# Properties

 Label 9408.2.a.bp.1.1 Level $9408$ Weight $2$ Character 9408.1 Self dual yes Analytic conductor $75.123$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$9408 = 2^{6} \cdot 3 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 9408.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$75.1232582216$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 672) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 9408.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{3} +4.00000 q^{5} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{3} +4.00000 q^{5} +1.00000 q^{9} +6.00000 q^{11} -5.00000 q^{13} -4.00000 q^{15} +2.00000 q^{17} +1.00000 q^{19} +6.00000 q^{23} +11.0000 q^{25} -1.00000 q^{27} +3.00000 q^{31} -6.00000 q^{33} -3.00000 q^{37} +5.00000 q^{39} -6.00000 q^{41} +5.00000 q^{43} +4.00000 q^{45} +4.00000 q^{47} -2.00000 q^{51} +6.00000 q^{53} +24.0000 q^{55} -1.00000 q^{57} -6.00000 q^{59} +2.00000 q^{61} -20.0000 q^{65} +7.00000 q^{67} -6.00000 q^{69} -16.0000 q^{71} -3.00000 q^{73} -11.0000 q^{75} -11.0000 q^{79} +1.00000 q^{81} +12.0000 q^{83} +8.00000 q^{85} +4.00000 q^{89} -3.00000 q^{93} +4.00000 q^{95} -6.00000 q^{97} +6.00000 q^{99} +O(q^{100})$$

## 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
$$3$$ −1.00000 −0.577350
$$4$$ 0 0
$$5$$ 4.00000 1.78885 0.894427 0.447214i $$-0.147584\pi$$
0.894427 + 0.447214i $$0.147584\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ 6.00000 1.80907 0.904534 0.426401i $$-0.140219\pi$$
0.904534 + 0.426401i $$0.140219\pi$$
$$12$$ 0 0
$$13$$ −5.00000 −1.38675 −0.693375 0.720577i $$-0.743877\pi$$
−0.693375 + 0.720577i $$0.743877\pi$$
$$14$$ 0 0
$$15$$ −4.00000 −1.03280
$$16$$ 0 0
$$17$$ 2.00000 0.485071 0.242536 0.970143i $$-0.422021\pi$$
0.242536 + 0.970143i $$0.422021\pi$$
$$18$$ 0 0
$$19$$ 1.00000 0.229416 0.114708 0.993399i $$-0.463407\pi$$
0.114708 + 0.993399i $$0.463407\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 6.00000 1.25109 0.625543 0.780189i $$-0.284877\pi$$
0.625543 + 0.780189i $$0.284877\pi$$
$$24$$ 0 0
$$25$$ 11.0000 2.20000
$$26$$ 0 0
$$27$$ −1.00000 −0.192450
$$28$$ 0 0
$$29$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$30$$ 0 0
$$31$$ 3.00000 0.538816 0.269408 0.963026i $$-0.413172\pi$$
0.269408 + 0.963026i $$0.413172\pi$$
$$32$$ 0 0
$$33$$ −6.00000 −1.04447
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −3.00000 −0.493197 −0.246598 0.969118i $$-0.579313\pi$$
−0.246598 + 0.969118i $$0.579313\pi$$
$$38$$ 0 0
$$39$$ 5.00000 0.800641
$$40$$ 0 0
$$41$$ −6.00000 −0.937043 −0.468521 0.883452i $$-0.655213\pi$$
−0.468521 + 0.883452i $$0.655213\pi$$
$$42$$ 0 0
$$43$$ 5.00000 0.762493 0.381246 0.924473i $$-0.375495\pi$$
0.381246 + 0.924473i $$0.375495\pi$$
$$44$$ 0 0
$$45$$ 4.00000 0.596285
$$46$$ 0 0
$$47$$ 4.00000 0.583460 0.291730 0.956501i $$-0.405769\pi$$
0.291730 + 0.956501i $$0.405769\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 0 0
$$51$$ −2.00000 −0.280056
$$52$$ 0 0
$$53$$ 6.00000 0.824163 0.412082 0.911147i $$-0.364802\pi$$
0.412082 + 0.911147i $$0.364802\pi$$
$$54$$ 0 0
$$55$$ 24.0000 3.23616
$$56$$ 0 0
$$57$$ −1.00000 −0.132453
$$58$$ 0 0
$$59$$ −6.00000 −0.781133 −0.390567 0.920575i $$-0.627721\pi$$
−0.390567 + 0.920575i $$0.627721\pi$$
$$60$$ 0 0
$$61$$ 2.00000 0.256074 0.128037 0.991769i $$-0.459132\pi$$
0.128037 + 0.991769i $$0.459132\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ −20.0000 −2.48069
$$66$$ 0 0
$$67$$ 7.00000 0.855186 0.427593 0.903971i $$-0.359362\pi$$
0.427593 + 0.903971i $$0.359362\pi$$
$$68$$ 0 0
$$69$$ −6.00000 −0.722315
$$70$$ 0 0
$$71$$ −16.0000 −1.89885 −0.949425 0.313993i $$-0.898333\pi$$
−0.949425 + 0.313993i $$0.898333\pi$$
$$72$$ 0 0
$$73$$ −3.00000 −0.351123 −0.175562 0.984468i $$-0.556174\pi$$
−0.175562 + 0.984468i $$0.556174\pi$$
$$74$$ 0 0
$$75$$ −11.0000 −1.27017
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −11.0000 −1.23760 −0.618798 0.785550i $$-0.712380\pi$$
−0.618798 + 0.785550i $$0.712380\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ 12.0000 1.31717 0.658586 0.752506i $$-0.271155\pi$$
0.658586 + 0.752506i $$0.271155\pi$$
$$84$$ 0 0
$$85$$ 8.00000 0.867722
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 4.00000 0.423999 0.212000 0.977270i $$-0.432002\pi$$
0.212000 + 0.977270i $$0.432002\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ −3.00000 −0.311086
$$94$$ 0 0
$$95$$ 4.00000 0.410391
$$96$$ 0 0
$$97$$ −6.00000 −0.609208 −0.304604 0.952479i $$-0.598524\pi$$
−0.304604 + 0.952479i $$0.598524\pi$$
$$98$$ 0 0
$$99$$ 6.00000 0.603023
$$100$$ 0 0
$$101$$ −2.00000 −0.199007 −0.0995037 0.995037i $$-0.531726\pi$$
−0.0995037 + 0.995037i $$0.531726\pi$$
$$102$$ 0 0
$$103$$ 11.0000 1.08386 0.541931 0.840423i $$-0.317693\pi$$
0.541931 + 0.840423i $$0.317693\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 10.0000 0.966736 0.483368 0.875417i $$-0.339413\pi$$
0.483368 + 0.875417i $$0.339413\pi$$
$$108$$ 0 0
$$109$$ 15.0000 1.43674 0.718370 0.695662i $$-0.244889\pi$$
0.718370 + 0.695662i $$0.244889\pi$$
$$110$$ 0 0
$$111$$ 3.00000 0.284747
$$112$$ 0 0
$$113$$ −16.0000 −1.50515 −0.752577 0.658505i $$-0.771189\pi$$
−0.752577 + 0.658505i $$0.771189\pi$$
$$114$$ 0 0
$$115$$ 24.0000 2.23801
$$116$$ 0 0
$$117$$ −5.00000 −0.462250
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 25.0000 2.27273
$$122$$ 0 0
$$123$$ 6.00000 0.541002
$$124$$ 0 0
$$125$$ 24.0000 2.14663
$$126$$ 0 0
$$127$$ 7.00000 0.621150 0.310575 0.950549i $$-0.399478\pi$$
0.310575 + 0.950549i $$0.399478\pi$$
$$128$$ 0 0
$$129$$ −5.00000 −0.440225
$$130$$ 0 0
$$131$$ 6.00000 0.524222 0.262111 0.965038i $$-0.415581\pi$$
0.262111 + 0.965038i $$0.415581\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ −4.00000 −0.344265
$$136$$ 0 0
$$137$$ −12.0000 −1.02523 −0.512615 0.858619i $$-0.671323\pi$$
−0.512615 + 0.858619i $$0.671323\pi$$
$$138$$ 0 0
$$139$$ 5.00000 0.424094 0.212047 0.977259i $$-0.431987\pi$$
0.212047 + 0.977259i $$0.431987\pi$$
$$140$$ 0 0
$$141$$ −4.00000 −0.336861
$$142$$ 0 0
$$143$$ −30.0000 −2.50873
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 4.00000 0.327693 0.163846 0.986486i $$-0.447610\pi$$
0.163846 + 0.986486i $$0.447610\pi$$
$$150$$ 0 0
$$151$$ 8.00000 0.651031 0.325515 0.945537i $$-0.394462\pi$$
0.325515 + 0.945537i $$0.394462\pi$$
$$152$$ 0 0
$$153$$ 2.00000 0.161690
$$154$$ 0 0
$$155$$ 12.0000 0.963863
$$156$$ 0 0
$$157$$ −10.0000 −0.798087 −0.399043 0.916932i $$-0.630658\pi$$
−0.399043 + 0.916932i $$0.630658\pi$$
$$158$$ 0 0
$$159$$ −6.00000 −0.475831
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 20.0000 1.56652 0.783260 0.621694i $$-0.213555\pi$$
0.783260 + 0.621694i $$0.213555\pi$$
$$164$$ 0 0
$$165$$ −24.0000 −1.86840
$$166$$ 0 0
$$167$$ 8.00000 0.619059 0.309529 0.950890i $$-0.399829\pi$$
0.309529 + 0.950890i $$0.399829\pi$$
$$168$$ 0 0
$$169$$ 12.0000 0.923077
$$170$$ 0 0
$$171$$ 1.00000 0.0764719
$$172$$ 0 0
$$173$$ −22.0000 −1.67263 −0.836315 0.548250i $$-0.815294\pi$$
−0.836315 + 0.548250i $$0.815294\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 6.00000 0.450988
$$178$$ 0 0
$$179$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$180$$ 0 0
$$181$$ −25.0000 −1.85824 −0.929118 0.369784i $$-0.879432\pi$$
−0.929118 + 0.369784i $$0.879432\pi$$
$$182$$ 0 0
$$183$$ −2.00000 −0.147844
$$184$$ 0 0
$$185$$ −12.0000 −0.882258
$$186$$ 0 0
$$187$$ 12.0000 0.877527
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$192$$ 0 0
$$193$$ −1.00000 −0.0719816 −0.0359908 0.999352i $$-0.511459\pi$$
−0.0359908 + 0.999352i $$0.511459\pi$$
$$194$$ 0 0
$$195$$ 20.0000 1.43223
$$196$$ 0 0
$$197$$ −18.0000 −1.28245 −0.641223 0.767354i $$-0.721573\pi$$
−0.641223 + 0.767354i $$0.721573\pi$$
$$198$$ 0 0
$$199$$ −20.0000 −1.41776 −0.708881 0.705328i $$-0.750800\pi$$
−0.708881 + 0.705328i $$0.750800\pi$$
$$200$$ 0 0
$$201$$ −7.00000 −0.493742
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ −24.0000 −1.67623
$$206$$ 0 0
$$207$$ 6.00000 0.417029
$$208$$ 0 0
$$209$$ 6.00000 0.415029
$$210$$ 0 0
$$211$$ 12.0000 0.826114 0.413057 0.910705i $$-0.364461\pi$$
0.413057 + 0.910705i $$0.364461\pi$$
$$212$$ 0 0
$$213$$ 16.0000 1.09630
$$214$$ 0 0
$$215$$ 20.0000 1.36399
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ 3.00000 0.202721
$$220$$ 0 0
$$221$$ −10.0000 −0.672673
$$222$$ 0 0
$$223$$ −16.0000 −1.07144 −0.535720 0.844396i $$-0.679960\pi$$
−0.535720 + 0.844396i $$0.679960\pi$$
$$224$$ 0 0
$$225$$ 11.0000 0.733333
$$226$$ 0 0
$$227$$ 22.0000 1.46019 0.730096 0.683345i $$-0.239475\pi$$
0.730096 + 0.683345i $$0.239475\pi$$
$$228$$ 0 0
$$229$$ 11.0000 0.726900 0.363450 0.931614i $$-0.381599\pi$$
0.363450 + 0.931614i $$0.381599\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −8.00000 −0.524097 −0.262049 0.965055i $$-0.584398\pi$$
−0.262049 + 0.965055i $$0.584398\pi$$
$$234$$ 0 0
$$235$$ 16.0000 1.04372
$$236$$ 0 0
$$237$$ 11.0000 0.714527
$$238$$ 0 0
$$239$$ −2.00000 −0.129369 −0.0646846 0.997906i $$-0.520604\pi$$
−0.0646846 + 0.997906i $$0.520604\pi$$
$$240$$ 0 0
$$241$$ 10.0000 0.644157 0.322078 0.946713i $$-0.395619\pi$$
0.322078 + 0.946713i $$0.395619\pi$$
$$242$$ 0 0
$$243$$ −1.00000 −0.0641500
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −5.00000 −0.318142
$$248$$ 0 0
$$249$$ −12.0000 −0.760469
$$250$$ 0 0
$$251$$ −14.0000 −0.883672 −0.441836 0.897096i $$-0.645673\pi$$
−0.441836 + 0.897096i $$0.645673\pi$$
$$252$$ 0 0
$$253$$ 36.0000 2.26330
$$254$$ 0 0
$$255$$ −8.00000 −0.500979
$$256$$ 0 0
$$257$$ −12.0000 −0.748539 −0.374270 0.927320i $$-0.622107\pi$$
−0.374270 + 0.927320i $$0.622107\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$264$$ 0 0
$$265$$ 24.0000 1.47431
$$266$$ 0 0
$$267$$ −4.00000 −0.244796
$$268$$ 0 0
$$269$$ 2.00000 0.121942 0.0609711 0.998140i $$-0.480580\pi$$
0.0609711 + 0.998140i $$0.480580\pi$$
$$270$$ 0 0
$$271$$ −16.0000 −0.971931 −0.485965 0.873978i $$-0.661532\pi$$
−0.485965 + 0.873978i $$0.661532\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 66.0000 3.97995
$$276$$ 0 0
$$277$$ −1.00000 −0.0600842 −0.0300421 0.999549i $$-0.509564\pi$$
−0.0300421 + 0.999549i $$0.509564\pi$$
$$278$$ 0 0
$$279$$ 3.00000 0.179605
$$280$$ 0 0
$$281$$ 6.00000 0.357930 0.178965 0.983855i $$-0.442725\pi$$
0.178965 + 0.983855i $$0.442725\pi$$
$$282$$ 0 0
$$283$$ −11.0000 −0.653882 −0.326941 0.945045i $$-0.606018\pi$$
−0.326941 + 0.945045i $$0.606018\pi$$
$$284$$ 0 0
$$285$$ −4.00000 −0.236940
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ −13.0000 −0.764706
$$290$$ 0 0
$$291$$ 6.00000 0.351726
$$292$$ 0 0
$$293$$ 12.0000 0.701047 0.350524 0.936554i $$-0.386004\pi$$
0.350524 + 0.936554i $$0.386004\pi$$
$$294$$ 0 0
$$295$$ −24.0000 −1.39733
$$296$$ 0 0
$$297$$ −6.00000 −0.348155
$$298$$ 0 0
$$299$$ −30.0000 −1.73494
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ 2.00000 0.114897
$$304$$ 0 0
$$305$$ 8.00000 0.458079
$$306$$ 0 0
$$307$$ 11.0000 0.627803 0.313902 0.949456i $$-0.398364\pi$$
0.313902 + 0.949456i $$0.398364\pi$$
$$308$$ 0 0
$$309$$ −11.0000 −0.625768
$$310$$ 0 0
$$311$$ 2.00000 0.113410 0.0567048 0.998391i $$-0.481941\pi$$
0.0567048 + 0.998391i $$0.481941\pi$$
$$312$$ 0 0
$$313$$ 31.0000 1.75222 0.876112 0.482108i $$-0.160129\pi$$
0.876112 + 0.482108i $$0.160129\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 20.0000 1.12331 0.561656 0.827371i $$-0.310164\pi$$
0.561656 + 0.827371i $$0.310164\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 0 0
$$321$$ −10.0000 −0.558146
$$322$$ 0 0
$$323$$ 2.00000 0.111283
$$324$$ 0 0
$$325$$ −55.0000 −3.05085
$$326$$ 0 0
$$327$$ −15.0000 −0.829502
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −5.00000 −0.274825 −0.137412 0.990514i $$-0.543879\pi$$
−0.137412 + 0.990514i $$0.543879\pi$$
$$332$$ 0 0
$$333$$ −3.00000 −0.164399
$$334$$ 0 0
$$335$$ 28.0000 1.52980
$$336$$ 0 0
$$337$$ 1.00000 0.0544735 0.0272367 0.999629i $$-0.491329\pi$$
0.0272367 + 0.999629i $$0.491329\pi$$
$$338$$ 0 0
$$339$$ 16.0000 0.869001
$$340$$ 0 0
$$341$$ 18.0000 0.974755
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ −24.0000 −1.29212
$$346$$ 0 0
$$347$$ −22.0000 −1.18102 −0.590511 0.807030i $$-0.701074\pi$$
−0.590511 + 0.807030i $$0.701074\pi$$
$$348$$ 0 0
$$349$$ −26.0000 −1.39175 −0.695874 0.718164i $$-0.744983\pi$$
−0.695874 + 0.718164i $$0.744983\pi$$
$$350$$ 0 0
$$351$$ 5.00000 0.266880
$$352$$ 0 0
$$353$$ −4.00000 −0.212899 −0.106449 0.994318i $$-0.533948\pi$$
−0.106449 + 0.994318i $$0.533948\pi$$
$$354$$ 0 0
$$355$$ −64.0000 −3.39677
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 24.0000 1.26667 0.633336 0.773877i $$-0.281685\pi$$
0.633336 + 0.773877i $$0.281685\pi$$
$$360$$ 0 0
$$361$$ −18.0000 −0.947368
$$362$$ 0 0
$$363$$ −25.0000 −1.31216
$$364$$ 0 0
$$365$$ −12.0000 −0.628109
$$366$$ 0 0
$$367$$ −27.0000 −1.40939 −0.704694 0.709511i $$-0.748916\pi$$
−0.704694 + 0.709511i $$0.748916\pi$$
$$368$$ 0 0
$$369$$ −6.00000 −0.312348
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 29.0000 1.50156 0.750782 0.660551i $$-0.229677\pi$$
0.750782 + 0.660551i $$0.229677\pi$$
$$374$$ 0 0
$$375$$ −24.0000 −1.23935
$$376$$ 0 0
$$377$$ 0 0
$$378$$ 0 0
$$379$$ 27.0000 1.38690 0.693448 0.720506i $$-0.256091\pi$$
0.693448 + 0.720506i $$0.256091\pi$$
$$380$$ 0 0
$$381$$ −7.00000 −0.358621
$$382$$ 0 0
$$383$$ −26.0000 −1.32854 −0.664269 0.747494i $$-0.731257\pi$$
−0.664269 + 0.747494i $$0.731257\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 5.00000 0.254164
$$388$$ 0 0
$$389$$ 4.00000 0.202808 0.101404 0.994845i $$-0.467667\pi$$
0.101404 + 0.994845i $$0.467667\pi$$
$$390$$ 0 0
$$391$$ 12.0000 0.606866
$$392$$ 0 0
$$393$$ −6.00000 −0.302660
$$394$$ 0 0
$$395$$ −44.0000 −2.21388
$$396$$ 0 0
$$397$$ 21.0000 1.05396 0.526980 0.849878i $$-0.323324\pi$$
0.526980 + 0.849878i $$0.323324\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −2.00000 −0.0998752 −0.0499376 0.998752i $$-0.515902\pi$$
−0.0499376 + 0.998752i $$0.515902\pi$$
$$402$$ 0 0
$$403$$ −15.0000 −0.747203
$$404$$ 0 0
$$405$$ 4.00000 0.198762
$$406$$ 0 0
$$407$$ −18.0000 −0.892227
$$408$$ 0 0
$$409$$ 13.0000 0.642809 0.321404 0.946942i $$-0.395845\pi$$
0.321404 + 0.946942i $$0.395845\pi$$
$$410$$ 0 0
$$411$$ 12.0000 0.591916
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 48.0000 2.35623
$$416$$ 0 0
$$417$$ −5.00000 −0.244851
$$418$$ 0 0
$$419$$ −26.0000 −1.27018 −0.635092 0.772437i $$-0.719038\pi$$
−0.635092 + 0.772437i $$0.719038\pi$$
$$420$$ 0 0
$$421$$ −17.0000 −0.828529 −0.414265 0.910156i $$-0.635961\pi$$
−0.414265 + 0.910156i $$0.635961\pi$$
$$422$$ 0 0
$$423$$ 4.00000 0.194487
$$424$$ 0 0
$$425$$ 22.0000 1.06716
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ 30.0000 1.44841
$$430$$ 0 0
$$431$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$432$$ 0 0
$$433$$ −25.0000 −1.20142 −0.600712 0.799466i $$-0.705116\pi$$
−0.600712 + 0.799466i $$0.705116\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 6.00000 0.287019
$$438$$ 0 0
$$439$$ −24.0000 −1.14546 −0.572729 0.819745i $$-0.694115\pi$$
−0.572729 + 0.819745i $$0.694115\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −26.0000 −1.23530 −0.617649 0.786454i $$-0.711915\pi$$
−0.617649 + 0.786454i $$0.711915\pi$$
$$444$$ 0 0
$$445$$ 16.0000 0.758473
$$446$$ 0 0
$$447$$ −4.00000 −0.189194
$$448$$ 0 0
$$449$$ 36.0000 1.69895 0.849473 0.527633i $$-0.176920\pi$$
0.849473 + 0.527633i $$0.176920\pi$$
$$450$$ 0 0
$$451$$ −36.0000 −1.69517
$$452$$ 0 0
$$453$$ −8.00000 −0.375873
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 25.0000 1.16945 0.584725 0.811231i $$-0.301202\pi$$
0.584725 + 0.811231i $$0.301202\pi$$
$$458$$ 0 0
$$459$$ −2.00000 −0.0933520
$$460$$ 0 0
$$461$$ 4.00000 0.186299 0.0931493 0.995652i $$-0.470307\pi$$
0.0931493 + 0.995652i $$0.470307\pi$$
$$462$$ 0 0
$$463$$ 5.00000 0.232370 0.116185 0.993228i $$-0.462933\pi$$
0.116185 + 0.993228i $$0.462933\pi$$
$$464$$ 0 0
$$465$$ −12.0000 −0.556487
$$466$$ 0 0
$$467$$ 8.00000 0.370196 0.185098 0.982720i $$-0.440740\pi$$
0.185098 + 0.982720i $$0.440740\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 10.0000 0.460776
$$472$$ 0 0
$$473$$ 30.0000 1.37940
$$474$$ 0 0
$$475$$ 11.0000 0.504715
$$476$$ 0 0
$$477$$ 6.00000 0.274721
$$478$$ 0 0
$$479$$ −2.00000 −0.0913823 −0.0456912 0.998956i $$-0.514549\pi$$
−0.0456912 + 0.998956i $$0.514549\pi$$
$$480$$ 0 0
$$481$$ 15.0000 0.683941
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ −24.0000 −1.08978
$$486$$ 0 0
$$487$$ 1.00000 0.0453143 0.0226572 0.999743i $$-0.492787\pi$$
0.0226572 + 0.999743i $$0.492787\pi$$
$$488$$ 0 0
$$489$$ −20.0000 −0.904431
$$490$$ 0 0
$$491$$ 8.00000 0.361035 0.180517 0.983572i $$-0.442223\pi$$
0.180517 + 0.983572i $$0.442223\pi$$
$$492$$ 0 0
$$493$$ 0 0
$$494$$ 0 0
$$495$$ 24.0000 1.07872
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 29.0000 1.29822 0.649109 0.760695i $$-0.275142\pi$$
0.649109 + 0.760695i $$0.275142\pi$$
$$500$$ 0 0
$$501$$ −8.00000 −0.357414
$$502$$ 0 0
$$503$$ −18.0000 −0.802580 −0.401290 0.915951i $$-0.631438\pi$$
−0.401290 + 0.915951i $$0.631438\pi$$
$$504$$ 0 0
$$505$$ −8.00000 −0.355995
$$506$$ 0 0
$$507$$ −12.0000 −0.532939
$$508$$ 0 0
$$509$$ 24.0000 1.06378 0.531891 0.846813i $$-0.321482\pi$$
0.531891 + 0.846813i $$0.321482\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ −1.00000 −0.0441511
$$514$$ 0 0
$$515$$ 44.0000 1.93887
$$516$$ 0 0
$$517$$ 24.0000 1.05552
$$518$$ 0 0
$$519$$ 22.0000 0.965693
$$520$$ 0 0
$$521$$ 24.0000 1.05146 0.525730 0.850652i $$-0.323792\pi$$
0.525730 + 0.850652i $$0.323792\pi$$
$$522$$ 0 0
$$523$$ −17.0000 −0.743358 −0.371679 0.928361i $$-0.621218\pi$$
−0.371679 + 0.928361i $$0.621218\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 6.00000 0.261364
$$528$$ 0 0
$$529$$ 13.0000 0.565217
$$530$$ 0 0
$$531$$ −6.00000 −0.260378
$$532$$ 0 0
$$533$$ 30.0000 1.29944
$$534$$ 0 0
$$535$$ 40.0000 1.72935
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −37.0000 −1.59075 −0.795377 0.606115i $$-0.792727\pi$$
−0.795377 + 0.606115i $$0.792727\pi$$
$$542$$ 0 0
$$543$$ 25.0000 1.07285
$$544$$ 0 0
$$545$$ 60.0000 2.57012
$$546$$ 0 0
$$547$$ 28.0000 1.19719 0.598597 0.801050i $$-0.295725\pi$$
0.598597 + 0.801050i $$0.295725\pi$$
$$548$$ 0 0
$$549$$ 2.00000 0.0853579
$$550$$ 0 0
$$551$$ 0 0
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ 12.0000 0.509372
$$556$$ 0 0
$$557$$ 6.00000 0.254228 0.127114 0.991888i $$-0.459429\pi$$
0.127114 + 0.991888i $$0.459429\pi$$
$$558$$ 0 0
$$559$$ −25.0000 −1.05739
$$560$$ 0 0
$$561$$ −12.0000 −0.506640
$$562$$ 0 0
$$563$$ 24.0000 1.01148 0.505740 0.862686i $$-0.331220\pi$$
0.505740 + 0.862686i $$0.331220\pi$$
$$564$$ 0 0
$$565$$ −64.0000 −2.69250
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 18.0000 0.754599 0.377300 0.926091i $$-0.376853\pi$$
0.377300 + 0.926091i $$0.376853\pi$$
$$570$$ 0 0
$$571$$ 29.0000 1.21361 0.606806 0.794850i $$-0.292450\pi$$
0.606806 + 0.794850i $$0.292450\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 66.0000 2.75239
$$576$$ 0 0
$$577$$ 23.0000 0.957503 0.478751 0.877951i $$-0.341090\pi$$
0.478751 + 0.877951i $$0.341090\pi$$
$$578$$ 0 0
$$579$$ 1.00000 0.0415586
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 36.0000 1.49097
$$584$$ 0 0
$$585$$ −20.0000 −0.826898
$$586$$ 0 0
$$587$$ 40.0000 1.65098 0.825488 0.564419i $$-0.190900\pi$$
0.825488 + 0.564419i $$0.190900\pi$$
$$588$$ 0 0
$$589$$ 3.00000 0.123613
$$590$$ 0 0
$$591$$ 18.0000 0.740421
$$592$$ 0 0
$$593$$ −26.0000 −1.06769 −0.533846 0.845582i $$-0.679254\pi$$
−0.533846 + 0.845582i $$0.679254\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 20.0000 0.818546
$$598$$ 0 0
$$599$$ −20.0000 −0.817178 −0.408589 0.912719i $$-0.633979\pi$$
−0.408589 + 0.912719i $$0.633979\pi$$
$$600$$ 0 0
$$601$$ −21.0000 −0.856608 −0.428304 0.903635i $$-0.640889\pi$$
−0.428304 + 0.903635i $$0.640889\pi$$
$$602$$ 0 0
$$603$$ 7.00000 0.285062
$$604$$ 0 0
$$605$$ 100.000 4.06558
$$606$$ 0 0
$$607$$ 5.00000 0.202944 0.101472 0.994838i $$-0.467645\pi$$
0.101472 + 0.994838i $$0.467645\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −20.0000 −0.809113
$$612$$ 0 0
$$613$$ 46.0000 1.85792 0.928961 0.370177i $$-0.120703\pi$$
0.928961 + 0.370177i $$0.120703\pi$$
$$614$$ 0 0
$$615$$ 24.0000 0.967773
$$616$$ 0 0
$$617$$ −14.0000 −0.563619 −0.281809 0.959470i $$-0.590935\pi$$
−0.281809 + 0.959470i $$0.590935\pi$$
$$618$$ 0 0
$$619$$ −25.0000 −1.00483 −0.502417 0.864625i $$-0.667556\pi$$
−0.502417 + 0.864625i $$0.667556\pi$$
$$620$$ 0 0
$$621$$ −6.00000 −0.240772
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 41.0000 1.64000
$$626$$ 0 0
$$627$$ −6.00000 −0.239617
$$628$$ 0 0
$$629$$ −6.00000 −0.239236
$$630$$ 0 0
$$631$$ −28.0000 −1.11466 −0.557331 0.830290i $$-0.688175\pi$$
−0.557331 + 0.830290i $$0.688175\pi$$
$$632$$ 0 0
$$633$$ −12.0000 −0.476957
$$634$$ 0 0
$$635$$ 28.0000 1.11115
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ −16.0000 −0.632950
$$640$$ 0 0
$$641$$ 38.0000 1.50091 0.750455 0.660922i $$-0.229834\pi$$
0.750455 + 0.660922i $$0.229834\pi$$
$$642$$ 0 0
$$643$$ −23.0000 −0.907031 −0.453516 0.891248i $$-0.649830\pi$$
−0.453516 + 0.891248i $$0.649830\pi$$
$$644$$ 0 0
$$645$$ −20.0000 −0.787499
$$646$$ 0 0
$$647$$ −10.0000 −0.393141 −0.196570 0.980490i $$-0.562980\pi$$
−0.196570 + 0.980490i $$0.562980\pi$$
$$648$$ 0 0
$$649$$ −36.0000 −1.41312
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −34.0000 −1.33052 −0.665261 0.746611i $$-0.731680\pi$$
−0.665261 + 0.746611i $$0.731680\pi$$
$$654$$ 0 0
$$655$$ 24.0000 0.937758
$$656$$ 0 0
$$657$$ −3.00000 −0.117041
$$658$$ 0 0
$$659$$ 6.00000 0.233727 0.116863 0.993148i $$-0.462716\pi$$
0.116863 + 0.993148i $$0.462716\pi$$
$$660$$ 0 0
$$661$$ −3.00000 −0.116686 −0.0583432 0.998297i $$-0.518582\pi$$
−0.0583432 + 0.998297i $$0.518582\pi$$
$$662$$ 0 0
$$663$$ 10.0000 0.388368
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 0 0
$$668$$ 0 0
$$669$$ 16.0000 0.618596
$$670$$ 0 0
$$671$$ 12.0000 0.463255
$$672$$ 0 0
$$673$$ −29.0000 −1.11787 −0.558934 0.829212i $$-0.688789\pi$$
−0.558934 + 0.829212i $$0.688789\pi$$
$$674$$ 0 0
$$675$$ −11.0000 −0.423390
$$676$$ 0 0
$$677$$ 12.0000 0.461197 0.230599 0.973049i $$-0.425932\pi$$
0.230599 + 0.973049i $$0.425932\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ −22.0000 −0.843042
$$682$$ 0 0
$$683$$ 18.0000 0.688751 0.344375 0.938832i $$-0.388091\pi$$
0.344375 + 0.938832i $$0.388091\pi$$
$$684$$ 0 0
$$685$$ −48.0000 −1.83399
$$686$$ 0 0
$$687$$ −11.0000 −0.419676
$$688$$ 0 0
$$689$$ −30.0000 −1.14291
$$690$$ 0 0
$$691$$ 23.0000 0.874961 0.437481 0.899228i $$-0.355871\pi$$
0.437481 + 0.899228i $$0.355871\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 20.0000 0.758643
$$696$$ 0 0
$$697$$ −12.0000 −0.454532
$$698$$ 0 0
$$699$$ 8.00000 0.302588
$$700$$ 0 0
$$701$$ −22.0000 −0.830929 −0.415464 0.909610i $$-0.636381\pi$$
−0.415464 + 0.909610i $$0.636381\pi$$
$$702$$ 0 0
$$703$$ −3.00000 −0.113147
$$704$$ 0 0
$$705$$ −16.0000 −0.602595
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 6.00000 0.225335 0.112667 0.993633i $$-0.464061\pi$$
0.112667 + 0.993633i $$0.464061\pi$$
$$710$$ 0 0
$$711$$ −11.0000 −0.412532
$$712$$ 0 0
$$713$$ 18.0000 0.674105
$$714$$ 0 0
$$715$$ −120.000 −4.48775
$$716$$ 0 0
$$717$$ 2.00000 0.0746914
$$718$$ 0 0
$$719$$ −30.0000 −1.11881 −0.559406 0.828894i $$-0.688971\pi$$
−0.559406 + 0.828894i $$0.688971\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ −10.0000 −0.371904
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 41.0000 1.52061 0.760303 0.649569i $$-0.225051\pi$$
0.760303 + 0.649569i $$0.225051\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ 10.0000 0.369863
$$732$$ 0 0
$$733$$ −21.0000 −0.775653 −0.387826 0.921732i $$-0.626774\pi$$
−0.387826 + 0.921732i $$0.626774\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 42.0000 1.54709
$$738$$ 0 0
$$739$$ −51.0000 −1.87607 −0.938033 0.346547i $$-0.887354\pi$$
−0.938033 + 0.346547i $$0.887354\pi$$
$$740$$ 0 0
$$741$$ 5.00000 0.183680
$$742$$ 0 0
$$743$$ 18.0000 0.660356 0.330178 0.943919i $$-0.392891\pi$$
0.330178 + 0.943919i $$0.392891\pi$$
$$744$$ 0 0
$$745$$ 16.0000 0.586195
$$746$$ 0 0
$$747$$ 12.0000 0.439057
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 7.00000 0.255434 0.127717 0.991811i $$-0.459235\pi$$
0.127717 + 0.991811i $$0.459235\pi$$
$$752$$ 0 0
$$753$$ 14.0000 0.510188
$$754$$ 0 0
$$755$$ 32.0000 1.16460
$$756$$ 0 0
$$757$$ 26.0000 0.944986 0.472493 0.881334i $$-0.343354\pi$$
0.472493 + 0.881334i $$0.343354\pi$$
$$758$$ 0 0
$$759$$ −36.0000 −1.30672
$$760$$ 0 0
$$761$$ 50.0000 1.81250 0.906249 0.422744i $$-0.138933\pi$$
0.906249 + 0.422744i $$0.138933\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ 8.00000 0.289241
$$766$$ 0 0
$$767$$ 30.0000 1.08324
$$768$$ 0 0
$$769$$ 31.0000 1.11789 0.558944 0.829205i $$-0.311207\pi$$
0.558944 + 0.829205i $$0.311207\pi$$
$$770$$ 0 0
$$771$$ 12.0000 0.432169
$$772$$ 0 0
$$773$$ 22.0000 0.791285 0.395643 0.918405i $$-0.370522\pi$$
0.395643 + 0.918405i $$0.370522\pi$$
$$774$$ 0 0
$$775$$ 33.0000 1.18539
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ −6.00000 −0.214972
$$780$$ 0 0
$$781$$ −96.0000 −3.43515
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ −40.0000 −1.42766
$$786$$ 0 0
$$787$$ 32.0000 1.14068 0.570338 0.821410i $$-0.306812\pi$$
0.570338 + 0.821410i $$0.306812\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ −10.0000 −0.355110
$$794$$ 0 0
$$795$$ −24.0000 −0.851192
$$796$$ 0 0
$$797$$ 46.0000 1.62940 0.814702 0.579880i $$-0.196901\pi$$
0.814702 + 0.579880i $$0.196901\pi$$
$$798$$ 0 0
$$799$$ 8.00000 0.283020
$$800$$ 0 0
$$801$$ 4.00000 0.141333
$$802$$ 0 0
$$803$$ −18.0000 −0.635206
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ −2.00000 −0.0704033
$$808$$ 0 0
$$809$$ 8.00000 0.281265 0.140633 0.990062i $$-0.455086\pi$$
0.140633 + 0.990062i $$0.455086\pi$$
$$810$$ 0 0
$$811$$ 28.0000 0.983213 0.491606 0.870817i $$-0.336410\pi$$
0.491606 + 0.870817i $$0.336410\pi$$
$$812$$ 0 0
$$813$$ 16.0000 0.561144
$$814$$ 0 0
$$815$$ 80.0000 2.80228
$$816$$ 0 0
$$817$$ 5.00000 0.174928
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −4.00000 −0.139601 −0.0698005 0.997561i $$-0.522236\pi$$
−0.0698005 + 0.997561i $$0.522236\pi$$
$$822$$ 0 0
$$823$$ 24.0000 0.836587 0.418294 0.908312i $$-0.362628\pi$$
0.418294 + 0.908312i $$0.362628\pi$$
$$824$$ 0 0
$$825$$ −66.0000 −2.29783
$$826$$ 0 0
$$827$$ 12.0000 0.417281 0.208640 0.977992i $$-0.433096\pi$$
0.208640 + 0.977992i $$0.433096\pi$$
$$828$$ 0 0
$$829$$ −5.00000 −0.173657 −0.0868286 0.996223i $$-0.527673\pi$$
−0.0868286 + 0.996223i $$0.527673\pi$$
$$830$$ 0 0
$$831$$ 1.00000 0.0346896
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 0 0
$$835$$ 32.0000 1.10741
$$836$$ 0 0
$$837$$ −3.00000 −0.103695
$$838$$ 0 0
$$839$$ −12.0000 −0.414286 −0.207143 0.978311i $$-0.566417\pi$$
−0.207143 + 0.978311i $$0.566417\pi$$
$$840$$ 0 0
$$841$$ −29.0000 −1.00000
$$842$$ 0 0
$$843$$ −6.00000 −0.206651
$$844$$ 0 0
$$845$$ 48.0000 1.65125
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ 11.0000 0.377519
$$850$$ 0 0
$$851$$ −18.0000 −0.617032
$$852$$ 0 0
$$853$$ 33.0000 1.12990 0.564949 0.825126i $$-0.308896\pi$$
0.564949 + 0.825126i $$0.308896\pi$$
$$854$$ 0 0
$$855$$ 4.00000 0.136797
$$856$$ 0 0
$$857$$ 24.0000 0.819824 0.409912 0.912125i $$-0.365559\pi$$
0.409912 + 0.912125i $$0.365559\pi$$
$$858$$ 0 0
$$859$$ 52.0000 1.77422 0.887109 0.461561i $$-0.152710\pi$$
0.887109 + 0.461561i $$0.152710\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ −54.0000 −1.83818 −0.919091 0.394046i $$-0.871075\pi$$
−0.919091 + 0.394046i $$0.871075\pi$$
$$864$$ 0 0
$$865$$ −88.0000 −2.99209
$$866$$ 0 0
$$867$$ 13.0000 0.441503
$$868$$ 0 0
$$869$$ −66.0000 −2.23890
$$870$$ 0 0
$$871$$ −35.0000 −1.18593
$$872$$ 0 0
$$873$$ −6.00000 −0.203069
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −2.00000 −0.0675352 −0.0337676 0.999430i $$-0.510751\pi$$
−0.0337676 + 0.999430i $$0.510751\pi$$
$$878$$ 0 0
$$879$$ −12.0000 −0.404750
$$880$$ 0 0
$$881$$ −42.0000 −1.41502 −0.707508 0.706705i $$-0.750181\pi$$
−0.707508 + 0.706705i $$0.750181\pi$$
$$882$$ 0 0
$$883$$ 7.00000 0.235569 0.117784 0.993039i $$-0.462421\pi$$
0.117784 + 0.993039i $$0.462421\pi$$
$$884$$ 0 0
$$885$$ 24.0000 0.806751
$$886$$ 0 0
$$887$$ −34.0000 −1.14161 −0.570804 0.821086i $$-0.693368\pi$$
−0.570804 + 0.821086i $$0.693368\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ 6.00000 0.201008
$$892$$ 0 0
$$893$$ 4.00000 0.133855
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 30.0000 1.00167
$$898$$ 0 0
$$899$$ 0 0
$$900$$ 0 0
$$901$$ 12.0000 0.399778
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ −100.000 −3.32411
$$906$$ 0 0
$$907$$ 43.0000 1.42779 0.713896 0.700252i $$-0.246929\pi$$
0.713896 + 0.700252i $$0.246929\pi$$
$$908$$ 0 0
$$909$$ −2.00000 −0.0663358
$$910$$ 0 0
$$911$$ 26.0000 0.861418 0.430709 0.902491i $$-0.358263\pi$$
0.430709 + 0.902491i $$0.358263\pi$$
$$912$$ 0 0
$$913$$ 72.0000 2.38285
$$914$$ 0 0
$$915$$ −8.00000 −0.264472
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ −43.0000 −1.41844 −0.709220 0.704988i $$-0.750953\pi$$
−0.709220 + 0.704988i $$0.750953\pi$$
$$920$$ 0 0
$$921$$ −11.0000 −0.362462
$$922$$ 0 0
$$923$$ 80.0000 2.63323
$$924$$ 0 0
$$925$$ −33.0000 −1.08503
$$926$$ 0 0
$$927$$ 11.0000 0.361287
$$928$$ 0 0
$$929$$ −36.0000 −1.18112 −0.590561 0.806993i $$-0.701093\pi$$
−0.590561 + 0.806993i $$0.701093\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 0 0
$$933$$ −2.00000 −0.0654771
$$934$$ 0 0
$$935$$ 48.0000 1.56977
$$936$$ 0 0
$$937$$ 51.0000 1.66610 0.833049 0.553200i $$-0.186593\pi$$
0.833049 + 0.553200i $$0.186593\pi$$
$$938$$ 0 0
$$939$$ −31.0000 −1.01165
$$940$$ 0 0
$$941$$ −54.0000 −1.76035 −0.880175 0.474650i $$-0.842575\pi$$
−0.880175 + 0.474650i $$0.842575\pi$$
$$942$$ 0 0
$$943$$ −36.0000 −1.17232
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −30.0000 −0.974869 −0.487435 0.873160i $$-0.662067\pi$$
−0.487435 + 0.873160i $$0.662067\pi$$
$$948$$ 0 0
$$949$$ 15.0000 0.486921
$$950$$ 0 0
$$951$$ −20.0000 −0.648544
$$952$$ 0 0
$$953$$ −6.00000 −0.194359 −0.0971795 0.995267i $$-0.530982\pi$$
−0.0971795 + 0.995267i $$0.530982\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −22.0000 −0.709677
$$962$$ 0 0
$$963$$ 10.0000 0.322245
$$964$$ 0 0
$$965$$ −4.00000 −0.128765
$$966$$ 0 0
$$967$$ 53.0000 1.70437 0.852183 0.523245i $$-0.175279\pi$$
0.852183 + 0.523245i $$0.175279\pi$$
$$968$$ 0 0
$$969$$ −2.00000 −0.0642493
$$970$$ 0 0
$$971$$ −26.0000 −0.834380 −0.417190 0.908819i $$-0.636985\pi$$
−0.417190 + 0.908819i $$0.636985\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 0 0
$$975$$ 55.0000 1.76141
$$976$$ 0 0
$$977$$ 30.0000 0.959785 0.479893 0.877327i $$-0.340676\pi$$
0.479893 + 0.877327i $$0.340676\pi$$
$$978$$ 0 0
$$979$$ 24.0000 0.767043
$$980$$ 0 0
$$981$$ 15.0000 0.478913
$$982$$ 0 0
$$983$$ −26.0000 −0.829271 −0.414636 0.909988i $$-0.636091\pi$$
−0.414636 + 0.909988i $$0.636091\pi$$
$$984$$ 0 0
$$985$$ −72.0000 −2.29411
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 30.0000 0.953945
$$990$$ 0 0
$$991$$ −25.0000 −0.794151 −0.397076 0.917786i $$-0.629975\pi$$
−0.397076 + 0.917786i $$0.629975\pi$$
$$992$$ 0 0
$$993$$ 5.00000 0.158670
$$994$$ 0 0
$$995$$ −80.0000 −2.53617
$$996$$ 0 0
$$997$$ 5.00000 0.158352 0.0791758 0.996861i $$-0.474771\pi$$
0.0791758 + 0.996861i $$0.474771\pi$$
$$998$$ 0 0
$$999$$ 3.00000 0.0949158
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 9408.2.a.bp.1.1 1
4.3 odd 2 9408.2.a.dd.1.1 1
7.2 even 3 1344.2.q.l.193.1 2
7.4 even 3 1344.2.q.l.961.1 2
7.6 odd 2 9408.2.a.bs.1.1 1
8.3 odd 2 4704.2.a.a.1.1 1
8.5 even 2 4704.2.a.r.1.1 1
28.11 odd 6 1344.2.q.a.961.1 2
28.23 odd 6 1344.2.q.a.193.1 2
28.27 even 2 9408.2.a.a.1.1 1
56.11 odd 6 672.2.q.j.289.1 yes 2
56.13 odd 2 4704.2.a.p.1.1 1
56.27 even 2 4704.2.a.bh.1.1 1
56.37 even 6 672.2.q.e.193.1 2
56.51 odd 6 672.2.q.j.193.1 yes 2
56.53 even 6 672.2.q.e.289.1 yes 2
168.11 even 6 2016.2.s.a.289.1 2
168.53 odd 6 2016.2.s.b.289.1 2
168.107 even 6 2016.2.s.a.865.1 2
168.149 odd 6 2016.2.s.b.865.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
672.2.q.e.193.1 2 56.37 even 6
672.2.q.e.289.1 yes 2 56.53 even 6
672.2.q.j.193.1 yes 2 56.51 odd 6
672.2.q.j.289.1 yes 2 56.11 odd 6
1344.2.q.a.193.1 2 28.23 odd 6
1344.2.q.a.961.1 2 28.11 odd 6
1344.2.q.l.193.1 2 7.2 even 3
1344.2.q.l.961.1 2 7.4 even 3
2016.2.s.a.289.1 2 168.11 even 6
2016.2.s.a.865.1 2 168.107 even 6
2016.2.s.b.289.1 2 168.53 odd 6
2016.2.s.b.865.1 2 168.149 odd 6
4704.2.a.a.1.1 1 8.3 odd 2
4704.2.a.p.1.1 1 56.13 odd 2
4704.2.a.r.1.1 1 8.5 even 2
4704.2.a.bh.1.1 1 56.27 even 2
9408.2.a.a.1.1 1 28.27 even 2
9408.2.a.bp.1.1 1 1.1 even 1 trivial
9408.2.a.bs.1.1 1 7.6 odd 2
9408.2.a.dd.1.1 1 4.3 odd 2