# Properties

 Label 5376.2.c.r.2689.2 Level $5376$ Weight $2$ Character 5376.2689 Analytic conductor $42.928$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$5376 = 2^{8} \cdot 3 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5376.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$42.9275761266$$ 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: no (minimal twist has level 21) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 2689.2 Root $$1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 5376.2689 Dual form 5376.2.c.r.2689.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000i q^{3} +2.00000i q^{5} +1.00000 q^{7} -1.00000 q^{9} +O(q^{10})$$ $$q+1.00000i q^{3} +2.00000i q^{5} +1.00000 q^{7} -1.00000 q^{9} -4.00000i q^{11} -2.00000i q^{13} -2.00000 q^{15} -6.00000 q^{17} +4.00000i q^{19} +1.00000i q^{21} +1.00000 q^{25} -1.00000i q^{27} -2.00000i q^{29} +4.00000 q^{33} +2.00000i q^{35} -6.00000i q^{37} +2.00000 q^{39} -2.00000 q^{41} +4.00000i q^{43} -2.00000i q^{45} +1.00000 q^{49} -6.00000i q^{51} -6.00000i q^{53} +8.00000 q^{55} -4.00000 q^{57} -12.0000i q^{59} -2.00000i q^{61} -1.00000 q^{63} +4.00000 q^{65} +4.00000i q^{67} +6.00000 q^{73} +1.00000i q^{75} -4.00000i q^{77} -16.0000 q^{79} +1.00000 q^{81} -12.0000i q^{83} -12.0000i q^{85} +2.00000 q^{87} +14.0000 q^{89} -2.00000i q^{91} -8.00000 q^{95} +18.0000 q^{97} +4.00000i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{7} - 2q^{9} + O(q^{10})$$ $$2q + 2q^{7} - 2q^{9} - 4q^{15} - 12q^{17} + 2q^{25} + 8q^{33} + 4q^{39} - 4q^{41} + 2q^{49} + 16q^{55} - 8q^{57} - 2q^{63} + 8q^{65} + 12q^{73} - 32q^{79} + 2q^{81} + 4q^{87} + 28q^{89} - 16q^{95} + 36q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/5376\mathbb{Z}\right)^\times$$.

 $$n$$ $$1793$$ $$2815$$ $$4609$$ $$5125$$ $$\chi(n)$$ $$1$$ $$1$$ $$1$$ $$-1$$

## 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
$$5$$ 2.00000i 0.894427i 0.894427 + 0.447214i $$0.147584\pi$$
−0.894427 + 0.447214i $$0.852416\pi$$
$$6$$ 0 0
$$7$$ 1.00000 0.377964
$$8$$ 0 0
$$9$$ −1.00000 −0.333333
$$10$$ 0 0
$$11$$ − 4.00000i − 1.20605i −0.797724 0.603023i $$-0.793963\pi$$
0.797724 0.603023i $$-0.206037\pi$$
$$12$$ 0 0
$$13$$ − 2.00000i − 0.554700i −0.960769 0.277350i $$-0.910544\pi$$
0.960769 0.277350i $$-0.0894562\pi$$
$$14$$ 0 0
$$15$$ −2.00000 −0.516398
$$16$$ 0 0
$$17$$ −6.00000 −1.45521 −0.727607 0.685994i $$-0.759367\pi$$
−0.727607 + 0.685994i $$0.759367\pi$$
$$18$$ 0 0
$$19$$ 4.00000i 0.917663i 0.888523 + 0.458831i $$0.151732\pi$$
−0.888523 + 0.458831i $$0.848268\pi$$
$$20$$ 0 0
$$21$$ 1.00000i 0.218218i
$$22$$ 0 0
$$23$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ − 1.00000i − 0.192450i
$$28$$ 0 0
$$29$$ − 2.00000i − 0.371391i −0.982607 0.185695i $$-0.940546\pi$$
0.982607 0.185695i $$-0.0594537\pi$$
$$30$$ 0 0
$$31$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$32$$ 0 0
$$33$$ 4.00000 0.696311
$$34$$ 0 0
$$35$$ 2.00000i 0.338062i
$$36$$ 0 0
$$37$$ − 6.00000i − 0.986394i −0.869918 0.493197i $$-0.835828\pi$$
0.869918 0.493197i $$-0.164172\pi$$
$$38$$ 0 0
$$39$$ 2.00000 0.320256
$$40$$ 0 0
$$41$$ −2.00000 −0.312348 −0.156174 0.987730i $$-0.549916\pi$$
−0.156174 + 0.987730i $$0.549916\pi$$
$$42$$ 0 0
$$43$$ 4.00000i 0.609994i 0.952353 + 0.304997i $$0.0986555\pi$$
−0.952353 + 0.304997i $$0.901344\pi$$
$$44$$ 0 0
$$45$$ − 2.00000i − 0.298142i
$$46$$ 0 0
$$47$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$48$$ 0 0
$$49$$ 1.00000 0.142857
$$50$$ 0 0
$$51$$ − 6.00000i − 0.840168i
$$52$$ 0 0
$$53$$ − 6.00000i − 0.824163i −0.911147 0.412082i $$-0.864802\pi$$
0.911147 0.412082i $$-0.135198\pi$$
$$54$$ 0 0
$$55$$ 8.00000 1.07872
$$56$$ 0 0
$$57$$ −4.00000 −0.529813
$$58$$ 0 0
$$59$$ − 12.0000i − 1.56227i −0.624364 0.781133i $$-0.714642\pi$$
0.624364 0.781133i $$-0.285358\pi$$
$$60$$ 0 0
$$61$$ − 2.00000i − 0.256074i −0.991769 0.128037i $$-0.959132\pi$$
0.991769 0.128037i $$-0.0408676\pi$$
$$62$$ 0 0
$$63$$ −1.00000 −0.125988
$$64$$ 0 0
$$65$$ 4.00000 0.496139
$$66$$ 0 0
$$67$$ 4.00000i 0.488678i 0.969690 + 0.244339i $$0.0785709\pi$$
−0.969690 + 0.244339i $$0.921429\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$72$$ 0 0
$$73$$ 6.00000 0.702247 0.351123 0.936329i $$-0.385800\pi$$
0.351123 + 0.936329i $$0.385800\pi$$
$$74$$ 0 0
$$75$$ 1.00000i 0.115470i
$$76$$ 0 0
$$77$$ − 4.00000i − 0.455842i
$$78$$ 0 0
$$79$$ −16.0000 −1.80014 −0.900070 0.435745i $$-0.856485\pi$$
−0.900070 + 0.435745i $$0.856485\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ − 12.0000i − 1.31717i −0.752506 0.658586i $$-0.771155\pi$$
0.752506 0.658586i $$-0.228845\pi$$
$$84$$ 0 0
$$85$$ − 12.0000i − 1.30158i
$$86$$ 0 0
$$87$$ 2.00000 0.214423
$$88$$ 0 0
$$89$$ 14.0000 1.48400 0.741999 0.670402i $$-0.233878\pi$$
0.741999 + 0.670402i $$0.233878\pi$$
$$90$$ 0 0
$$91$$ − 2.00000i − 0.209657i
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ −8.00000 −0.820783
$$96$$ 0 0
$$97$$ 18.0000 1.82762 0.913812 0.406138i $$-0.133125\pi$$
0.913812 + 0.406138i $$0.133125\pi$$
$$98$$ 0 0
$$99$$ 4.00000i 0.402015i
$$100$$ 0 0
$$101$$ − 14.0000i − 1.39305i −0.717532 0.696526i $$-0.754728\pi$$
0.717532 0.696526i $$-0.245272\pi$$
$$102$$ 0 0
$$103$$ −8.00000 −0.788263 −0.394132 0.919054i $$-0.628955\pi$$
−0.394132 + 0.919054i $$0.628955\pi$$
$$104$$ 0 0
$$105$$ −2.00000 −0.195180
$$106$$ 0 0
$$107$$ − 4.00000i − 0.386695i −0.981130 0.193347i $$-0.938066\pi$$
0.981130 0.193347i $$-0.0619344\pi$$
$$108$$ 0 0
$$109$$ − 18.0000i − 1.72409i −0.506834 0.862044i $$-0.669184\pi$$
0.506834 0.862044i $$-0.330816\pi$$
$$110$$ 0 0
$$111$$ 6.00000 0.569495
$$112$$ 0 0
$$113$$ −14.0000 −1.31701 −0.658505 0.752577i $$-0.728811\pi$$
−0.658505 + 0.752577i $$0.728811\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 2.00000i 0.184900i
$$118$$ 0 0
$$119$$ −6.00000 −0.550019
$$120$$ 0 0
$$121$$ −5.00000 −0.454545
$$122$$ 0 0
$$123$$ − 2.00000i − 0.180334i
$$124$$ 0 0
$$125$$ 12.0000i 1.07331i
$$126$$ 0 0
$$127$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$128$$ 0 0
$$129$$ −4.00000 −0.352180
$$130$$ 0 0
$$131$$ 4.00000i 0.349482i 0.984614 + 0.174741i $$0.0559088\pi$$
−0.984614 + 0.174741i $$0.944091\pi$$
$$132$$ 0 0
$$133$$ 4.00000i 0.346844i
$$134$$ 0 0
$$135$$ 2.00000 0.172133
$$136$$ 0 0
$$137$$ 6.00000 0.512615 0.256307 0.966595i $$-0.417494\pi$$
0.256307 + 0.966595i $$0.417494\pi$$
$$138$$ 0 0
$$139$$ − 12.0000i − 1.01783i −0.860818 0.508913i $$-0.830047\pi$$
0.860818 0.508913i $$-0.169953\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ −8.00000 −0.668994
$$144$$ 0 0
$$145$$ 4.00000 0.332182
$$146$$ 0 0
$$147$$ 1.00000i 0.0824786i
$$148$$ 0 0
$$149$$ − 6.00000i − 0.491539i −0.969328 0.245770i $$-0.920959\pi$$
0.969328 0.245770i $$-0.0790407\pi$$
$$150$$ 0 0
$$151$$ −8.00000 −0.651031 −0.325515 0.945537i $$-0.605538\pi$$
−0.325515 + 0.945537i $$0.605538\pi$$
$$152$$ 0 0
$$153$$ 6.00000 0.485071
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ − 2.00000i − 0.159617i −0.996810 0.0798087i $$-0.974569\pi$$
0.996810 0.0798087i $$-0.0254309\pi$$
$$158$$ 0 0
$$159$$ 6.00000 0.475831
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 4.00000i 0.313304i 0.987654 + 0.156652i $$0.0500701\pi$$
−0.987654 + 0.156652i $$0.949930\pi$$
$$164$$ 0 0
$$165$$ 8.00000i 0.622799i
$$166$$ 0 0
$$167$$ 8.00000 0.619059 0.309529 0.950890i $$-0.399829\pi$$
0.309529 + 0.950890i $$0.399829\pi$$
$$168$$ 0 0
$$169$$ 9.00000 0.692308
$$170$$ 0 0
$$171$$ − 4.00000i − 0.305888i
$$172$$ 0 0
$$173$$ − 10.0000i − 0.760286i −0.924928 0.380143i $$-0.875875\pi$$
0.924928 0.380143i $$-0.124125\pi$$
$$174$$ 0 0
$$175$$ 1.00000 0.0755929
$$176$$ 0 0
$$177$$ 12.0000 0.901975
$$178$$ 0 0
$$179$$ − 4.00000i − 0.298974i −0.988764 0.149487i $$-0.952238\pi$$
0.988764 0.149487i $$-0.0477622\pi$$
$$180$$ 0 0
$$181$$ 26.0000i 1.93256i 0.257485 + 0.966282i $$0.417106\pi$$
−0.257485 + 0.966282i $$0.582894\pi$$
$$182$$ 0 0
$$183$$ 2.00000 0.147844
$$184$$ 0 0
$$185$$ 12.0000 0.882258
$$186$$ 0 0
$$187$$ 24.0000i 1.75505i
$$188$$ 0 0
$$189$$ − 1.00000i − 0.0727393i
$$190$$ 0 0
$$191$$ −8.00000 −0.578860 −0.289430 0.957199i $$-0.593466\pi$$
−0.289430 + 0.957199i $$0.593466\pi$$
$$192$$ 0 0
$$193$$ 2.00000 0.143963 0.0719816 0.997406i $$-0.477068\pi$$
0.0719816 + 0.997406i $$0.477068\pi$$
$$194$$ 0 0
$$195$$ 4.00000i 0.286446i
$$196$$ 0 0
$$197$$ − 22.0000i − 1.56744i −0.621117 0.783718i $$-0.713321\pi$$
0.621117 0.783718i $$-0.286679\pi$$
$$198$$ 0 0
$$199$$ −24.0000 −1.70131 −0.850657 0.525720i $$-0.823796\pi$$
−0.850657 + 0.525720i $$0.823796\pi$$
$$200$$ 0 0
$$201$$ −4.00000 −0.282138
$$202$$ 0 0
$$203$$ − 2.00000i − 0.140372i
$$204$$ 0 0
$$205$$ − 4.00000i − 0.279372i
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 16.0000 1.10674
$$210$$ 0 0
$$211$$ 4.00000i 0.275371i 0.990476 + 0.137686i $$0.0439664\pi$$
−0.990476 + 0.137686i $$0.956034\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ −8.00000 −0.545595
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ 6.00000i 0.405442i
$$220$$ 0 0
$$221$$ 12.0000i 0.807207i
$$222$$ 0 0
$$223$$ 16.0000 1.07144 0.535720 0.844396i $$-0.320040\pi$$
0.535720 + 0.844396i $$0.320040\pi$$
$$224$$ 0 0
$$225$$ −1.00000 −0.0666667
$$226$$ 0 0
$$227$$ − 12.0000i − 0.796468i −0.917284 0.398234i $$-0.869623\pi$$
0.917284 0.398234i $$-0.130377\pi$$
$$228$$ 0 0
$$229$$ 10.0000i 0.660819i 0.943838 + 0.330409i $$0.107187\pi$$
−0.943838 + 0.330409i $$0.892813\pi$$
$$230$$ 0 0
$$231$$ 4.00000 0.263181
$$232$$ 0 0
$$233$$ 6.00000 0.393073 0.196537 0.980497i $$-0.437031\pi$$
0.196537 + 0.980497i $$0.437031\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ − 16.0000i − 1.03931i
$$238$$ 0 0
$$239$$ 24.0000 1.55243 0.776215 0.630468i $$-0.217137\pi$$
0.776215 + 0.630468i $$0.217137\pi$$
$$240$$ 0 0
$$241$$ 2.00000 0.128831 0.0644157 0.997923i $$-0.479482\pi$$
0.0644157 + 0.997923i $$0.479482\pi$$
$$242$$ 0 0
$$243$$ 1.00000i 0.0641500i
$$244$$ 0 0
$$245$$ 2.00000i 0.127775i
$$246$$ 0 0
$$247$$ 8.00000 0.509028
$$248$$ 0 0
$$249$$ 12.0000 0.760469
$$250$$ 0 0
$$251$$ 20.0000i 1.26239i 0.775625 + 0.631194i $$0.217435\pi$$
−0.775625 + 0.631194i $$0.782565\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 0 0
$$255$$ 12.0000 0.751469
$$256$$ 0 0
$$257$$ 26.0000 1.62184 0.810918 0.585160i $$-0.198968\pi$$
0.810918 + 0.585160i $$0.198968\pi$$
$$258$$ 0 0
$$259$$ − 6.00000i − 0.372822i
$$260$$ 0 0
$$261$$ 2.00000i 0.123797i
$$262$$ 0 0
$$263$$ −16.0000 −0.986602 −0.493301 0.869859i $$-0.664210\pi$$
−0.493301 + 0.869859i $$0.664210\pi$$
$$264$$ 0 0
$$265$$ 12.0000 0.737154
$$266$$ 0 0
$$267$$ 14.0000i 0.856786i
$$268$$ 0 0
$$269$$ 6.00000i 0.365826i 0.983129 + 0.182913i $$0.0585527\pi$$
−0.983129 + 0.182913i $$0.941447\pi$$
$$270$$ 0 0
$$271$$ 16.0000 0.971931 0.485965 0.873978i $$-0.338468\pi$$
0.485965 + 0.873978i $$0.338468\pi$$
$$272$$ 0 0
$$273$$ 2.00000 0.121046
$$274$$ 0 0
$$275$$ − 4.00000i − 0.241209i
$$276$$ 0 0
$$277$$ − 22.0000i − 1.32185i −0.750451 0.660926i $$-0.770164\pi$$
0.750451 0.660926i $$-0.229836\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 22.0000 1.31241 0.656205 0.754583i $$-0.272161\pi$$
0.656205 + 0.754583i $$0.272161\pi$$
$$282$$ 0 0
$$283$$ 20.0000i 1.18888i 0.804141 + 0.594438i $$0.202626\pi$$
−0.804141 + 0.594438i $$0.797374\pi$$
$$284$$ 0 0
$$285$$ − 8.00000i − 0.473879i
$$286$$ 0 0
$$287$$ −2.00000 −0.118056
$$288$$ 0 0
$$289$$ 19.0000 1.11765
$$290$$ 0 0
$$291$$ 18.0000i 1.05518i
$$292$$ 0 0
$$293$$ − 14.0000i − 0.817889i −0.912559 0.408944i $$-0.865897\pi$$
0.912559 0.408944i $$-0.134103\pi$$
$$294$$ 0 0
$$295$$ 24.0000 1.39733
$$296$$ 0 0
$$297$$ −4.00000 −0.232104
$$298$$ 0 0
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 4.00000i 0.230556i
$$302$$ 0 0
$$303$$ 14.0000 0.804279
$$304$$ 0 0
$$305$$ 4.00000 0.229039
$$306$$ 0 0
$$307$$ 4.00000i 0.228292i 0.993464 + 0.114146i $$0.0364132\pi$$
−0.993464 + 0.114146i $$0.963587\pi$$
$$308$$ 0 0
$$309$$ − 8.00000i − 0.455104i
$$310$$ 0 0
$$311$$ 24.0000 1.36092 0.680458 0.732787i $$-0.261781\pi$$
0.680458 + 0.732787i $$0.261781\pi$$
$$312$$ 0 0
$$313$$ −26.0000 −1.46961 −0.734803 0.678280i $$-0.762726\pi$$
−0.734803 + 0.678280i $$0.762726\pi$$
$$314$$ 0 0
$$315$$ − 2.00000i − 0.112687i
$$316$$ 0 0
$$317$$ − 18.0000i − 1.01098i −0.862832 0.505490i $$-0.831312\pi$$
0.862832 0.505490i $$-0.168688\pi$$
$$318$$ 0 0
$$319$$ −8.00000 −0.447914
$$320$$ 0 0
$$321$$ 4.00000 0.223258
$$322$$ 0 0
$$323$$ − 24.0000i − 1.33540i
$$324$$ 0 0
$$325$$ − 2.00000i − 0.110940i
$$326$$ 0 0
$$327$$ 18.0000 0.995402
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 4.00000i 0.219860i 0.993939 + 0.109930i $$0.0350627\pi$$
−0.993939 + 0.109930i $$0.964937\pi$$
$$332$$ 0 0
$$333$$ 6.00000i 0.328798i
$$334$$ 0 0
$$335$$ −8.00000 −0.437087
$$336$$ 0 0
$$337$$ −14.0000 −0.762629 −0.381314 0.924445i $$-0.624528\pi$$
−0.381314 + 0.924445i $$0.624528\pi$$
$$338$$ 0 0
$$339$$ − 14.0000i − 0.760376i
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ 1.00000 0.0539949
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 28.0000i 1.50312i 0.659665 + 0.751559i $$0.270698\pi$$
−0.659665 + 0.751559i $$0.729302\pi$$
$$348$$ 0 0
$$349$$ − 2.00000i − 0.107058i −0.998566 0.0535288i $$-0.982953\pi$$
0.998566 0.0535288i $$-0.0170469\pi$$
$$350$$ 0 0
$$351$$ −2.00000 −0.106752
$$352$$ 0 0
$$353$$ 10.0000 0.532246 0.266123 0.963939i $$-0.414257\pi$$
0.266123 + 0.963939i $$0.414257\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ − 6.00000i − 0.317554i
$$358$$ 0 0
$$359$$ −32.0000 −1.68890 −0.844448 0.535638i $$-0.820071\pi$$
−0.844448 + 0.535638i $$0.820071\pi$$
$$360$$ 0 0
$$361$$ 3.00000 0.157895
$$362$$ 0 0
$$363$$ − 5.00000i − 0.262432i
$$364$$ 0 0
$$365$$ 12.0000i 0.628109i
$$366$$ 0 0
$$367$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$368$$ 0 0
$$369$$ 2.00000 0.104116
$$370$$ 0 0
$$371$$ − 6.00000i − 0.311504i
$$372$$ 0 0
$$373$$ 10.0000i 0.517780i 0.965907 + 0.258890i $$0.0833568\pi$$
−0.965907 + 0.258890i $$0.916643\pi$$
$$374$$ 0 0
$$375$$ −12.0000 −0.619677
$$376$$ 0 0
$$377$$ −4.00000 −0.206010
$$378$$ 0 0
$$379$$ − 12.0000i − 0.616399i −0.951322 0.308199i $$-0.900274\pi$$
0.951322 0.308199i $$-0.0997264\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$384$$ 0 0
$$385$$ 8.00000 0.407718
$$386$$ 0 0
$$387$$ − 4.00000i − 0.203331i
$$388$$ 0 0
$$389$$ − 6.00000i − 0.304212i −0.988364 0.152106i $$-0.951394\pi$$
0.988364 0.152106i $$-0.0486055\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 0 0
$$393$$ −4.00000 −0.201773
$$394$$ 0 0
$$395$$ − 32.0000i − 1.61009i
$$396$$ 0 0
$$397$$ − 18.0000i − 0.903394i −0.892171 0.451697i $$-0.850819\pi$$
0.892171 0.451697i $$-0.149181\pi$$
$$398$$ 0 0
$$399$$ −4.00000 −0.200250
$$400$$ 0 0
$$401$$ −30.0000 −1.49813 −0.749064 0.662497i $$-0.769497\pi$$
−0.749064 + 0.662497i $$0.769497\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 0 0
$$405$$ 2.00000i 0.0993808i
$$406$$ 0 0
$$407$$ −24.0000 −1.18964
$$408$$ 0 0
$$409$$ 22.0000 1.08783 0.543915 0.839140i $$-0.316941\pi$$
0.543915 + 0.839140i $$0.316941\pi$$
$$410$$ 0 0
$$411$$ 6.00000i 0.295958i
$$412$$ 0 0
$$413$$ − 12.0000i − 0.590481i
$$414$$ 0 0
$$415$$ 24.0000 1.17811
$$416$$ 0 0
$$417$$ 12.0000 0.587643
$$418$$ 0 0
$$419$$ − 12.0000i − 0.586238i −0.956076 0.293119i $$-0.905307\pi$$
0.956076 0.293119i $$-0.0946933\pi$$
$$420$$ 0 0
$$421$$ − 38.0000i − 1.85201i −0.377515 0.926003i $$-0.623221\pi$$
0.377515 0.926003i $$-0.376779\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ −6.00000 −0.291043
$$426$$ 0 0
$$427$$ − 2.00000i − 0.0967868i
$$428$$ 0 0
$$429$$ − 8.00000i − 0.386244i
$$430$$ 0 0
$$431$$ −24.0000 −1.15604 −0.578020 0.816023i $$-0.696174\pi$$
−0.578020 + 0.816023i $$0.696174\pi$$
$$432$$ 0 0
$$433$$ −14.0000 −0.672797 −0.336399 0.941720i $$-0.609209\pi$$
−0.336399 + 0.941720i $$0.609209\pi$$
$$434$$ 0 0
$$435$$ 4.00000i 0.191785i
$$436$$ 0 0
$$437$$ 0 0
$$438$$ 0 0
$$439$$ 24.0000 1.14546 0.572729 0.819745i $$-0.305885\pi$$
0.572729 + 0.819745i $$0.305885\pi$$
$$440$$ 0 0
$$441$$ −1.00000 −0.0476190
$$442$$ 0 0
$$443$$ − 36.0000i − 1.71041i −0.518289 0.855206i $$-0.673431\pi$$
0.518289 0.855206i $$-0.326569\pi$$
$$444$$ 0 0
$$445$$ 28.0000i 1.32733i
$$446$$ 0 0
$$447$$ 6.00000 0.283790
$$448$$ 0 0
$$449$$ −30.0000 −1.41579 −0.707894 0.706319i $$-0.750354\pi$$
−0.707894 + 0.706319i $$0.750354\pi$$
$$450$$ 0 0
$$451$$ 8.00000i 0.376705i
$$452$$ 0 0
$$453$$ − 8.00000i − 0.375873i
$$454$$ 0 0
$$455$$ 4.00000 0.187523
$$456$$ 0 0
$$457$$ −10.0000 −0.467780 −0.233890 0.972263i $$-0.575146\pi$$
−0.233890 + 0.972263i $$0.575146\pi$$
$$458$$ 0 0
$$459$$ 6.00000i 0.280056i
$$460$$ 0 0
$$461$$ − 10.0000i − 0.465746i −0.972507 0.232873i $$-0.925187\pi$$
0.972507 0.232873i $$-0.0748127\pi$$
$$462$$ 0 0
$$463$$ 16.0000 0.743583 0.371792 0.928316i $$-0.378744\pi$$
0.371792 + 0.928316i $$0.378744\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 36.0000i 1.66588i 0.553362 + 0.832941i $$0.313345\pi$$
−0.553362 + 0.832941i $$0.686655\pi$$
$$468$$ 0 0
$$469$$ 4.00000i 0.184703i
$$470$$ 0 0
$$471$$ 2.00000 0.0921551
$$472$$ 0 0
$$473$$ 16.0000 0.735681
$$474$$ 0 0
$$475$$ 4.00000i 0.183533i
$$476$$ 0 0
$$477$$ 6.00000i 0.274721i
$$478$$ 0 0
$$479$$ −16.0000 −0.731059 −0.365529 0.930800i $$-0.619112\pi$$
−0.365529 + 0.930800i $$0.619112\pi$$
$$480$$ 0 0
$$481$$ −12.0000 −0.547153
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 36.0000i 1.63468i
$$486$$ 0 0
$$487$$ 8.00000 0.362515 0.181257 0.983436i $$-0.441983\pi$$
0.181257 + 0.983436i $$0.441983\pi$$
$$488$$ 0 0
$$489$$ −4.00000 −0.180886
$$490$$ 0 0
$$491$$ − 20.0000i − 0.902587i −0.892375 0.451294i $$-0.850963\pi$$
0.892375 0.451294i $$-0.149037\pi$$
$$492$$ 0 0
$$493$$ 12.0000i 0.540453i
$$494$$ 0 0
$$495$$ −8.00000 −0.359573
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 4.00000i 0.179065i 0.995984 + 0.0895323i $$0.0285372\pi$$
−0.995984 + 0.0895323i $$0.971463\pi$$
$$500$$ 0 0
$$501$$ 8.00000i 0.357414i
$$502$$ 0 0
$$503$$ −24.0000 −1.07011 −0.535054 0.844818i $$-0.679709\pi$$
−0.535054 + 0.844818i $$0.679709\pi$$
$$504$$ 0 0
$$505$$ 28.0000 1.24598
$$506$$ 0 0
$$507$$ 9.00000i 0.399704i
$$508$$ 0 0
$$509$$ − 10.0000i − 0.443242i −0.975133 0.221621i $$-0.928865\pi$$
0.975133 0.221621i $$-0.0711348\pi$$
$$510$$ 0 0
$$511$$ 6.00000 0.265424
$$512$$ 0 0
$$513$$ 4.00000 0.176604
$$514$$ 0 0
$$515$$ − 16.0000i − 0.705044i
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 0 0
$$519$$ 10.0000 0.438951
$$520$$ 0 0
$$521$$ −18.0000 −0.788594 −0.394297 0.918983i $$-0.629012\pi$$
−0.394297 + 0.918983i $$0.629012\pi$$
$$522$$ 0 0
$$523$$ 20.0000i 0.874539i 0.899331 + 0.437269i $$0.144054\pi$$
−0.899331 + 0.437269i $$0.855946\pi$$
$$524$$ 0 0
$$525$$ 1.00000i 0.0436436i
$$526$$ 0 0
$$527$$ 0 0
$$528$$ 0 0
$$529$$ −23.0000 −1.00000
$$530$$ 0 0
$$531$$ 12.0000i 0.520756i
$$532$$ 0 0
$$533$$ 4.00000i 0.173259i
$$534$$ 0 0
$$535$$ 8.00000 0.345870
$$536$$ 0 0
$$537$$ 4.00000 0.172613
$$538$$ 0 0
$$539$$ − 4.00000i − 0.172292i
$$540$$ 0 0
$$541$$ − 34.0000i − 1.46177i −0.682498 0.730887i $$-0.739107\pi$$
0.682498 0.730887i $$-0.260893\pi$$
$$542$$ 0 0
$$543$$ −26.0000 −1.11577
$$544$$ 0 0
$$545$$ 36.0000 1.54207
$$546$$ 0 0
$$547$$ 4.00000i 0.171028i 0.996337 + 0.0855138i $$0.0272532\pi$$
−0.996337 + 0.0855138i $$0.972747\pi$$
$$548$$ 0 0
$$549$$ 2.00000i 0.0853579i
$$550$$ 0 0
$$551$$ 8.00000 0.340811
$$552$$ 0 0
$$553$$ −16.0000 −0.680389
$$554$$ 0 0
$$555$$ 12.0000i 0.509372i
$$556$$ 0 0
$$557$$ − 2.00000i − 0.0847427i −0.999102 0.0423714i $$-0.986509\pi$$
0.999102 0.0423714i $$-0.0134913\pi$$
$$558$$ 0 0
$$559$$ 8.00000 0.338364
$$560$$ 0 0
$$561$$ −24.0000 −1.01328
$$562$$ 0 0
$$563$$ 4.00000i 0.168580i 0.996441 + 0.0842900i $$0.0268622\pi$$
−0.996441 + 0.0842900i $$0.973138\pi$$
$$564$$ 0 0
$$565$$ − 28.0000i − 1.17797i
$$566$$ 0 0
$$567$$ 1.00000 0.0419961
$$568$$ 0 0
$$569$$ −10.0000 −0.419222 −0.209611 0.977785i $$-0.567220\pi$$
−0.209611 + 0.977785i $$0.567220\pi$$
$$570$$ 0 0
$$571$$ 4.00000i 0.167395i 0.996491 + 0.0836974i $$0.0266729\pi$$
−0.996491 + 0.0836974i $$0.973327\pi$$
$$572$$ 0 0
$$573$$ − 8.00000i − 0.334205i
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 34.0000 1.41544 0.707719 0.706494i $$-0.249724\pi$$
0.707719 + 0.706494i $$0.249724\pi$$
$$578$$ 0 0
$$579$$ 2.00000i 0.0831172i
$$580$$ 0 0
$$581$$ − 12.0000i − 0.497844i
$$582$$ 0 0
$$583$$ −24.0000 −0.993978
$$584$$ 0 0
$$585$$ −4.00000 −0.165380
$$586$$ 0 0
$$587$$ − 28.0000i − 1.15568i −0.816149 0.577842i $$-0.803895\pi$$
0.816149 0.577842i $$-0.196105\pi$$
$$588$$ 0 0
$$589$$ 0 0
$$590$$ 0 0
$$591$$ 22.0000 0.904959
$$592$$ 0 0
$$593$$ −6.00000 −0.246390 −0.123195 0.992382i $$-0.539314\pi$$
−0.123195 + 0.992382i $$0.539314\pi$$
$$594$$ 0 0
$$595$$ − 12.0000i − 0.491952i
$$596$$ 0 0
$$597$$ − 24.0000i − 0.982255i
$$598$$ 0 0
$$599$$ −48.0000 −1.96123 −0.980613 0.195952i $$-0.937220\pi$$
−0.980613 + 0.195952i $$0.937220\pi$$
$$600$$ 0 0
$$601$$ 6.00000 0.244745 0.122373 0.992484i $$-0.460950\pi$$
0.122373 + 0.992484i $$0.460950\pi$$
$$602$$ 0 0
$$603$$ − 4.00000i − 0.162893i
$$604$$ 0 0
$$605$$ − 10.0000i − 0.406558i
$$606$$ 0 0
$$607$$ −16.0000 −0.649420 −0.324710 0.945814i $$-0.605267\pi$$
−0.324710 + 0.945814i $$0.605267\pi$$
$$608$$ 0 0
$$609$$ 2.00000 0.0810441
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 0 0
$$613$$ 26.0000i 1.05013i 0.851062 + 0.525065i $$0.175959\pi$$
−0.851062 + 0.525065i $$0.824041\pi$$
$$614$$ 0 0
$$615$$ 4.00000 0.161296
$$616$$ 0 0
$$617$$ 6.00000 0.241551 0.120775 0.992680i $$-0.461462\pi$$
0.120775 + 0.992680i $$0.461462\pi$$
$$618$$ 0 0
$$619$$ 20.0000i 0.803868i 0.915669 + 0.401934i $$0.131662\pi$$
−0.915669 + 0.401934i $$0.868338\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 14.0000 0.560898
$$624$$ 0 0
$$625$$ −19.0000 −0.760000
$$626$$ 0 0
$$627$$ 16.0000i 0.638978i
$$628$$ 0 0
$$629$$ 36.0000i 1.43541i
$$630$$ 0 0
$$631$$ 40.0000 1.59237 0.796187 0.605050i $$-0.206847\pi$$
0.796187 + 0.605050i $$0.206847\pi$$
$$632$$ 0 0
$$633$$ −4.00000 −0.158986
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ − 2.00000i − 0.0792429i
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 18.0000 0.710957 0.355479 0.934684i $$-0.384318\pi$$
0.355479 + 0.934684i $$0.384318\pi$$
$$642$$ 0 0
$$643$$ 20.0000i 0.788723i 0.918955 + 0.394362i $$0.129034\pi$$
−0.918955 + 0.394362i $$0.870966\pi$$
$$644$$ 0 0
$$645$$ − 8.00000i − 0.315000i
$$646$$ 0 0
$$647$$ 40.0000 1.57256 0.786281 0.617869i $$-0.212004\pi$$
0.786281 + 0.617869i $$0.212004\pi$$
$$648$$ 0 0
$$649$$ −48.0000 −1.88416
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ − 18.0000i − 0.704394i −0.935926 0.352197i $$-0.885435\pi$$
0.935926 0.352197i $$-0.114565\pi$$
$$654$$ 0 0
$$655$$ −8.00000 −0.312586
$$656$$ 0 0
$$657$$ −6.00000 −0.234082
$$658$$ 0 0
$$659$$ 12.0000i 0.467454i 0.972302 + 0.233727i $$0.0750921\pi$$
−0.972302 + 0.233727i $$0.924908\pi$$
$$660$$ 0 0
$$661$$ − 22.0000i − 0.855701i −0.903850 0.427850i $$-0.859271\pi$$
0.903850 0.427850i $$-0.140729\pi$$
$$662$$ 0 0
$$663$$ −12.0000 −0.466041
$$664$$ 0 0
$$665$$ −8.00000 −0.310227
$$666$$ 0 0
$$667$$ 0 0
$$668$$ 0 0
$$669$$ 16.0000i 0.618596i
$$670$$ 0 0
$$671$$ −8.00000 −0.308837
$$672$$ 0 0
$$673$$ 34.0000 1.31060 0.655302 0.755367i $$-0.272541\pi$$
0.655302 + 0.755367i $$0.272541\pi$$
$$674$$ 0 0
$$675$$ − 1.00000i − 0.0384900i
$$676$$ 0 0
$$677$$ 18.0000i 0.691796i 0.938272 + 0.345898i $$0.112426\pi$$
−0.938272 + 0.345898i $$0.887574\pi$$
$$678$$ 0 0
$$679$$ 18.0000 0.690777
$$680$$ 0 0
$$681$$ 12.0000 0.459841
$$682$$ 0 0
$$683$$ 12.0000i 0.459167i 0.973289 + 0.229584i $$0.0737364\pi$$
−0.973289 + 0.229584i $$0.926264\pi$$
$$684$$ 0 0
$$685$$ 12.0000i 0.458496i
$$686$$ 0 0
$$687$$ −10.0000 −0.381524
$$688$$ 0 0
$$689$$ −12.0000 −0.457164
$$690$$ 0 0
$$691$$ 20.0000i 0.760836i 0.924815 + 0.380418i $$0.124220\pi$$
−0.924815 + 0.380418i $$0.875780\pi$$
$$692$$ 0 0
$$693$$ 4.00000i 0.151947i
$$694$$ 0 0
$$695$$ 24.0000 0.910372
$$696$$ 0 0
$$697$$ 12.0000 0.454532
$$698$$ 0 0
$$699$$ 6.00000i 0.226941i
$$700$$ 0 0
$$701$$ 30.0000i 1.13308i 0.824033 + 0.566542i $$0.191719\pi$$
−0.824033 + 0.566542i $$0.808281\pi$$
$$702$$ 0 0
$$703$$ 24.0000 0.905177
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ − 14.0000i − 0.526524i
$$708$$ 0 0
$$709$$ − 6.00000i − 0.225335i −0.993633 0.112667i $$-0.964061\pi$$
0.993633 0.112667i $$-0.0359394\pi$$
$$710$$ 0 0
$$711$$ 16.0000 0.600047
$$712$$ 0 0
$$713$$ 0 0
$$714$$ 0 0
$$715$$ − 16.0000i − 0.598366i
$$716$$ 0 0
$$717$$ 24.0000i 0.896296i
$$718$$ 0 0
$$719$$ −48.0000 −1.79010 −0.895049 0.445968i $$-0.852860\pi$$
−0.895049 + 0.445968i $$0.852860\pi$$
$$720$$ 0 0
$$721$$ −8.00000 −0.297936
$$722$$ 0 0
$$723$$ 2.00000i 0.0743808i
$$724$$ 0 0
$$725$$ − 2.00000i − 0.0742781i
$$726$$ 0 0
$$727$$ 40.0000 1.48352 0.741759 0.670667i $$-0.233992\pi$$
0.741759 + 0.670667i $$0.233992\pi$$
$$728$$ 0 0
$$729$$ −1.00000 −0.0370370
$$730$$ 0 0
$$731$$ − 24.0000i − 0.887672i
$$732$$ 0 0
$$733$$ − 18.0000i − 0.664845i −0.943131 0.332423i $$-0.892134\pi$$
0.943131 0.332423i $$-0.107866\pi$$
$$734$$ 0 0
$$735$$ −2.00000 −0.0737711
$$736$$ 0 0
$$737$$ 16.0000 0.589368
$$738$$ 0 0
$$739$$ 36.0000i 1.32428i 0.749380 + 0.662141i $$0.230352\pi$$
−0.749380 + 0.662141i $$0.769648\pi$$
$$740$$ 0 0
$$741$$ 8.00000i 0.293887i
$$742$$ 0 0
$$743$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$744$$ 0 0
$$745$$ 12.0000 0.439646
$$746$$ 0 0
$$747$$ 12.0000i 0.439057i
$$748$$ 0 0
$$749$$ − 4.00000i − 0.146157i
$$750$$ 0 0
$$751$$ −32.0000 −1.16770 −0.583848 0.811863i $$-0.698454\pi$$
−0.583848 + 0.811863i $$0.698454\pi$$
$$752$$ 0 0
$$753$$ −20.0000 −0.728841
$$754$$ 0 0
$$755$$ − 16.0000i − 0.582300i
$$756$$ 0 0
$$757$$ 10.0000i 0.363456i 0.983349 + 0.181728i $$0.0581691\pi$$
−0.983349 + 0.181728i $$0.941831\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −18.0000 −0.652499 −0.326250 0.945284i $$-0.605785\pi$$
−0.326250 + 0.945284i $$0.605785\pi$$
$$762$$ 0 0
$$763$$ − 18.0000i − 0.651644i
$$764$$ 0 0
$$765$$ 12.0000i 0.433861i
$$766$$ 0 0
$$767$$ −24.0000 −0.866590
$$768$$ 0 0
$$769$$ 2.00000 0.0721218 0.0360609 0.999350i $$-0.488519\pi$$
0.0360609 + 0.999350i $$0.488519\pi$$
$$770$$ 0 0
$$771$$ 26.0000i 0.936367i
$$772$$ 0 0
$$773$$ − 14.0000i − 0.503545i −0.967786 0.251773i $$-0.918987\pi$$
0.967786 0.251773i $$-0.0810135\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 6.00000 0.215249
$$778$$ 0 0
$$779$$ − 8.00000i − 0.286630i
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ −2.00000 −0.0714742
$$784$$ 0 0
$$785$$ 4.00000 0.142766
$$786$$ 0 0
$$787$$ − 44.0000i − 1.56843i −0.620489 0.784215i $$-0.713066\pi$$
0.620489 0.784215i $$-0.286934\pi$$
$$788$$ 0 0
$$789$$ − 16.0000i − 0.569615i
$$790$$ 0 0
$$791$$ −14.0000 −0.497783
$$792$$ 0 0
$$793$$ −4.00000 −0.142044
$$794$$ 0 0
$$795$$ 12.0000i 0.425596i
$$796$$ 0 0
$$797$$ − 26.0000i − 0.920967i −0.887668 0.460484i $$-0.847676\pi$$
0.887668 0.460484i $$-0.152324\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ 0 0
$$801$$ −14.0000 −0.494666
$$802$$ 0 0
$$803$$ − 24.0000i − 0.846942i
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ −6.00000 −0.211210
$$808$$ 0 0
$$809$$ −42.0000 −1.47664 −0.738321 0.674450i $$-0.764381\pi$$
−0.738321 + 0.674450i $$0.764381\pi$$
$$810$$ 0 0
$$811$$ − 44.0000i − 1.54505i −0.634985 0.772524i $$-0.718994\pi$$
0.634985 0.772524i $$-0.281006\pi$$
$$812$$ 0 0
$$813$$ 16.0000i 0.561144i
$$814$$ 0 0
$$815$$ −8.00000 −0.280228
$$816$$ 0 0
$$817$$ −16.0000 −0.559769
$$818$$ 0 0
$$819$$ 2.00000i 0.0698857i
$$820$$ 0 0
$$821$$ − 38.0000i − 1.32621i −0.748527 0.663105i $$-0.769238\pi$$
0.748527 0.663105i $$-0.230762\pi$$
$$822$$ 0 0
$$823$$ −24.0000 −0.836587 −0.418294 0.908312i $$-0.637372\pi$$
−0.418294 + 0.908312i $$0.637372\pi$$
$$824$$ 0 0
$$825$$ 4.00000 0.139262
$$826$$ 0 0
$$827$$ 12.0000i 0.417281i 0.977992 + 0.208640i $$0.0669038\pi$$
−0.977992 + 0.208640i $$0.933096\pi$$
$$828$$ 0 0
$$829$$ 14.0000i 0.486240i 0.969996 + 0.243120i $$0.0781709\pi$$
−0.969996 + 0.243120i $$0.921829\pi$$
$$830$$ 0 0
$$831$$ 22.0000 0.763172
$$832$$ 0 0
$$833$$ −6.00000 −0.207888
$$834$$ 0 0
$$835$$ 16.0000i 0.553703i
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ 8.00000 0.276191 0.138095 0.990419i $$-0.455902\pi$$
0.138095 + 0.990419i $$0.455902\pi$$
$$840$$ 0 0
$$841$$ 25.0000 0.862069
$$842$$ 0 0
$$843$$ 22.0000i 0.757720i
$$844$$ 0 0
$$845$$ 18.0000i 0.619219i
$$846$$ 0 0
$$847$$ −5.00000 −0.171802
$$848$$ 0 0
$$849$$ −20.0000 −0.686398
$$850$$ 0 0
$$851$$ 0 0
$$852$$ 0 0
$$853$$ 10.0000i 0.342393i 0.985237 + 0.171197i $$0.0547634\pi$$
−0.985237 + 0.171197i $$0.945237\pi$$
$$854$$ 0 0
$$855$$ 8.00000 0.273594
$$856$$ 0 0
$$857$$ 14.0000 0.478231 0.239115 0.970991i $$-0.423143\pi$$
0.239115 + 0.970991i $$0.423143\pi$$
$$858$$ 0 0
$$859$$ − 44.0000i − 1.50126i −0.660722 0.750630i $$-0.729750\pi$$
0.660722 0.750630i $$-0.270250\pi$$
$$860$$ 0 0
$$861$$ − 2.00000i − 0.0681598i
$$862$$ 0 0
$$863$$ −24.0000 −0.816970 −0.408485 0.912765i $$-0.633943\pi$$
−0.408485 + 0.912765i $$0.633943\pi$$
$$864$$ 0 0
$$865$$ 20.0000 0.680020
$$866$$ 0 0
$$867$$ 19.0000i 0.645274i
$$868$$ 0 0
$$869$$ 64.0000i 2.17105i
$$870$$ 0 0
$$871$$ 8.00000 0.271070
$$872$$ 0 0
$$873$$ −18.0000 −0.609208
$$874$$ 0 0
$$875$$ 12.0000i 0.405674i
$$876$$ 0 0
$$877$$ 46.0000i 1.55331i 0.629926 + 0.776655i $$0.283085\pi$$
−0.629926 + 0.776655i $$0.716915\pi$$
$$878$$ 0 0
$$879$$ 14.0000 0.472208
$$880$$ 0 0
$$881$$ −6.00000 −0.202145 −0.101073 0.994879i $$-0.532227\pi$$
−0.101073 + 0.994879i $$0.532227\pi$$
$$882$$ 0 0
$$883$$ − 28.0000i − 0.942275i −0.882060 0.471138i $$-0.843844\pi$$
0.882060 0.471138i $$-0.156156\pi$$
$$884$$ 0 0
$$885$$ 24.0000i 0.806751i
$$886$$ 0 0
$$887$$ −8.00000 −0.268614 −0.134307 0.990940i $$-0.542881\pi$$
−0.134307 + 0.990940i $$0.542881\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ − 4.00000i − 0.134005i
$$892$$ 0 0
$$893$$ 0 0
$$894$$ 0 0
$$895$$ 8.00000 0.267411
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 0 0
$$900$$ 0 0
$$901$$ 36.0000i 1.19933i
$$902$$ 0 0
$$903$$ −4.00000 −0.133112
$$904$$ 0 0
$$905$$ −52.0000 −1.72854
$$906$$ 0 0
$$907$$ 4.00000i 0.132818i 0.997792 + 0.0664089i $$0.0211542\pi$$
−0.997792 + 0.0664089i $$0.978846\pi$$
$$908$$ 0 0
$$909$$ 14.0000i 0.464351i
$$910$$ 0 0
$$911$$ −24.0000 −0.795155 −0.397578 0.917568i $$-0.630149\pi$$
−0.397578 + 0.917568i $$0.630149\pi$$
$$912$$ 0 0
$$913$$ −48.0000 −1.58857
$$914$$ 0 0
$$915$$ 4.00000i 0.132236i
$$916$$ 0 0
$$917$$ 4.00000i 0.132092i
$$918$$ 0 0
$$919$$ −8.00000 −0.263896 −0.131948 0.991257i $$-0.542123\pi$$
−0.131948 + 0.991257i $$0.542123\pi$$
$$920$$ 0 0
$$921$$ −4.00000 −0.131804
$$922$$ 0 0
$$923$$ 0 0
$$924$$ 0 0
$$925$$ − 6.00000i − 0.197279i
$$926$$ 0 0
$$927$$ 8.00000 0.262754
$$928$$ 0 0
$$929$$ 26.0000 0.853032 0.426516 0.904480i $$-0.359741\pi$$
0.426516 + 0.904480i $$0.359741\pi$$
$$930$$ 0 0
$$931$$ 4.00000i 0.131095i
$$932$$ 0 0
$$933$$ 24.0000i 0.785725i
$$934$$ 0 0
$$935$$ −48.0000 −1.56977
$$936$$ 0 0
$$937$$ −42.0000 −1.37208 −0.686040 0.727564i $$-0.740653\pi$$
−0.686040 + 0.727564i $$0.740653\pi$$
$$938$$ 0 0
$$939$$ − 26.0000i − 0.848478i
$$940$$ 0 0
$$941$$ 38.0000i 1.23876i 0.785090 + 0.619382i $$0.212617\pi$$
−0.785090 + 0.619382i $$0.787383\pi$$
$$942$$ 0 0
$$943$$ 0 0
$$944$$ 0 0
$$945$$ 2.00000 0.0650600
$$946$$ 0 0
$$947$$ 44.0000i 1.42981i 0.699223 + 0.714904i $$0.253530\pi$$
−0.699223 + 0.714904i $$0.746470\pi$$
$$948$$ 0 0
$$949$$ − 12.0000i − 0.389536i
$$950$$ 0 0
$$951$$ 18.0000 0.583690
$$952$$ 0 0
$$953$$ −26.0000 −0.842223 −0.421111 0.907009i $$-0.638360\pi$$
−0.421111 + 0.907009i $$0.638360\pi$$
$$954$$ 0 0
$$955$$ − 16.0000i − 0.517748i
$$956$$ 0 0
$$957$$ − 8.00000i − 0.258603i
$$958$$ 0 0
$$959$$ 6.00000 0.193750
$$960$$ 0 0
$$961$$ −31.0000 −1.00000
$$962$$ 0 0
$$963$$ 4.00000i 0.128898i
$$964$$ 0 0
$$965$$ 4.00000i 0.128765i
$$966$$ 0 0
$$967$$ −40.0000 −1.28631 −0.643157 0.765735i $$-0.722376\pi$$
−0.643157 + 0.765735i $$0.722376\pi$$
$$968$$ 0 0
$$969$$ 24.0000 0.770991
$$970$$ 0 0
$$971$$ − 12.0000i − 0.385098i −0.981287 0.192549i $$-0.938325\pi$$
0.981287 0.192549i $$-0.0616755\pi$$
$$972$$ 0 0
$$973$$ − 12.0000i − 0.384702i
$$974$$ 0 0
$$975$$ 2.00000 0.0640513
$$976$$ 0 0
$$977$$ −30.0000 −0.959785 −0.479893 0.877327i $$-0.659324\pi$$
−0.479893 + 0.877327i $$0.659324\pi$$
$$978$$ 0 0
$$979$$ − 56.0000i − 1.78977i
$$980$$ 0 0
$$981$$ 18.0000i 0.574696i
$$982$$ 0 0
$$983$$ −24.0000 −0.765481 −0.382741 0.923856i $$-0.625020\pi$$
−0.382741 + 0.923856i $$0.625020\pi$$
$$984$$ 0 0
$$985$$ 44.0000 1.40196
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 0 0
$$990$$ 0 0
$$991$$ −16.0000 −0.508257 −0.254128 0.967170i $$-0.581789\pi$$
−0.254128 + 0.967170i $$0.581789\pi$$
$$992$$ 0 0
$$993$$ −4.00000 −0.126936
$$994$$ 0 0
$$995$$ − 48.0000i − 1.52170i
$$996$$ 0 0
$$997$$ 26.0000i 0.823428i 0.911313 + 0.411714i $$0.135070\pi$$
−0.911313 + 0.411714i $$0.864930\pi$$
$$998$$ 0 0
$$999$$ −6.00000 −0.189832
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 5376.2.c.r.2689.2 2
4.3 odd 2 5376.2.c.l.2689.1 2
8.3 odd 2 5376.2.c.l.2689.2 2
8.5 even 2 inner 5376.2.c.r.2689.1 2
16.3 odd 4 1344.2.a.s.1.1 1
16.5 even 4 21.2.a.a.1.1 1
16.11 odd 4 336.2.a.a.1.1 1
16.13 even 4 1344.2.a.g.1.1 1
48.5 odd 4 63.2.a.a.1.1 1
48.11 even 4 1008.2.a.l.1.1 1
48.29 odd 4 4032.2.a.h.1.1 1
48.35 even 4 4032.2.a.k.1.1 1
80.37 odd 4 525.2.d.a.274.1 2
80.53 odd 4 525.2.d.a.274.2 2
80.59 odd 4 8400.2.a.bn.1.1 1
80.69 even 4 525.2.a.d.1.1 1
112.5 odd 12 147.2.e.c.67.1 2
112.11 odd 12 2352.2.q.x.961.1 2
112.13 odd 4 9408.2.a.bv.1.1 1
112.27 even 4 2352.2.a.v.1.1 1
112.37 even 12 147.2.e.b.67.1 2
112.53 even 12 147.2.e.b.79.1 2
112.59 even 12 2352.2.q.e.961.1 2
112.69 odd 4 147.2.a.a.1.1 1
112.75 even 12 2352.2.q.e.1537.1 2
112.83 even 4 9408.2.a.m.1.1 1
112.101 odd 12 147.2.e.c.79.1 2
112.107 odd 12 2352.2.q.x.1537.1 2
144.5 odd 12 567.2.f.b.379.1 2
144.85 even 12 567.2.f.g.379.1 2
144.101 odd 12 567.2.f.b.190.1 2
144.133 even 12 567.2.f.g.190.1 2
176.21 odd 4 2541.2.a.j.1.1 1
208.181 even 4 3549.2.a.c.1.1 1
240.53 even 4 1575.2.d.a.1324.1 2
240.149 odd 4 1575.2.a.c.1.1 1
240.197 even 4 1575.2.d.a.1324.2 2
272.101 even 4 6069.2.a.b.1.1 1
304.37 odd 4 7581.2.a.d.1.1 1
336.5 even 12 441.2.e.b.361.1 2
336.53 odd 12 441.2.e.a.226.1 2
336.101 even 12 441.2.e.b.226.1 2
336.149 odd 12 441.2.e.a.361.1 2
336.251 odd 4 7056.2.a.p.1.1 1
336.293 even 4 441.2.a.f.1.1 1
528.197 even 4 7623.2.a.g.1.1 1
560.69 odd 4 3675.2.a.n.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
21.2.a.a.1.1 1 16.5 even 4
63.2.a.a.1.1 1 48.5 odd 4
147.2.a.a.1.1 1 112.69 odd 4
147.2.e.b.67.1 2 112.37 even 12
147.2.e.b.79.1 2 112.53 even 12
147.2.e.c.67.1 2 112.5 odd 12
147.2.e.c.79.1 2 112.101 odd 12
336.2.a.a.1.1 1 16.11 odd 4
441.2.a.f.1.1 1 336.293 even 4
441.2.e.a.226.1 2 336.53 odd 12
441.2.e.a.361.1 2 336.149 odd 12
441.2.e.b.226.1 2 336.101 even 12
441.2.e.b.361.1 2 336.5 even 12
525.2.a.d.1.1 1 80.69 even 4
525.2.d.a.274.1 2 80.37 odd 4
525.2.d.a.274.2 2 80.53 odd 4
567.2.f.b.190.1 2 144.101 odd 12
567.2.f.b.379.1 2 144.5 odd 12
567.2.f.g.190.1 2 144.133 even 12
567.2.f.g.379.1 2 144.85 even 12
1008.2.a.l.1.1 1 48.11 even 4
1344.2.a.g.1.1 1 16.13 even 4
1344.2.a.s.1.1 1 16.3 odd 4
1575.2.a.c.1.1 1 240.149 odd 4
1575.2.d.a.1324.1 2 240.53 even 4
1575.2.d.a.1324.2 2 240.197 even 4
2352.2.a.v.1.1 1 112.27 even 4
2352.2.q.e.961.1 2 112.59 even 12
2352.2.q.e.1537.1 2 112.75 even 12
2352.2.q.x.961.1 2 112.11 odd 12
2352.2.q.x.1537.1 2 112.107 odd 12
2541.2.a.j.1.1 1 176.21 odd 4
3549.2.a.c.1.1 1 208.181 even 4
3675.2.a.n.1.1 1 560.69 odd 4
4032.2.a.h.1.1 1 48.29 odd 4
4032.2.a.k.1.1 1 48.35 even 4
5376.2.c.l.2689.1 2 4.3 odd 2
5376.2.c.l.2689.2 2 8.3 odd 2
5376.2.c.r.2689.1 2 8.5 even 2 inner
5376.2.c.r.2689.2 2 1.1 even 1 trivial
6069.2.a.b.1.1 1 272.101 even 4
7056.2.a.p.1.1 1 336.251 odd 4
7581.2.a.d.1.1 1 304.37 odd 4
7623.2.a.g.1.1 1 528.197 even 4
8400.2.a.bn.1.1 1 80.59 odd 4
9408.2.a.m.1.1 1 112.83 even 4
9408.2.a.bv.1.1 1 112.13 odd 4