607\pi$$
$$410$$ 0 0
$$411$$ 2.00000 0.0986527
$$412$$ 0 0
$$413$$ 4.00000i 0.196827i
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ − 10.0000i − 0.489702i
$$418$$ 0 0
$$419$$ −30.0000 −1.46560 −0.732798 0.680446i $$-0.761786\pi$$
−0.732798 + 0.680446i $$0.761786\pi$$
$$420$$ 0 0
$$421$$ −17.0000 −0.828529 −0.414265 0.910156i $$-0.635961\pi$$
−0.414265 + 0.910156i $$0.635961\pi$$
$$422$$ 0 0
$$423$$ 4.00000i 0.194487i
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ − 4.00000i − 0.193574i
$$428$$ 0 0
$$429$$ 4.00000 0.193122
$$430$$ 0 0
$$431$$ 24.0000 1.15604 0.578020 0.816023i $$-0.303826\pi$$
0.578020 + 0.816023i $$0.303826\pi$$
$$432$$ 0 0
$$433$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 28.0000i 1.33942i
$$438$$ 0 0
$$439$$ −20.0000 −0.954548 −0.477274 0.878755i $$-0.658375\pi$$
−0.477274 + 0.878755i $$0.658375\pi$$
$$440$$ 0 0
$$441$$ 1.00000 0.0476190
$$442$$ 0 0
$$443$$ 20.0000i 0.950229i 0.879924 + 0.475114i $$0.157593\pi$$
−0.879924 + 0.475114i $$0.842407\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ − 3.00000i − 0.141895i
$$448$$ 0 0
$$449$$ −13.0000 −0.613508 −0.306754 0.951789i $$-0.599243\pi$$
−0.306754 + 0.951789i $$0.599243\pi$$
$$450$$ 0 0
$$451$$ 8.00000 0.376705
$$452$$ 0 0
$$453$$ − 11.0000i − 0.516825i
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 5.00000i 0.233890i 0.993138 + 0.116945i $$0.0373101\pi$$
−0.993138 + 0.116945i $$0.962690\pi$$
$$458$$ 0 0
$$459$$ 2.00000 0.0933520
$$460$$ 0 0
$$461$$ −40.0000 −1.86299 −0.931493 0.363760i $$-0.881493\pi$$
−0.931493 + 0.363760i $$0.881493\pi$$
$$462$$ 0 0
$$463$$ − 4.00000i − 0.185896i −0.995671 0.0929479i $$-0.970371\pi$$
0.995671 0.0929479i $$-0.0296290\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 38.0000i 1.75843i 0.476425 + 0.879215i $$0.341932\pi$$
−0.476425 + 0.879215i $$0.658068\pi$$
$$468$$ 0 0
$$469$$ −9.00000 −0.415581
$$470$$ 0 0
$$471$$ −20.0000 −0.921551
$$472$$ 0 0
$$473$$ − 9.00000i − 0.413820i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ − 6.00000i − 0.274721i
$$478$$ 0 0
$$479$$ −36.0000 −1.64488 −0.822441 0.568850i $$-0.807388\pi$$
−0.822441 + 0.568850i $$0.807388\pi$$
$$480$$ 0 0
$$481$$ 4.00000 0.182384
$$482$$ 0 0
$$483$$ − 7.00000i − 0.318511i
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ − 3.00000i − 0.135943i −0.997687 0.0679715i $$-0.978347\pi$$
0.997687 0.0679715i $$-0.0216527\pi$$
$$488$$ 0 0
$$489$$ 12.0000 0.542659
$$490$$ 0 0
$$491$$ −3.00000 −0.135388 −0.0676941 0.997706i $$-0.521564\pi$$
−0.0676941 + 0.997706i $$0.521564\pi$$
$$492$$ 0 0
$$493$$ − 18.0000i − 0.810679i
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ − 5.00000i − 0.224281i
$$498$$ 0 0
$$499$$ −4.00000 −0.179065 −0.0895323 0.995984i $$-0.528537\pi$$
−0.0895323 + 0.995984i $$0.528537\pi$$
$$500$$ 0 0
$$501$$ 16.0000 0.714827
$$502$$ 0 0
$$503$$ − 14.0000i − 0.624229i −0.950044 0.312115i $$-0.898963\pi$$
0.950044 0.312115i $$-0.101037\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 3.00000i 0.133235i
$$508$$ 0 0
$$509$$ 28.0000 1.24108 0.620539 0.784176i $$-0.286914\pi$$
0.620539 + 0.784176i $$0.286914\pi$$
$$510$$ 0 0
$$511$$ 10.0000 0.442374
$$512$$ 0 0
$$513$$ 4.00000i 0.176604i
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ − 4.00000i − 0.175920i
$$518$$ 0 0
$$519$$ 8.00000 0.351161
$$520$$ 0 0
$$521$$ 6.00000 0.262865 0.131432 0.991325i $$-0.458042\pi$$
0.131432 + 0.991325i $$0.458042\pi$$
$$522$$ 0 0
$$523$$ − 40.0000i − 1.74908i −0.484955 0.874539i $$-0.661164\pi$$
0.484955 0.874539i $$-0.338836\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 4.00000i 0.174243i
$$528$$ 0 0
$$529$$ −26.0000 −1.13043
$$530$$ 0 0
$$531$$ 4.00000 0.173585
$$532$$ 0 0
$$533$$ 32.0000i 1.38607i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 4.00000i 0.172613i
$$538$$ 0 0
$$539$$ −1.00000 −0.0430730
$$540$$ 0 0
$$541$$ −25.0000 −1.07483 −0.537417 0.843317i $$-0.680600\pi$$
−0.537417 + 0.843317i $$0.680600\pi$$
$$542$$ 0 0
$$543$$ 10.0000i 0.429141i
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 35.0000i 1.49649i 0.663421 + 0.748246i $$0.269104\pi$$
−0.663421 + 0.748246i $$0.730896\pi$$
$$548$$ 0 0
$$549$$ −4.00000 −0.170716
$$550$$ 0 0
$$551$$ 36.0000 1.53365
$$552$$ 0 0
$$553$$ − 15.0000i − 0.637865i
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ − 1.00000i − 0.0423714i −0.999776 0.0211857i $$-0.993256\pi$$
0.999776 0.0211857i $$-0.00674412\pi$$
$$558$$ 0 0
$$559$$ 36.0000 1.52264
$$560$$ 0 0
$$561$$ −2.00000 −0.0844401
$$562$$ 0 0
$$563$$ − 30.0000i − 1.26435i −0.774826 0.632175i $$-0.782163\pi$$
0.774826 0.632175i $$-0.217837\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ − 1.00000i − 0.0419961i
$$568$$ 0 0
$$569$$ 45.0000 1.88650 0.943249 0.332086i $$-0.107752\pi$$
0.943249 + 0.332086i $$0.107752\pi$$
$$570$$ 0 0
$$571$$ −23.0000 −0.962520 −0.481260 0.876578i $$-0.659821\pi$$
−0.481260 + 0.876578i $$0.659821\pi$$
$$572$$ 0 0
$$573$$ − 20.0000i − 0.835512i
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ − 8.00000i − 0.333044i −0.986038 0.166522i $$-0.946746\pi$$
0.986038 0.166522i $$-0.0532537\pi$$
$$578$$ 0 0
$$579$$ −17.0000 −0.706496
$$580$$ 0 0
$$581$$ −6.00000 −0.248922
$$582$$ 0 0
$$583$$ 6.00000i 0.248495i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 30.0000i 1.23823i 0.785299 + 0.619116i $$0.212509\pi$$
−0.785299 + 0.619116i $$0.787491\pi$$
$$588$$ 0 0
$$589$$ −8.00000 −0.329634
$$590$$ 0 0
$$591$$ 15.0000 0.617018
$$592$$ 0 0
$$593$$ − 12.0000i − 0.492781i −0.969171 0.246390i $$-0.920755\pi$$
0.969171 0.246390i $$-0.0792446\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 8.00000i 0.327418i
$$598$$ 0 0
$$599$$ −35.0000 −1.43006 −0.715031 0.699093i $$-0.753587\pi$$
−0.715031 + 0.699093i $$0.753587\pi$$
$$600$$ 0 0
$$601$$ −4.00000 −0.163163 −0.0815817 0.996667i $$-0.525997\pi$$
−0.0815817 + 0.996667i $$0.525997\pi$$
$$602$$ 0 0
$$603$$ 9.00000i 0.366508i
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$608$$ 0 0
$$609$$ −9.00000 −0.364698
$$610$$ 0 0
$$611$$ 16.0000 0.647291
$$612$$ 0 0
$$613$$ 29.0000i 1.17130i 0.810564 + 0.585649i $$0.199160\pi$$
−0.810564 + 0.585649i $$0.800840\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ − 31.0000i − 1.24801i −0.781419 0.624007i $$-0.785504\pi$$
0.781419 0.624007i $$-0.214496\pi$$
$$618$$ 0 0
$$619$$ −16.0000 −0.643094 −0.321547 0.946894i $$-0.604203\pi$$
−0.321547 + 0.946894i $$0.604203\pi$$
$$620$$ 0 0
$$621$$ −7.00000 −0.280900
$$622$$ 0 0
$$623$$ 8.00000i 0.320513i
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ − 4.00000i − 0.159745i
$$628$$ 0 0
$$629$$ −2.00000 −0.0797452
$$630$$ 0 0
$$631$$ −27.0000 −1.07485 −0.537427 0.843311i $$-0.680603\pi$$
−0.537427 + 0.843311i $$0.680603\pi$$
$$632$$ 0 0
$$633$$ 4.00000i 0.158986i
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ − 4.00000i − 0.158486i
$$638$$ 0 0
$$639$$ −5.00000 −0.197797
$$640$$ 0 0
$$641$$ −33.0000 −1.30342 −0.651711 0.758468i $$-0.725948\pi$$
−0.651711 + 0.758468i $$0.725948\pi$$
$$642$$ 0 0
$$643$$ 28.0000i 1.10421i 0.833774 + 0.552106i $$0.186176\pi$$
−0.833774 + 0.552106i $$0.813824\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ − 46.0000i − 1.80845i −0.427060 0.904223i $$-0.640451\pi$$
0.427060 0.904223i $$-0.359549\pi$$
$$648$$ 0 0
$$649$$ −4.00000 −0.157014
$$650$$ 0 0
$$651$$ 2.00000 0.0783862
$$652$$ 0 0
$$653$$ 26.0000i 1.01746i 0.860927 + 0.508729i $$0.169885\pi$$
−0.860927 + 0.508729i $$0.830115\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ − 10.0000i − 0.390137i
$$658$$ 0 0
$$659$$ −4.00000 −0.155818 −0.0779089 0.996960i $$-0.524824\pi$$
−0.0779089 + 0.996960i $$0.524824\pi$$
$$660$$ 0 0
$$661$$ −16.0000 −0.622328 −0.311164 0.950356i $$-0.600719\pi$$
−0.311164 + 0.950356i $$0.600719\pi$$
$$662$$ 0 0
$$663$$ − 8.00000i − 0.310694i
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 63.0000i 2.43937i
$$668$$ 0 0
$$669$$ −6.00000 −0.231973
$$670$$ 0 0
$$671$$ 4.00000 0.154418
$$672$$ 0 0
$$673$$ 2.00000i 0.0770943i 0.999257 + 0.0385472i $$0.0122730\pi$$
−0.999257 + 0.0385472i $$0.987727\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ − 18.0000i − 0.691796i −0.938272 0.345898i $$-0.887574\pi$$
0.938272 0.345898i $$-0.112426\pi$$
$$678$$ 0 0
$$679$$ −10.0000 −0.383765
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ − 19.0000i − 0.727015i −0.931591 0.363507i $$-0.881579\pi$$
0.931591 0.363507i $$-0.118421\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 22.0000i 0.839352i
$$688$$ 0 0
$$689$$ −24.0000 −0.914327
$$690$$ 0 0
$$691$$ 28.0000 1.06517 0.532585 0.846376i $$-0.321221\pi$$
0.532585 + 0.846376i $$0.321221\pi$$
$$692$$ 0 0
$$693$$ 1.00000i 0.0379869i
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ − 16.0000i − 0.606043i
$$698$$ 0 0
$$699$$ 19.0000 0.718646
$$700$$ 0 0
$$701$$ 18.0000 0.679851 0.339925 0.940452i $$-0.389598\pi$$
0.339925 + 0.940452i $$0.389598\pi$$
$$702$$ 0 0
$$703$$ − 4.00000i − 0.150863i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ − 6.00000i − 0.225653i
$$708$$ 0 0
$$709$$ 26.0000 0.976450 0.488225 0.872718i $$-0.337644\pi$$
0.488225 + 0.872718i $$0.337644\pi$$
$$710$$ 0 0
$$711$$ −15.0000 −0.562544
$$712$$ 0 0
$$713$$ − 14.0000i − 0.524304i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ − 16.0000i − 0.597531i
$$718$$ 0 0
$$719$$ −24.0000 −0.895049 −0.447524 0.894272i $$-0.647694\pi$$
−0.447524 + 0.894272i $$0.647694\pi$$
$$720$$ 0 0
$$721$$ −4.00000 −0.148968
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ − 30.0000i − 1.11264i −0.830969 0.556319i $$-0.812213\pi$$
0.830969 0.556319i $$-0.187787\pi$$
$$728$$ 0 0
$$729$$ −1.00000 −0.0370370
$$730$$ 0 0
$$731$$ −18.0000 −0.665754
$$732$$ 0 0
$$733$$ − 36.0000i − 1.32969i −0.746981 0.664845i $$-0.768498\pi$$
0.746981 0.664845i $$-0.231502\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ − 9.00000i − 0.331519i
$$738$$ 0 0
$$739$$ 9.00000 0.331070 0.165535 0.986204i $$-0.447065\pi$$
0.165535 + 0.986204i $$0.447065\pi$$
$$740$$ 0 0
$$741$$ 16.0000 0.587775
$$742$$ 0 0
$$743$$ − 24.0000i − 0.880475i −0.897881 0.440237i $$-0.854894\pi$$
0.897881 0.440237i $$-0.145106\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 6.00000i 0.219529i
$$748$$ 0 0
$$749$$ −4.00000 −0.146157
$$750$$ 0 0
$$751$$ 40.0000 1.45962 0.729810 0.683650i $$-0.239608\pi$$
0.729810 + 0.683650i $$0.239608\pi$$
$$752$$ 0 0
$$753$$ 2.00000i 0.0728841i
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 7.00000i 0.254419i 0.991876 + 0.127210i $$0.0406021\pi$$
−0.991876 + 0.127210i $$0.959398\pi$$
$$758$$ 0 0
$$759$$ 7.00000 0.254084
$$760$$ 0 0
$$761$$ −4.00000 −0.145000 −0.0724999 0.997368i $$-0.523098\pi$$
−0.0724999 + 0.997368i $$0.523098\pi$$
$$762$$ 0 0
$$763$$ 1.00000i 0.0362024i
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ − 16.0000i − 0.577727i
$$768$$ 0 0
$$769$$ −40.0000 −1.44244 −0.721218 0.692708i $$-0.756418\pi$$
−0.721218 + 0.692708i $$0.756418\pi$$
$$770$$ 0 0
$$771$$ −26.0000 −0.936367
$$772$$ 0 0
$$773$$ − 26.0000i − 0.935155i −0.883952 0.467578i $$-0.845127\pi$$
0.883952 0.467578i $$-0.154873\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 1.00000i 0.0358748i
$$778$$ 0 0
$$779$$ 32.0000 1.14652
$$780$$ 0 0
$$781$$ 5.00000 0.178914
$$782$$ 0 0
$$783$$ 9.00000i 0.321634i
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ − 26.0000i − 0.926800i −0.886149 0.463400i $$-0.846629\pi$$
0.886149 0.463400i $$-0.153371\pi$$
$$788$$ 0 0
$$789$$ −15.0000 −0.534014
$$790$$ 0 0
$$791$$ 13.0000 0.462227
$$792$$ 0 0
$$793$$ 16.0000i 0.568177i
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 54.0000i 1.91278i 0.292096 + 0.956389i $$0.405647\pi$$
−0.292096 + 0.956389i $$0.594353\pi$$
$$798$$ 0 0
$$799$$ −8.00000 −0.283020
$$800$$ 0 0
$$801$$ 8.00000 0.282666
$$802$$ 0 0
$$803$$ 10.0000i 0.352892i
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 18.0000i 0.633630i
$$808$$ 0 0
$$809$$ 23.0000 0.808637 0.404318 0.914618i $$-0.367509\pi$$
0.404318 + 0.914618i $$0.367509\pi$$
$$810$$ 0 0
$$811$$ −6.00000 −0.210688 −0.105344 0.994436i $$-0.533594\pi$$
−0.105344 + 0.994436i $$0.533594\pi$$
$$812$$ 0 0
$$813$$ 24.0000i 0.841717i
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ − 36.0000i − 1.25948i
$$818$$ 0 0
$$819$$ −4.00000 −0.139771
$$820$$ 0 0
$$821$$ 10.0000 0.349002 0.174501 0.984657i $$-0.444169\pi$$
0.174501 + 0.984657i $$0.444169\pi$$
$$822$$ 0 0
$$823$$ − 11.0000i − 0.383436i −0.981450 0.191718i $$-0.938594\pi$$
0.981450 0.191718i $$-0.0614059\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ − 33.0000i − 1.14752i −0.819023 0.573761i $$-0.805484\pi$$
0.819023 0.573761i $$-0.194516\pi$$
$$828$$ 0 0
$$829$$ 34.0000 1.18087 0.590434 0.807086i $$-0.298956\pi$$
0.590434 + 0.807086i $$0.298956\pi$$
$$830$$ 0 0
$$831$$ −2.00000 −0.0693792
$$832$$ 0 0
$$833$$ 2.00000i 0.0692959i
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ − 2.00000i − 0.0691301i
$$838$$ 0 0
$$839$$ 42.0000 1.45000 0.725001 0.688748i $$-0.241839\pi$$
0.725001 + 0.688748i $$0.241839\pi$$
$$840$$ 0 0
$$841$$ 52.0000 1.79310
$$842$$ 0 0
$$843$$ − 19.0000i − 0.654395i
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 10.0000i 0.343604i
$$848$$ 0 0
$$849$$ −22.0000 −0.755038
$$850$$ 0 0
$$851$$ 7.00000 0.239957
$$852$$ 0 0
$$853$$ 28.0000i 0.958702i 0.877623 + 0.479351i $$0.159128\pi$$
−0.877623 + 0.479351i $$0.840872\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 4.00000i 0.136637i 0.997664 + 0.0683187i $$0.0217635\pi$$
−0.997664 + 0.0683187i $$0.978237\pi$$
$$858$$ 0 0
$$859$$ −14.0000 −0.477674 −0.238837 0.971060i $$-0.576766\pi$$
−0.238837 + 0.971060i $$0.576766\pi$$
$$860$$ 0 0
$$861$$ −8.00000 −0.272639
$$862$$ 0 0
$$863$$ 33.0000i 1.12333i 0.827364 + 0.561667i $$0.189840\pi$$
−0.827364 + 0.561667i $$0.810160\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ − 13.0000i − 0.441503i
$$868$$ 0 0
$$869$$ 15.0000 0.508840
$$870$$ 0 0
$$871$$ 36.0000 1.21981
$$872$$ 0 0
$$873$$ 10.0000i 0.338449i
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 34.0000i 1.14810i 0.818821 + 0.574049i $$0.194628\pi$$
−0.818821 + 0.574049i $$0.805372\pi$$
$$878$$ 0 0
$$879$$ 24.0000 0.809500
$$880$$ 0 0
$$881$$ −54.0000 −1.81931 −0.909653 0.415369i $$-0.863653\pi$$
−0.909653 + 0.415369i $$0.863653\pi$$
$$882$$ 0 0
$$883$$ 1.00000i 0.0336527i 0.999858 + 0.0168263i $$0.00535624\pi$$
−0.999858 + 0.0168263i $$0.994644\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ − 52.0000i − 1.74599i −0.487730 0.872995i $$-0.662175\pi$$
0.487730 0.872995i $$-0.337825\pi$$
$$888$$ 0 0
$$889$$ −9.00000 −0.301850
$$890$$ 0 0
$$891$$ 1.00000 0.0335013
$$892$$ 0 0
$$893$$ − 16.0000i − 0.535420i
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 28.0000i 0.934893i
$$898$$ 0 0
$$899$$ −18.0000 −0.600334
$$900$$ 0 0
$$901$$ 12.0000 0.399778
$$902$$ 0 0
$$903$$ 9.00000i 0.299501i
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ − 24.0000i − 0.796907i −0.917189 0.398453i $$-0.869547\pi$$
0.917189 0.398453i $$-0.130453\pi$$
$$908$$ 0 0
$$909$$ −6.00000 −0.199007
$$910$$ 0 0
$$911$$ −9.00000 −0.298183 −0.149092 0.988823i $$-0.547635\pi$$
−0.149092 + 0.988823i $$0.547635\pi$$
$$912$$ 0 0
$$913$$ − 6.00000i − 0.198571i
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ − 18.0000i − 0.594412i
$$918$$ 0 0
$$919$$ 59.0000 1.94623 0.973115 0.230319i $$-0.0739769\pi$$
0.973115 + 0.230319i $$0.0739769\pi$$
$$920$$ 0 0
$$921$$ 22.0000 0.724925
$$922$$ 0 0
$$923$$ 20.0000i 0.658308i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ 4.00000i 0.131377i
$$928$$ 0 0
$$929$$ 26.0000 0.853032 0.426516 0.904480i $$-0.359741\pi$$
0.426516 + 0.904480i $$0.359741\pi$$
$$930$$ 0 0
$$931$$ −4.00000 −0.131095
$$932$$ 0 0
$$933$$ − 2.00000i − 0.0654771i
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ − 36.0000i − 1.17607i −0.808836 0.588034i $$-0.799902\pi$$
0.808836 0.588034i $$-0.200098\pi$$
$$938$$ 0 0
$$939$$ 4.00000 0.130535
$$940$$ 0 0
$$941$$ 54.0000 1.76035 0.880175 0.474650i $$-0.157425\pi$$
0.880175 + 0.474650i $$0.157425\pi$$
$$942$$ 0 0
$$943$$ 56.0000i 1.82361i
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ − 36.0000i − 1.16984i −0.811090 0.584921i $$-0.801125\pi$$
0.811090 0.584921i $$-0.198875\pi$$
$$948$$ 0 0
$$949$$ −40.0000 −1.29845
$$950$$ 0 0
$$951$$ 31.0000 1.00524
$$952$$ 0 0
$$953$$ 21.0000i 0.680257i 0.940379 + 0.340128i $$0.110471\pi$$
−0.940379 + 0.340128i $$0.889529\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ − 9.00000i − 0.290929i
$$958$$ 0 0
$$959$$ 2.00000 0.0645834
$$960$$ 0 0
$$961$$ −27.0000 −0.870968
$$962$$ 0 0
$$963$$ 4.00000i 0.128898i
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ − 40.0000i − 1.28631i −0.765735 0.643157i $$-0.777624\pi$$
0.765735 0.643157i $$-0.222376\pi$$
$$968$$ 0 0
$$969$$ −8.00000 −0.256997
$$970$$ 0 0
$$971$$ −24.0000 −0.770197 −0.385098 0.922876i $$-0.625832\pi$$
−0.385098 + 0.922876i $$0.625832\pi$$
$$972$$ 0 0
$$973$$ − 10.0000i − 0.320585i
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 33.0000i 1.05576i 0.849318 + 0.527882i $$0.177014\pi$$
−0.849318 + 0.527882i $$0.822986\pi$$
$$978$$ 0 0
$$979$$ −8.00000 −0.255681
$$980$$ 0 0
$$981$$ 1.00000 0.0319275
$$982$$ 0 0
$$983$$ − 58.0000i − 1.84991i −0.380073 0.924956i $$-0.624101\pi$$
0.380073 0.924956i $$-0.375899\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 4.00000i 0.127321i
$$988$$ 0 0
$$989$$ 63.0000 2.00328
$$990$$ 0 0
$$991$$ 37.0000 1.17534 0.587672 0.809099i $$-0.300045\pi$$
0.587672 + 0.809099i $$0.300045\pi$$
$$992$$ 0 0
$$993$$ 23.0000i 0.729883i
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 28.0000i 0.886769i 0.896332 + 0.443384i $$0.146222\pi$$
−0.896332 + 0.443384i $$0.853778\pi$$
$$998$$ 0 0
$$999$$ 1.00000 0.0316386
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 2100.2.k.f.1849.1 2
3.2 odd 2 6300.2.k.h.6049.1 2
5.2 odd 4 2100.2.a.g.1.1 1
5.3 odd 4 2100.2.a.m.1.1 yes 1
5.4 even 2 inner 2100.2.k.f.1849.2 2
15.2 even 4 6300.2.a.x.1.1 1
15.8 even 4 6300.2.a.f.1.1 1
15.14 odd 2 6300.2.k.h.6049.2 2
20.3 even 4 8400.2.a.x.1.1 1
20.7 even 4 8400.2.a.bv.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
2100.2.a.g.1.1 1 5.2 odd 4
2100.2.a.m.1.1 yes 1 5.3 odd 4
2100.2.k.f.1849.1 2 1.1 even 1 trivial
2100.2.k.f.1849.2 2 5.4 even 2 inner
6300.2.a.f.1.1 1 15.8 even 4
6300.2.a.x.1.1 1 15.2 even 4
6300.2.k.h.6049.1 2 3.2 odd 2
6300.2.k.h.6049.2 2 15.14 odd 2
8400.2.a.x.1.1 1 20.3 even 4
8400.2.a.bv.1.1 1 20.7 even 4