# Properties

 Label 2736.2.a.bb.1.1 Level $2736$ Weight $2$ Character 2736.1 Self dual yes Analytic conductor $21.847$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2736,2,Mod(1,2736)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2736, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2736.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2736 = 2^{4} \cdot 3^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2736.a (trivial)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$21.8470699930$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{41})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 10$$ x^2 - x - 10 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 456) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-2.70156$$ of defining polynomial Character $$\chi$$ $$=$$ 2736.1

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-2.70156 q^{5} +4.70156 q^{7} +O(q^{10})$$ $$q-2.70156 q^{5} +4.70156 q^{7} +4.70156 q^{11} +6.00000 q^{13} +2.70156 q^{17} -1.00000 q^{19} +4.00000 q^{23} +2.29844 q^{25} -2.00000 q^{29} -9.40312 q^{31} -12.7016 q^{35} -3.40312 q^{37} +3.40312 q^{41} -10.1047 q^{43} -0.701562 q^{47} +15.1047 q^{49} +6.00000 q^{53} -12.7016 q^{55} -4.00000 q^{59} +1.29844 q^{61} -16.2094 q^{65} +12.0000 q^{67} +6.70156 q^{73} +22.1047 q^{77} +10.8062 q^{79} -10.8062 q^{83} -7.29844 q^{85} +12.8062 q^{89} +28.2094 q^{91} +2.70156 q^{95} -6.00000 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{5} + 3 q^{7}+O(q^{10})$$ 2 * q + q^5 + 3 * q^7 $$2 q + q^{5} + 3 q^{7} + 3 q^{11} + 12 q^{13} - q^{17} - 2 q^{19} + 8 q^{23} + 11 q^{25} - 4 q^{29} - 6 q^{31} - 19 q^{35} + 6 q^{37} - 6 q^{41} - q^{43} + 5 q^{47} + 11 q^{49} + 12 q^{53} - 19 q^{55} - 8 q^{59} + 9 q^{61} + 6 q^{65} + 24 q^{67} + 7 q^{73} + 25 q^{77} - 4 q^{79} + 4 q^{83} - 21 q^{85} + 18 q^{91} - q^{95} - 12 q^{97}+O(q^{100})$$ 2 * q + q^5 + 3 * q^7 + 3 * q^11 + 12 * q^13 - q^17 - 2 * q^19 + 8 * q^23 + 11 * q^25 - 4 * q^29 - 6 * q^31 - 19 * q^35 + 6 * q^37 - 6 * q^41 - q^43 + 5 * q^47 + 11 * q^49 + 12 * q^53 - 19 * q^55 - 8 * q^59 + 9 * q^61 + 6 * q^65 + 24 * q^67 + 7 * q^73 + 25 * q^77 - 4 * q^79 + 4 * q^83 - 21 * q^85 + 18 * q^91 - q^95 - 12 * q^97

## 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
$$3$$ 0 0
$$4$$ 0 0
$$5$$ −2.70156 −1.20818 −0.604088 0.796918i $$-0.706462\pi$$
−0.604088 + 0.796918i $$0.706462\pi$$
$$6$$ 0 0
$$7$$ 4.70156 1.77702 0.888512 0.458854i $$-0.151740\pi$$
0.888512 + 0.458854i $$0.151740\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 4.70156 1.41757 0.708787 0.705422i $$-0.249243\pi$$
0.708787 + 0.705422i $$0.249243\pi$$
$$12$$ 0 0
$$13$$ 6.00000 1.66410 0.832050 0.554700i $$-0.187167\pi$$
0.832050 + 0.554700i $$0.187167\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 2.70156 0.655225 0.327613 0.944812i $$-0.393756\pi$$
0.327613 + 0.944812i $$0.393756\pi$$
$$18$$ 0 0
$$19$$ −1.00000 −0.229416
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 4.00000 0.834058 0.417029 0.908893i $$-0.363071\pi$$
0.417029 + 0.908893i $$0.363071\pi$$
$$24$$ 0 0
$$25$$ 2.29844 0.459688
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ −2.00000 −0.371391 −0.185695 0.982607i $$-0.559454\pi$$
−0.185695 + 0.982607i $$0.559454\pi$$
$$30$$ 0 0
$$31$$ −9.40312 −1.68885 −0.844425 0.535673i $$-0.820058\pi$$
−0.844425 + 0.535673i $$0.820058\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ −12.7016 −2.14696
$$36$$ 0 0
$$37$$ −3.40312 −0.559470 −0.279735 0.960077i $$-0.590247\pi$$
−0.279735 + 0.960077i $$0.590247\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 3.40312 0.531479 0.265739 0.964045i $$-0.414384\pi$$
0.265739 + 0.964045i $$0.414384\pi$$
$$42$$ 0 0
$$43$$ −10.1047 −1.54095 −0.770475 0.637470i $$-0.779981\pi$$
−0.770475 + 0.637470i $$0.779981\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ −0.701562 −0.102333 −0.0511667 0.998690i $$-0.516294\pi$$
−0.0511667 + 0.998690i $$0.516294\pi$$
$$48$$ 0 0
$$49$$ 15.1047 2.15781
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 6.00000 0.824163 0.412082 0.911147i $$-0.364802\pi$$
0.412082 + 0.911147i $$0.364802\pi$$
$$54$$ 0 0
$$55$$ −12.7016 −1.71268
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ −4.00000 −0.520756 −0.260378 0.965507i $$-0.583847\pi$$
−0.260378 + 0.965507i $$0.583847\pi$$
$$60$$ 0 0
$$61$$ 1.29844 0.166248 0.0831240 0.996539i $$-0.473510\pi$$
0.0831240 + 0.996539i $$0.473510\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ −16.2094 −2.01053
$$66$$ 0 0
$$67$$ 12.0000 1.46603 0.733017 0.680211i $$-0.238112\pi$$
0.733017 + 0.680211i $$0.238112\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$72$$ 0 0
$$73$$ 6.70156 0.784359 0.392179 0.919889i $$-0.371721\pi$$
0.392179 + 0.919889i $$0.371721\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 22.1047 2.51906
$$78$$ 0 0
$$79$$ 10.8062 1.21580 0.607899 0.794014i $$-0.292013\pi$$
0.607899 + 0.794014i $$0.292013\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ −10.8062 −1.18614 −0.593070 0.805151i $$-0.702084\pi$$
−0.593070 + 0.805151i $$0.702084\pi$$
$$84$$ 0 0
$$85$$ −7.29844 −0.791627
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 12.8062 1.35746 0.678730 0.734388i $$-0.262531\pi$$
0.678730 + 0.734388i $$0.262531\pi$$
$$90$$ 0 0
$$91$$ 28.2094 2.95715
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 2.70156 0.277174
$$96$$ 0 0
$$97$$ −6.00000 −0.609208 −0.304604 0.952479i $$-0.598524\pi$$
−0.304604 + 0.952479i $$0.598524\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −16.8062 −1.67228 −0.836142 0.548513i $$-0.815194\pi$$
−0.836142 + 0.548513i $$0.815194\pi$$
$$102$$ 0 0
$$103$$ 9.40312 0.926517 0.463259 0.886223i $$-0.346680\pi$$
0.463259 + 0.886223i $$0.346680\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 5.40312 0.522340 0.261170 0.965293i $$-0.415892\pi$$
0.261170 + 0.965293i $$0.415892\pi$$
$$108$$ 0 0
$$109$$ −3.40312 −0.325960 −0.162980 0.986629i $$-0.552111\pi$$
−0.162980 + 0.986629i $$0.552111\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 18.0000 1.69330 0.846649 0.532152i $$-0.178617\pi$$
0.846649 + 0.532152i $$0.178617\pi$$
$$114$$ 0 0
$$115$$ −10.8062 −1.00769
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 12.7016 1.16435
$$120$$ 0 0
$$121$$ 11.1047 1.00952
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 7.29844 0.652792
$$126$$ 0 0
$$127$$ −9.40312 −0.834392 −0.417196 0.908816i $$-0.636987\pi$$
−0.417196 + 0.908816i $$0.636987\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −12.7016 −1.10974 −0.554870 0.831937i $$-0.687232\pi$$
−0.554870 + 0.831937i $$0.687232\pi$$
$$132$$ 0 0
$$133$$ −4.70156 −0.407677
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 10.7016 0.914296 0.457148 0.889391i $$-0.348871\pi$$
0.457148 + 0.889391i $$0.348871\pi$$
$$138$$ 0 0
$$139$$ −15.2984 −1.29760 −0.648798 0.760960i $$-0.724728\pi$$
−0.648798 + 0.760960i $$0.724728\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 28.2094 2.35899
$$144$$ 0 0
$$145$$ 5.40312 0.448705
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ −20.1047 −1.64704 −0.823520 0.567287i $$-0.807993\pi$$
−0.823520 + 0.567287i $$0.807993\pi$$
$$150$$ 0 0
$$151$$ −2.80625 −0.228369 −0.114185 0.993460i $$-0.536426\pi$$
−0.114185 + 0.993460i $$0.536426\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 25.4031 2.04043
$$156$$ 0 0
$$157$$ −2.00000 −0.159617 −0.0798087 0.996810i $$-0.525431\pi$$
−0.0798087 + 0.996810i $$0.525431\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 18.8062 1.48214
$$162$$ 0 0
$$163$$ 14.8062 1.15971 0.579857 0.814718i $$-0.303108\pi$$
0.579857 + 0.814718i $$0.303108\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 20.2094 1.56385 0.781924 0.623374i $$-0.214238\pi$$
0.781924 + 0.623374i $$0.214238\pi$$
$$168$$ 0 0
$$169$$ 23.0000 1.76923
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ 8.80625 0.669527 0.334763 0.942302i $$-0.391344\pi$$
0.334763 + 0.942302i $$0.391344\pi$$
$$174$$ 0 0
$$175$$ 10.8062 0.816876
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ −5.40312 −0.403848 −0.201924 0.979401i $$-0.564719\pi$$
−0.201924 + 0.979401i $$0.564719\pi$$
$$180$$ 0 0
$$181$$ −4.80625 −0.357246 −0.178623 0.983918i $$-0.557164\pi$$
−0.178623 + 0.983918i $$0.557164\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 9.19375 0.675938
$$186$$ 0 0
$$187$$ 12.7016 0.928830
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −20.9109 −1.51306 −0.756531 0.653958i $$-0.773107\pi$$
−0.756531 + 0.653958i $$0.773107\pi$$
$$192$$ 0 0
$$193$$ 11.4031 0.820815 0.410407 0.911902i $$-0.365386\pi$$
0.410407 + 0.911902i $$0.365386\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −22.0000 −1.56744 −0.783718 0.621117i $$-0.786679\pi$$
−0.783718 + 0.621117i $$0.786679\pi$$
$$198$$ 0 0
$$199$$ −3.29844 −0.233820 −0.116910 0.993143i $$-0.537299\pi$$
−0.116910 + 0.993143i $$0.537299\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ −9.40312 −0.659970
$$204$$ 0 0
$$205$$ −9.19375 −0.642119
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ −4.70156 −0.325214
$$210$$ 0 0
$$211$$ −6.80625 −0.468561 −0.234281 0.972169i $$-0.575273\pi$$
−0.234281 + 0.972169i $$0.575273\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 27.2984 1.86174
$$216$$ 0 0
$$217$$ −44.2094 −3.00113
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 16.2094 1.09036
$$222$$ 0 0
$$223$$ −26.8062 −1.79508 −0.897540 0.440934i $$-0.854647\pi$$
−0.897540 + 0.440934i $$0.854647\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ −10.5969 −0.703339 −0.351670 0.936124i $$-0.614386\pi$$
−0.351670 + 0.936124i $$0.614386\pi$$
$$228$$ 0 0
$$229$$ 5.50781 0.363966 0.181983 0.983302i $$-0.441748\pi$$
0.181983 + 0.983302i $$0.441748\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 12.1047 0.793004 0.396502 0.918034i $$-0.370224\pi$$
0.396502 + 0.918034i $$0.370224\pi$$
$$234$$ 0 0
$$235$$ 1.89531 0.123637
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −2.10469 −0.136141 −0.0680704 0.997681i $$-0.521684\pi$$
−0.0680704 + 0.997681i $$0.521684\pi$$
$$240$$ 0 0
$$241$$ −15.4031 −0.992202 −0.496101 0.868265i $$-0.665236\pi$$
−0.496101 + 0.868265i $$0.665236\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ −40.8062 −2.60702
$$246$$ 0 0
$$247$$ −6.00000 −0.381771
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 19.2984 1.21811 0.609053 0.793129i $$-0.291550\pi$$
0.609053 + 0.793129i $$0.291550\pi$$
$$252$$ 0 0
$$253$$ 18.8062 1.18234
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 19.4031 1.21033 0.605167 0.796099i $$-0.293106\pi$$
0.605167 + 0.796099i $$0.293106\pi$$
$$258$$ 0 0
$$259$$ −16.0000 −0.994192
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ −0.701562 −0.0432602 −0.0216301 0.999766i $$-0.506886\pi$$
−0.0216301 + 0.999766i $$0.506886\pi$$
$$264$$ 0 0
$$265$$ −16.2094 −0.995734
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ −10.0000 −0.609711 −0.304855 0.952399i $$-0.598608\pi$$
−0.304855 + 0.952399i $$0.598608\pi$$
$$270$$ 0 0
$$271$$ −10.8062 −0.656433 −0.328216 0.944603i $$-0.606448\pi$$
−0.328216 + 0.944603i $$0.606448\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 10.8062 0.651641
$$276$$ 0 0
$$277$$ 2.70156 0.162321 0.0811606 0.996701i $$-0.474137\pi$$
0.0811606 + 0.996701i $$0.474137\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −0.806248 −0.0480968 −0.0240484 0.999711i $$-0.507656\pi$$
−0.0240484 + 0.999711i $$0.507656\pi$$
$$282$$ 0 0
$$283$$ 8.70156 0.517254 0.258627 0.965977i $$-0.416730\pi$$
0.258627 + 0.965977i $$0.416730\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 16.0000 0.944450
$$288$$ 0 0
$$289$$ −9.70156 −0.570680
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ 23.4031 1.36723 0.683613 0.729845i $$-0.260408\pi$$
0.683613 + 0.729845i $$0.260408\pi$$
$$294$$ 0 0
$$295$$ 10.8062 0.629164
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 24.0000 1.38796
$$300$$ 0 0
$$301$$ −47.5078 −2.73830
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ −3.50781 −0.200857
$$306$$ 0 0
$$307$$ 1.19375 0.0681310 0.0340655 0.999420i $$-0.489155\pi$$
0.0340655 + 0.999420i $$0.489155\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −27.5078 −1.55982 −0.779912 0.625889i $$-0.784736\pi$$
−0.779912 + 0.625889i $$0.784736\pi$$
$$312$$ 0 0
$$313$$ 28.8062 1.62823 0.814113 0.580707i $$-0.197224\pi$$
0.814113 + 0.580707i $$0.197224\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −3.40312 −0.191138 −0.0955692 0.995423i $$-0.530467\pi$$
−0.0955692 + 0.995423i $$0.530467\pi$$
$$318$$ 0 0
$$319$$ −9.40312 −0.526474
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ −2.70156 −0.150319
$$324$$ 0 0
$$325$$ 13.7906 0.764966
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ −3.29844 −0.181849
$$330$$ 0 0
$$331$$ 30.8062 1.69326 0.846632 0.532178i $$-0.178626\pi$$
0.846632 + 0.532178i $$0.178626\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ −32.4187 −1.77123
$$336$$ 0 0
$$337$$ 12.8062 0.697601 0.348800 0.937197i $$-0.386589\pi$$
0.348800 + 0.937197i $$0.386589\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −44.2094 −2.39407
$$342$$ 0 0
$$343$$ 38.1047 2.05746
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −20.7016 −1.11132 −0.555659 0.831410i $$-0.687534\pi$$
−0.555659 + 0.831410i $$0.687534\pi$$
$$348$$ 0 0
$$349$$ −24.1047 −1.29029 −0.645147 0.764058i $$-0.723204\pi$$
−0.645147 + 0.764058i $$0.723204\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ −7.19375 −0.382885 −0.191442 0.981504i $$-0.561317\pi$$
−0.191442 + 0.981504i $$0.561317\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −4.49219 −0.237089 −0.118544 0.992949i $$-0.537823\pi$$
−0.118544 + 0.992949i $$0.537823\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ −18.1047 −0.947643
$$366$$ 0 0
$$367$$ −8.00000 −0.417597 −0.208798 0.977959i $$-0.566955\pi$$
−0.208798 + 0.977959i $$0.566955\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 28.2094 1.46456
$$372$$ 0 0
$$373$$ −3.40312 −0.176207 −0.0881035 0.996111i $$-0.528081\pi$$
−0.0881035 + 0.996111i $$0.528081\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −12.0000 −0.618031
$$378$$ 0 0
$$379$$ 21.4031 1.09940 0.549702 0.835361i $$-0.314741\pi$$
0.549702 + 0.835361i $$0.314741\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ 20.2094 1.03265 0.516325 0.856393i $$-0.327300\pi$$
0.516325 + 0.856393i $$0.327300\pi$$
$$384$$ 0 0
$$385$$ −59.7172 −3.04347
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ 34.9109 1.77005 0.885027 0.465539i $$-0.154140\pi$$
0.885027 + 0.465539i $$0.154140\pi$$
$$390$$ 0 0
$$391$$ 10.8062 0.546495
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ −29.1938 −1.46890
$$396$$ 0 0
$$397$$ 29.5078 1.48095 0.740477 0.672081i $$-0.234600\pi$$
0.740477 + 0.672081i $$0.234600\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −10.2094 −0.509832 −0.254916 0.966963i $$-0.582048\pi$$
−0.254916 + 0.966963i $$0.582048\pi$$
$$402$$ 0 0
$$403$$ −56.4187 −2.81042
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −16.0000 −0.793091
$$408$$ 0 0
$$409$$ 2.00000 0.0988936 0.0494468 0.998777i $$-0.484254\pi$$
0.0494468 + 0.998777i $$0.484254\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ −18.8062 −0.925395
$$414$$ 0 0
$$415$$ 29.1938 1.43306
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 18.8062 0.918745 0.459373 0.888244i $$-0.348074\pi$$
0.459373 + 0.888244i $$0.348074\pi$$
$$420$$ 0 0
$$421$$ 0.806248 0.0392941 0.0196471 0.999807i $$-0.493746\pi$$
0.0196471 + 0.999807i $$0.493746\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 6.20937 0.301199
$$426$$ 0 0
$$427$$ 6.10469 0.295426
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −14.5969 −0.703107 −0.351553 0.936168i $$-0.614346\pi$$
−0.351553 + 0.936168i $$0.614346\pi$$
$$432$$ 0 0
$$433$$ 23.6125 1.13474 0.567372 0.823462i $$-0.307960\pi$$
0.567372 + 0.823462i $$0.307960\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ −4.00000 −0.191346
$$438$$ 0 0
$$439$$ −6.59688 −0.314852 −0.157426 0.987531i $$-0.550320\pi$$
−0.157426 + 0.987531i $$0.550320\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −7.50781 −0.356707 −0.178353 0.983966i $$-0.557077\pi$$
−0.178353 + 0.983966i $$0.557077\pi$$
$$444$$ 0 0
$$445$$ −34.5969 −1.64005
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 0.596876 0.0281683 0.0140842 0.999901i $$-0.495517\pi$$
0.0140842 + 0.999901i $$0.495517\pi$$
$$450$$ 0 0
$$451$$ 16.0000 0.753411
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ −76.2094 −3.57275
$$456$$ 0 0
$$457$$ −15.8953 −0.743551 −0.371776 0.928323i $$-0.621251\pi$$
−0.371776 + 0.928323i $$0.621251\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −28.1047 −1.30897 −0.654483 0.756077i $$-0.727114\pi$$
−0.654483 + 0.756077i $$0.727114\pi$$
$$462$$ 0 0
$$463$$ −7.50781 −0.348918 −0.174459 0.984664i $$-0.555818\pi$$
−0.174459 + 0.984664i $$0.555818\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −17.8953 −0.828096 −0.414048 0.910255i $$-0.635886\pi$$
−0.414048 + 0.910255i $$0.635886\pi$$
$$468$$ 0 0
$$469$$ 56.4187 2.60518
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ −47.5078 −2.18441
$$474$$ 0 0
$$475$$ −2.29844 −0.105460
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ 22.8062 1.04204 0.521022 0.853543i $$-0.325551\pi$$
0.521022 + 0.853543i $$0.325551\pi$$
$$480$$ 0 0
$$481$$ −20.4187 −0.931015
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 16.2094 0.736030
$$486$$ 0 0
$$487$$ −5.19375 −0.235351 −0.117676 0.993052i $$-0.537544\pi$$
−0.117676 + 0.993052i $$0.537544\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −8.00000 −0.361035 −0.180517 0.983572i $$-0.557777\pi$$
−0.180517 + 0.983572i $$0.557777\pi$$
$$492$$ 0 0
$$493$$ −5.40312 −0.243344
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 20.9109 0.936102 0.468051 0.883701i $$-0.344956\pi$$
0.468051 + 0.883701i $$0.344956\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ −38.8062 −1.73029 −0.865143 0.501526i $$-0.832772\pi$$
−0.865143 + 0.501526i $$0.832772\pi$$
$$504$$ 0 0
$$505$$ 45.4031 2.02041
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ 43.6125 1.93309 0.966545 0.256497i $$-0.0825684\pi$$
0.966545 + 0.256497i $$0.0825684\pi$$
$$510$$ 0 0
$$511$$ 31.5078 1.39382
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ −25.4031 −1.11940
$$516$$ 0 0
$$517$$ −3.29844 −0.145065
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −12.5969 −0.551879 −0.275940 0.961175i $$-0.588989\pi$$
−0.275940 + 0.961175i $$0.588989\pi$$
$$522$$ 0 0
$$523$$ 16.2094 0.708786 0.354393 0.935097i $$-0.384687\pi$$
0.354393 + 0.935097i $$0.384687\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −25.4031 −1.10658
$$528$$ 0 0
$$529$$ −7.00000 −0.304348
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 20.4187 0.884434
$$534$$ 0 0
$$535$$ −14.5969 −0.631078
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 71.0156 3.05886
$$540$$ 0 0
$$541$$ 25.7172 1.10567 0.552834 0.833291i $$-0.313546\pi$$
0.552834 + 0.833291i $$0.313546\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 9.19375 0.393817
$$546$$ 0 0
$$547$$ 19.0156 0.813049 0.406525 0.913640i $$-0.366741\pi$$
0.406525 + 0.913640i $$0.366741\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 2.00000 0.0852029
$$552$$ 0 0
$$553$$ 50.8062 2.16050
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 38.7016 1.63984 0.819919 0.572480i $$-0.194018\pi$$
0.819919 + 0.572480i $$0.194018\pi$$
$$558$$ 0 0
$$559$$ −60.6281 −2.56430
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ −4.00000 −0.168580 −0.0842900 0.996441i $$-0.526862\pi$$
−0.0842900 + 0.996441i $$0.526862\pi$$
$$564$$ 0 0
$$565$$ −48.6281 −2.04580
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −0.806248 −0.0337997 −0.0168998 0.999857i $$-0.505380\pi$$
−0.0168998 + 0.999857i $$0.505380\pi$$
$$570$$ 0 0
$$571$$ −1.19375 −0.0499569 −0.0249785 0.999688i $$-0.507952\pi$$
−0.0249785 + 0.999688i $$0.507952\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 9.19375 0.383406
$$576$$ 0 0
$$577$$ 2.49219 0.103751 0.0518756 0.998654i $$-0.483480\pi$$
0.0518756 + 0.998654i $$0.483480\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −50.8062 −2.10780
$$582$$ 0 0
$$583$$ 28.2094 1.16831
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 3.29844 0.136141 0.0680706 0.997681i $$-0.478316\pi$$
0.0680706 + 0.997681i $$0.478316\pi$$
$$588$$ 0 0
$$589$$ 9.40312 0.387449
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ −2.00000 −0.0821302 −0.0410651 0.999156i $$-0.513075\pi$$
−0.0410651 + 0.999156i $$0.513075\pi$$
$$594$$ 0 0
$$595$$ −34.3141 −1.40674
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ −10.8062 −0.441531 −0.220766 0.975327i $$-0.570856\pi$$
−0.220766 + 0.975327i $$0.570856\pi$$
$$600$$ 0 0
$$601$$ 22.2094 0.905939 0.452970 0.891526i $$-0.350365\pi$$
0.452970 + 0.891526i $$0.350365\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ −30.0000 −1.21967
$$606$$ 0 0
$$607$$ −28.2094 −1.14498 −0.572492 0.819911i $$-0.694023\pi$$
−0.572492 + 0.819911i $$0.694023\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −4.20937 −0.170293
$$612$$ 0 0
$$613$$ −10.4922 −0.423776 −0.211888 0.977294i $$-0.567961\pi$$
−0.211888 + 0.977294i $$0.567961\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 29.5078 1.18794 0.593970 0.804487i $$-0.297560\pi$$
0.593970 + 0.804487i $$0.297560\pi$$
$$618$$ 0 0
$$619$$ −12.0000 −0.482321 −0.241160 0.970485i $$-0.577528\pi$$
−0.241160 + 0.970485i $$0.577528\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 60.2094 2.41224
$$624$$ 0 0
$$625$$ −31.2094 −1.24837
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ −9.19375 −0.366579
$$630$$ 0 0
$$631$$ −8.49219 −0.338069 −0.169034 0.985610i $$-0.554065\pi$$
−0.169034 + 0.985610i $$0.554065\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ 25.4031 1.00809
$$636$$ 0 0
$$637$$ 90.6281 3.59082
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 2.00000 0.0789953 0.0394976 0.999220i $$-0.487424\pi$$
0.0394976 + 0.999220i $$0.487424\pi$$
$$642$$ 0 0
$$643$$ 19.5078 0.769313 0.384656 0.923060i $$-0.374320\pi$$
0.384656 + 0.923060i $$0.374320\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −31.2984 −1.23047 −0.615234 0.788344i $$-0.710939\pi$$
−0.615234 + 0.788344i $$0.710939\pi$$
$$648$$ 0 0
$$649$$ −18.8062 −0.738210
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −28.1047 −1.09982 −0.549911 0.835223i $$-0.685338\pi$$
−0.549911 + 0.835223i $$0.685338\pi$$
$$654$$ 0 0
$$655$$ 34.3141 1.34076
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 18.5969 0.724431 0.362216 0.932094i $$-0.382020\pi$$
0.362216 + 0.932094i $$0.382020\pi$$
$$660$$ 0 0
$$661$$ −38.2094 −1.48617 −0.743086 0.669196i $$-0.766639\pi$$
−0.743086 + 0.669196i $$0.766639\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 12.7016 0.492545
$$666$$ 0 0
$$667$$ −8.00000 −0.309761
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 6.10469 0.235669
$$672$$ 0 0
$$673$$ −7.40312 −0.285369 −0.142685 0.989768i $$-0.545574\pi$$
−0.142685 + 0.989768i $$0.545574\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ −12.8062 −0.492184 −0.246092 0.969246i $$-0.579147\pi$$
−0.246092 + 0.969246i $$0.579147\pi$$
$$678$$ 0 0
$$679$$ −28.2094 −1.08258
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ −48.2094 −1.84468 −0.922340 0.386379i $$-0.873726\pi$$
−0.922340 + 0.386379i $$0.873726\pi$$
$$684$$ 0 0
$$685$$ −28.9109 −1.10463
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ 36.0000 1.37149
$$690$$ 0 0
$$691$$ −11.5078 −0.437778 −0.218889 0.975750i $$-0.570243\pi$$
−0.218889 + 0.975750i $$0.570243\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 41.3297 1.56772
$$696$$ 0 0
$$697$$ 9.19375 0.348238
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −3.19375 −0.120626 −0.0603132 0.998180i $$-0.519210\pi$$
−0.0603132 + 0.998180i $$0.519210\pi$$
$$702$$ 0 0
$$703$$ 3.40312 0.128351
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ −79.0156 −2.97169
$$708$$ 0 0
$$709$$ 27.6125 1.03701 0.518505 0.855075i $$-0.326489\pi$$
0.518505 + 0.855075i $$0.326489\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ −37.6125 −1.40860
$$714$$ 0 0
$$715$$ −76.2094 −2.85007
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 24.7016 0.921213 0.460606 0.887604i $$-0.347632\pi$$
0.460606 + 0.887604i $$0.347632\pi$$
$$720$$ 0 0
$$721$$ 44.2094 1.64644
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ −4.59688 −0.170724
$$726$$ 0 0
$$727$$ −15.5078 −0.575153 −0.287576 0.957758i $$-0.592850\pi$$
−0.287576 + 0.957758i $$0.592850\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ −27.2984 −1.00967
$$732$$ 0 0
$$733$$ −20.8062 −0.768496 −0.384248 0.923230i $$-0.625539\pi$$
−0.384248 + 0.923230i $$0.625539\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 56.4187 2.07821
$$738$$ 0 0
$$739$$ −23.2984 −0.857047 −0.428523 0.903531i $$-0.640966\pi$$
−0.428523 + 0.903531i $$0.640966\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 2.80625 0.102951 0.0514756 0.998674i $$-0.483608\pi$$
0.0514756 + 0.998674i $$0.483608\pi$$
$$744$$ 0 0
$$745$$ 54.3141 1.98991
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 25.4031 0.928210
$$750$$ 0 0
$$751$$ −29.6125 −1.08058 −0.540288 0.841480i $$-0.681685\pi$$
−0.540288 + 0.841480i $$0.681685\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 7.58125 0.275910
$$756$$ 0 0
$$757$$ 25.2984 0.919487 0.459744 0.888052i $$-0.347941\pi$$
0.459744 + 0.888052i $$0.347941\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −26.9109 −0.975521 −0.487760 0.872978i $$-0.662186\pi$$
−0.487760 + 0.872978i $$0.662186\pi$$
$$762$$ 0 0
$$763$$ −16.0000 −0.579239
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ −24.0000 −0.866590
$$768$$ 0 0
$$769$$ −35.1203 −1.26647 −0.633235 0.773959i $$-0.718273\pi$$
−0.633235 + 0.773959i $$0.718273\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ −33.0156 −1.18749 −0.593745 0.804654i $$-0.702351\pi$$
−0.593745 + 0.804654i $$0.702351\pi$$
$$774$$ 0 0
$$775$$ −21.6125 −0.776344
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ −3.40312 −0.121930
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 5.40312 0.192846
$$786$$ 0 0
$$787$$ −38.8062 −1.38329 −0.691647 0.722236i $$-0.743114\pi$$
−0.691647 + 0.722236i $$0.743114\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 84.6281 3.00903
$$792$$ 0 0
$$793$$ 7.79063 0.276653
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 12.5969 0.446204 0.223102 0.974795i $$-0.428382\pi$$
0.223102 + 0.974795i $$0.428382\pi$$
$$798$$ 0 0
$$799$$ −1.89531 −0.0670514
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 31.5078 1.11189
$$804$$ 0 0
$$805$$ −50.8062 −1.79068
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ 49.7172 1.74796 0.873982 0.485959i $$-0.161530\pi$$
0.873982 + 0.485959i $$0.161530\pi$$
$$810$$ 0 0
$$811$$ 2.59688 0.0911886 0.0455943 0.998960i $$-0.485482\pi$$
0.0455943 + 0.998960i $$0.485482\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ −40.0000 −1.40114
$$816$$ 0 0
$$817$$ 10.1047 0.353518
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −14.4922 −0.505781 −0.252890 0.967495i $$-0.581381\pi$$
−0.252890 + 0.967495i $$0.581381\pi$$
$$822$$ 0 0
$$823$$ 10.3141 0.359525 0.179763 0.983710i $$-0.442467\pi$$
0.179763 + 0.983710i $$0.442467\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −44.4187 −1.54459 −0.772296 0.635263i $$-0.780892\pi$$
−0.772296 + 0.635263i $$0.780892\pi$$
$$828$$ 0 0
$$829$$ −22.2094 −0.771363 −0.385682 0.922632i $$-0.626034\pi$$
−0.385682 + 0.922632i $$0.626034\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ 40.8062 1.41385
$$834$$ 0 0
$$835$$ −54.5969 −1.88940
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ −31.0156 −1.07078 −0.535389 0.844606i $$-0.679835\pi$$
−0.535389 + 0.844606i $$0.679835\pi$$
$$840$$ 0 0
$$841$$ −25.0000 −0.862069
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ −62.1359 −2.13754
$$846$$ 0 0
$$847$$ 52.2094 1.79394
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −13.6125 −0.466630
$$852$$ 0 0
$$853$$ 3.19375 0.109352 0.0546760 0.998504i $$-0.482587\pi$$
0.0546760 + 0.998504i $$0.482587\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ −53.0156 −1.81098 −0.905490 0.424369i $$-0.860496\pi$$
−0.905490 + 0.424369i $$0.860496\pi$$
$$858$$ 0 0
$$859$$ −31.2984 −1.06789 −0.533944 0.845520i $$-0.679291\pi$$
−0.533944 + 0.845520i $$0.679291\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ −26.8062 −0.912495 −0.456248 0.889853i $$-0.650807\pi$$
−0.456248 + 0.889853i $$0.650807\pi$$
$$864$$ 0 0
$$865$$ −23.7906 −0.808906
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 50.8062 1.72348
$$870$$ 0 0
$$871$$ 72.0000 2.43963
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 34.3141 1.16003
$$876$$ 0 0
$$877$$ −52.8062 −1.78314 −0.891570 0.452883i $$-0.850396\pi$$
−0.891570 + 0.452883i $$0.850396\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −32.1047 −1.08163 −0.540817 0.841140i $$-0.681885\pi$$
−0.540817 + 0.841140i $$0.681885\pi$$
$$882$$ 0 0
$$883$$ −44.9109 −1.51137 −0.755687 0.654933i $$-0.772697\pi$$
−0.755687 + 0.654933i $$0.772697\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −29.6125 −0.994290 −0.497145 0.867667i $$-0.665618\pi$$
−0.497145 + 0.867667i $$0.665618\pi$$
$$888$$ 0 0
$$889$$ −44.2094 −1.48273
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ 0.701562 0.0234769
$$894$$ 0 0
$$895$$ 14.5969 0.487920
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 18.8062 0.627224
$$900$$ 0 0
$$901$$ 16.2094 0.540013
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 12.9844 0.431615
$$906$$ 0 0
$$907$$ −27.0156 −0.897039 −0.448519 0.893773i $$-0.648049\pi$$
−0.448519 + 0.893773i $$0.648049\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −28.2094 −0.934618 −0.467309 0.884094i $$-0.654776\pi$$
−0.467309 + 0.884094i $$0.654776\pi$$
$$912$$ 0 0
$$913$$ −50.8062 −1.68144
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ −59.7172 −1.97204
$$918$$ 0 0
$$919$$ 29.6125 0.976826 0.488413 0.872613i $$-0.337576\pi$$
0.488413 + 0.872613i $$0.337576\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ 0 0
$$924$$ 0 0
$$925$$ −7.82187 −0.257181
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ −36.8062 −1.20757 −0.603787 0.797146i $$-0.706342\pi$$
−0.603787 + 0.797146i $$0.706342\pi$$
$$930$$ 0 0
$$931$$ −15.1047 −0.495036
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ −34.3141 −1.12219
$$936$$ 0 0
$$937$$ 6.70156 0.218930 0.109465 0.993991i $$-0.465086\pi$$
0.109465 + 0.993991i $$0.465086\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −10.0000 −0.325991 −0.162995 0.986627i $$-0.552116\pi$$
−0.162995 + 0.986627i $$0.552116\pi$$
$$942$$ 0 0
$$943$$ 13.6125 0.443284
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 18.8062 0.611121 0.305560 0.952173i $$-0.401156\pi$$
0.305560 + 0.952173i $$0.401156\pi$$
$$948$$ 0 0
$$949$$ 40.2094 1.30525
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ −7.40312 −0.239811 −0.119905 0.992785i $$-0.538259\pi$$
−0.119905 + 0.992785i $$0.538259\pi$$
$$954$$ 0 0
$$955$$ 56.4922 1.82804
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 50.3141 1.62473
$$960$$ 0 0
$$961$$ 57.4187 1.85222
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ −30.8062 −0.991688
$$966$$ 0 0
$$967$$ −40.0000 −1.28631 −0.643157 0.765735i $$-0.722376\pi$$
−0.643157 + 0.765735i $$0.722376\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 20.0000 0.641831 0.320915 0.947108i $$-0.396010\pi$$
0.320915 + 0.947108i $$0.396010\pi$$
$$972$$ 0 0
$$973$$ −71.9266 −2.30586
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 18.0000 0.575871 0.287936 0.957650i $$-0.407031\pi$$
0.287936 + 0.957650i $$0.407031\pi$$
$$978$$ 0 0
$$979$$ 60.2094 1.92430
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ 32.4187 1.03400 0.516999 0.855986i $$-0.327049\pi$$
0.516999 + 0.855986i $$0.327049\pi$$
$$984$$ 0 0
$$985$$ 59.4344 1.89374
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −40.4187 −1.28524
$$990$$ 0 0
$$991$$ 48.0000 1.52477 0.762385 0.647124i $$-0.224028\pi$$
0.762385 + 0.647124i $$0.224028\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 8.91093 0.282496
$$996$$ 0 0
$$997$$ 15.8953 0.503410 0.251705 0.967804i $$-0.419009\pi$$
0.251705 + 0.967804i $$0.419009\pi$$
$$998$$ 0 0
$$999$$ 0 0
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 2736.2.a.bb.1.1 2
3.2 odd 2 912.2.a.o.1.2 2
4.3 odd 2 1368.2.a.l.1.1 2
12.11 even 2 456.2.a.e.1.2 2
24.5 odd 2 3648.2.a.bn.1.1 2
24.11 even 2 3648.2.a.bs.1.1 2
228.227 odd 2 8664.2.a.v.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
456.2.a.e.1.2 2 12.11 even 2
912.2.a.o.1.2 2 3.2 odd 2
1368.2.a.l.1.1 2 4.3 odd 2
2736.2.a.bb.1.1 2 1.1 even 1 trivial
3648.2.a.bn.1.1 2 24.5 odd 2
3648.2.a.bs.1.1 2 24.11 even 2
8664.2.a.v.1.2 2 228.227 odd 2