# Properties

 Label 2496.4.a.c.1.1 Level $2496$ Weight $4$ Character 2496.1 Self dual yes Analytic conductor $147.269$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2496,4,Mod(1,2496)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2496, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("2496.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2496 = 2^{6} \cdot 3 \cdot 13$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 2496.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: $$147.268767374$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 78) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 2496.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-3.00000 q^{3} -4.00000 q^{5} +4.00000 q^{7} +9.00000 q^{9} +O(q^{10})$$ $$q-3.00000 q^{3} -4.00000 q^{5} +4.00000 q^{7} +9.00000 q^{9} -2.00000 q^{11} +13.0000 q^{13} +12.0000 q^{15} -6.00000 q^{17} +36.0000 q^{19} -12.0000 q^{21} -20.0000 q^{23} -109.000 q^{25} -27.0000 q^{27} +14.0000 q^{29} -152.000 q^{31} +6.00000 q^{33} -16.0000 q^{35} +258.000 q^{37} -39.0000 q^{39} +84.0000 q^{41} +188.000 q^{43} -36.0000 q^{45} +254.000 q^{47} -327.000 q^{49} +18.0000 q^{51} -366.000 q^{53} +8.00000 q^{55} -108.000 q^{57} -550.000 q^{59} +14.0000 q^{61} +36.0000 q^{63} -52.0000 q^{65} -448.000 q^{67} +60.0000 q^{69} +926.000 q^{71} +254.000 q^{73} +327.000 q^{75} -8.00000 q^{77} +1328.00 q^{79} +81.0000 q^{81} -186.000 q^{83} +24.0000 q^{85} -42.0000 q^{87} -336.000 q^{89} +52.0000 q^{91} +456.000 q^{93} -144.000 q^{95} +614.000 q^{97} -18.0000 q^{99} +O(q^{100})$$

## 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
$$3$$ −3.00000 −0.577350
$$4$$ 0 0
$$5$$ −4.00000 −0.357771 −0.178885 0.983870i $$-0.557249\pi$$
−0.178885 + 0.983870i $$0.557249\pi$$
$$6$$ 0 0
$$7$$ 4.00000 0.215980 0.107990 0.994152i $$-0.465559\pi$$
0.107990 + 0.994152i $$0.465559\pi$$
$$8$$ 0 0
$$9$$ 9.00000 0.333333
$$10$$ 0 0
$$11$$ −2.00000 −0.0548202 −0.0274101 0.999624i $$-0.508726\pi$$
−0.0274101 + 0.999624i $$0.508726\pi$$
$$12$$ 0 0
$$13$$ 13.0000 0.277350
$$14$$ 0 0
$$15$$ 12.0000 0.206559
$$16$$ 0 0
$$17$$ −6.00000 −0.0856008 −0.0428004 0.999084i $$-0.513628\pi$$
−0.0428004 + 0.999084i $$0.513628\pi$$
$$18$$ 0 0
$$19$$ 36.0000 0.434682 0.217341 0.976096i $$-0.430262\pi$$
0.217341 + 0.976096i $$0.430262\pi$$
$$20$$ 0 0
$$21$$ −12.0000 −0.124696
$$22$$ 0 0
$$23$$ −20.0000 −0.181317 −0.0906584 0.995882i $$-0.528897\pi$$
−0.0906584 + 0.995882i $$0.528897\pi$$
$$24$$ 0 0
$$25$$ −109.000 −0.872000
$$26$$ 0 0
$$27$$ −27.0000 −0.192450
$$28$$ 0 0
$$29$$ 14.0000 0.0896460 0.0448230 0.998995i $$-0.485728\pi$$
0.0448230 + 0.998995i $$0.485728\pi$$
$$30$$ 0 0
$$31$$ −152.000 −0.880645 −0.440323 0.897840i $$-0.645136\pi$$
−0.440323 + 0.897840i $$0.645136\pi$$
$$32$$ 0 0
$$33$$ 6.00000 0.0316505
$$34$$ 0 0
$$35$$ −16.0000 −0.0772712
$$36$$ 0 0
$$37$$ 258.000 1.14635 0.573175 0.819433i $$-0.305712\pi$$
0.573175 + 0.819433i $$0.305712\pi$$
$$38$$ 0 0
$$39$$ −39.0000 −0.160128
$$40$$ 0 0
$$41$$ 84.0000 0.319966 0.159983 0.987120i $$-0.448856\pi$$
0.159983 + 0.987120i $$0.448856\pi$$
$$42$$ 0 0
$$43$$ 188.000 0.666738 0.333369 0.942796i $$-0.391815\pi$$
0.333369 + 0.942796i $$0.391815\pi$$
$$44$$ 0 0
$$45$$ −36.0000 −0.119257
$$46$$ 0 0
$$47$$ 254.000 0.788292 0.394146 0.919048i $$-0.371040\pi$$
0.394146 + 0.919048i $$0.371040\pi$$
$$48$$ 0 0
$$49$$ −327.000 −0.953353
$$50$$ 0 0
$$51$$ 18.0000 0.0494217
$$52$$ 0 0
$$53$$ −366.000 −0.948565 −0.474283 0.880373i $$-0.657293\pi$$
−0.474283 + 0.880373i $$0.657293\pi$$
$$54$$ 0 0
$$55$$ 8.00000 0.0196131
$$56$$ 0 0
$$57$$ −108.000 −0.250964
$$58$$ 0 0
$$59$$ −550.000 −1.21363 −0.606813 0.794845i $$-0.707552\pi$$
−0.606813 + 0.794845i $$0.707552\pi$$
$$60$$ 0 0
$$61$$ 14.0000 0.0293855 0.0146928 0.999892i $$-0.495323\pi$$
0.0146928 + 0.999892i $$0.495323\pi$$
$$62$$ 0 0
$$63$$ 36.0000 0.0719932
$$64$$ 0 0
$$65$$ −52.0000 −0.0992278
$$66$$ 0 0
$$67$$ −448.000 −0.816894 −0.408447 0.912782i $$-0.633930\pi$$
−0.408447 + 0.912782i $$0.633930\pi$$
$$68$$ 0 0
$$69$$ 60.0000 0.104683
$$70$$ 0 0
$$71$$ 926.000 1.54783 0.773915 0.633289i $$-0.218296\pi$$
0.773915 + 0.633289i $$0.218296\pi$$
$$72$$ 0 0
$$73$$ 254.000 0.407239 0.203620 0.979050i $$-0.434729\pi$$
0.203620 + 0.979050i $$0.434729\pi$$
$$74$$ 0 0
$$75$$ 327.000 0.503449
$$76$$ 0 0
$$77$$ −8.00000 −0.0118401
$$78$$ 0 0
$$79$$ 1328.00 1.89129 0.945644 0.325205i $$-0.105433\pi$$
0.945644 + 0.325205i $$0.105433\pi$$
$$80$$ 0 0
$$81$$ 81.0000 0.111111
$$82$$ 0 0
$$83$$ −186.000 −0.245978 −0.122989 0.992408i $$-0.539248\pi$$
−0.122989 + 0.992408i $$0.539248\pi$$
$$84$$ 0 0
$$85$$ 24.0000 0.0306255
$$86$$ 0 0
$$87$$ −42.0000 −0.0517572
$$88$$ 0 0
$$89$$ −336.000 −0.400179 −0.200089 0.979778i $$-0.564123\pi$$
−0.200089 + 0.979778i $$0.564123\pi$$
$$90$$ 0 0
$$91$$ 52.0000 0.0599020
$$92$$ 0 0
$$93$$ 456.000 0.508441
$$94$$ 0 0
$$95$$ −144.000 −0.155517
$$96$$ 0 0
$$97$$ 614.000 0.642704 0.321352 0.946960i $$-0.395863\pi$$
0.321352 + 0.946960i $$0.395863\pi$$
$$98$$ 0 0
$$99$$ −18.0000 −0.0182734
$$100$$ 0 0
$$101$$ 1606.00 1.58221 0.791104 0.611682i $$-0.209507\pi$$
0.791104 + 0.611682i $$0.209507\pi$$
$$102$$ 0 0
$$103$$ 208.000 0.198979 0.0994896 0.995039i $$-0.468279\pi$$
0.0994896 + 0.995039i $$0.468279\pi$$
$$104$$ 0 0
$$105$$ 48.0000 0.0446126
$$106$$ 0 0
$$107$$ 248.000 0.224066 0.112033 0.993704i $$-0.464264\pi$$
0.112033 + 0.993704i $$0.464264\pi$$
$$108$$ 0 0
$$109$$ 542.000 0.476277 0.238138 0.971231i $$-0.423463\pi$$
0.238138 + 0.971231i $$0.423463\pi$$
$$110$$ 0 0
$$111$$ −774.000 −0.661845
$$112$$ 0 0
$$113$$ −2042.00 −1.69996 −0.849979 0.526817i $$-0.823385\pi$$
−0.849979 + 0.526817i $$0.823385\pi$$
$$114$$ 0 0
$$115$$ 80.0000 0.0648699
$$116$$ 0 0
$$117$$ 117.000 0.0924500
$$118$$ 0 0
$$119$$ −24.0000 −0.0184880
$$120$$ 0 0
$$121$$ −1327.00 −0.996995
$$122$$ 0 0
$$123$$ −252.000 −0.184732
$$124$$ 0 0
$$125$$ 936.000 0.669747
$$126$$ 0 0
$$127$$ −488.000 −0.340968 −0.170484 0.985360i $$-0.554533\pi$$
−0.170484 + 0.985360i $$0.554533\pi$$
$$128$$ 0 0
$$129$$ −564.000 −0.384941
$$130$$ 0 0
$$131$$ −1744.00 −1.16316 −0.581580 0.813489i $$-0.697565\pi$$
−0.581580 + 0.813489i $$0.697565\pi$$
$$132$$ 0 0
$$133$$ 144.000 0.0938826
$$134$$ 0 0
$$135$$ 108.000 0.0688530
$$136$$ 0 0
$$137$$ −828.000 −0.516356 −0.258178 0.966097i $$-0.583122\pi$$
−0.258178 + 0.966097i $$0.583122\pi$$
$$138$$ 0 0
$$139$$ 404.000 0.246524 0.123262 0.992374i $$-0.460664\pi$$
0.123262 + 0.992374i $$0.460664\pi$$
$$140$$ 0 0
$$141$$ −762.000 −0.455120
$$142$$ 0 0
$$143$$ −26.0000 −0.0152044
$$144$$ 0 0
$$145$$ −56.0000 −0.0320727
$$146$$ 0 0
$$147$$ 981.000 0.550418
$$148$$ 0 0
$$149$$ −2928.00 −1.60987 −0.804937 0.593361i $$-0.797801\pi$$
−0.804937 + 0.593361i $$0.797801\pi$$
$$150$$ 0 0
$$151$$ 1944.00 1.04769 0.523843 0.851815i $$-0.324498\pi$$
0.523843 + 0.851815i $$0.324498\pi$$
$$152$$ 0 0
$$153$$ −54.0000 −0.0285336
$$154$$ 0 0
$$155$$ 608.000 0.315069
$$156$$ 0 0
$$157$$ −3590.00 −1.82492 −0.912462 0.409161i $$-0.865822\pi$$
−0.912462 + 0.409161i $$0.865822\pi$$
$$158$$ 0 0
$$159$$ 1098.00 0.547654
$$160$$ 0 0
$$161$$ −80.0000 −0.0391608
$$162$$ 0 0
$$163$$ 2284.00 1.09753 0.548763 0.835978i $$-0.315099\pi$$
0.548763 + 0.835978i $$0.315099\pi$$
$$164$$ 0 0
$$165$$ −24.0000 −0.0113236
$$166$$ 0 0
$$167$$ 3174.00 1.47073 0.735364 0.677673i $$-0.237011\pi$$
0.735364 + 0.677673i $$0.237011\pi$$
$$168$$ 0 0
$$169$$ 169.000 0.0769231
$$170$$ 0 0
$$171$$ 324.000 0.144894
$$172$$ 0 0
$$173$$ 1358.00 0.596802 0.298401 0.954441i $$-0.403547\pi$$
0.298401 + 0.954441i $$0.403547\pi$$
$$174$$ 0 0
$$175$$ −436.000 −0.188334
$$176$$ 0 0
$$177$$ 1650.00 0.700687
$$178$$ 0 0
$$179$$ −708.000 −0.295634 −0.147817 0.989015i $$-0.547225\pi$$
−0.147817 + 0.989015i $$0.547225\pi$$
$$180$$ 0 0
$$181$$ 546.000 0.224220 0.112110 0.993696i $$-0.464239\pi$$
0.112110 + 0.993696i $$0.464239\pi$$
$$182$$ 0 0
$$183$$ −42.0000 −0.0169657
$$184$$ 0 0
$$185$$ −1032.00 −0.410131
$$186$$ 0 0
$$187$$ 12.0000 0.00469266
$$188$$ 0 0
$$189$$ −108.000 −0.0415653
$$190$$ 0 0
$$191$$ −3472.00 −1.31531 −0.657657 0.753317i $$-0.728453\pi$$
−0.657657 + 0.753317i $$0.728453\pi$$
$$192$$ 0 0
$$193$$ −310.000 −0.115618 −0.0578090 0.998328i $$-0.518411\pi$$
−0.0578090 + 0.998328i $$0.518411\pi$$
$$194$$ 0 0
$$195$$ 156.000 0.0572892
$$196$$ 0 0
$$197$$ −1020.00 −0.368893 −0.184447 0.982843i $$-0.559049\pi$$
−0.184447 + 0.982843i $$0.559049\pi$$
$$198$$ 0 0
$$199$$ −3256.00 −1.15986 −0.579929 0.814667i $$-0.696920\pi$$
−0.579929 + 0.814667i $$0.696920\pi$$
$$200$$ 0 0
$$201$$ 1344.00 0.471634
$$202$$ 0 0
$$203$$ 56.0000 0.0193617
$$204$$ 0 0
$$205$$ −336.000 −0.114474
$$206$$ 0 0
$$207$$ −180.000 −0.0604390
$$208$$ 0 0
$$209$$ −72.0000 −0.0238294
$$210$$ 0 0
$$211$$ 4564.00 1.48909 0.744547 0.667570i $$-0.232666\pi$$
0.744547 + 0.667570i $$0.232666\pi$$
$$212$$ 0 0
$$213$$ −2778.00 −0.893640
$$214$$ 0 0
$$215$$ −752.000 −0.238539
$$216$$ 0 0
$$217$$ −608.000 −0.190202
$$218$$ 0 0
$$219$$ −762.000 −0.235120
$$220$$ 0 0
$$221$$ −78.0000 −0.0237414
$$222$$ 0 0
$$223$$ −72.0000 −0.0216210 −0.0108105 0.999942i $$-0.503441\pi$$
−0.0108105 + 0.999942i $$0.503441\pi$$
$$224$$ 0 0
$$225$$ −981.000 −0.290667
$$226$$ 0 0
$$227$$ −2694.00 −0.787696 −0.393848 0.919176i $$-0.628856\pi$$
−0.393848 + 0.919176i $$0.628856\pi$$
$$228$$ 0 0
$$229$$ −5922.00 −1.70889 −0.854447 0.519538i $$-0.826104\pi$$
−0.854447 + 0.519538i $$0.826104\pi$$
$$230$$ 0 0
$$231$$ 24.0000 0.00683586
$$232$$ 0 0
$$233$$ −5122.00 −1.44014 −0.720072 0.693900i $$-0.755891\pi$$
−0.720072 + 0.693900i $$0.755891\pi$$
$$234$$ 0 0
$$235$$ −1016.00 −0.282028
$$236$$ 0 0
$$237$$ −3984.00 −1.09194
$$238$$ 0 0
$$239$$ 5022.00 1.35919 0.679595 0.733588i $$-0.262156\pi$$
0.679595 + 0.733588i $$0.262156\pi$$
$$240$$ 0 0
$$241$$ −1218.00 −0.325553 −0.162777 0.986663i $$-0.552045\pi$$
−0.162777 + 0.986663i $$0.552045\pi$$
$$242$$ 0 0
$$243$$ −243.000 −0.0641500
$$244$$ 0 0
$$245$$ 1308.00 0.341082
$$246$$ 0 0
$$247$$ 468.000 0.120559
$$248$$ 0 0
$$249$$ 558.000 0.142015
$$250$$ 0 0
$$251$$ 2112.00 0.531109 0.265554 0.964096i $$-0.414445\pi$$
0.265554 + 0.964096i $$0.414445\pi$$
$$252$$ 0 0
$$253$$ 40.0000 0.00993984
$$254$$ 0 0
$$255$$ −72.0000 −0.0176816
$$256$$ 0 0
$$257$$ 2814.00 0.683006 0.341503 0.939881i $$-0.389064\pi$$
0.341503 + 0.939881i $$0.389064\pi$$
$$258$$ 0 0
$$259$$ 1032.00 0.247588
$$260$$ 0 0
$$261$$ 126.000 0.0298820
$$262$$ 0 0
$$263$$ −4044.00 −0.948151 −0.474076 0.880484i $$-0.657218\pi$$
−0.474076 + 0.880484i $$0.657218\pi$$
$$264$$ 0 0
$$265$$ 1464.00 0.339369
$$266$$ 0 0
$$267$$ 1008.00 0.231043
$$268$$ 0 0
$$269$$ 1470.00 0.333188 0.166594 0.986026i $$-0.446723\pi$$
0.166594 + 0.986026i $$0.446723\pi$$
$$270$$ 0 0
$$271$$ −1844.00 −0.413340 −0.206670 0.978411i $$-0.566263\pi$$
−0.206670 + 0.978411i $$0.566263\pi$$
$$272$$ 0 0
$$273$$ −156.000 −0.0345844
$$274$$ 0 0
$$275$$ 218.000 0.0478033
$$276$$ 0 0
$$277$$ −5766.00 −1.25071 −0.625353 0.780342i $$-0.715045\pi$$
−0.625353 + 0.780342i $$0.715045\pi$$
$$278$$ 0 0
$$279$$ −1368.00 −0.293548
$$280$$ 0 0
$$281$$ −7468.00 −1.58542 −0.792711 0.609598i $$-0.791331\pi$$
−0.792711 + 0.609598i $$0.791331\pi$$
$$282$$ 0 0
$$283$$ −1228.00 −0.257940 −0.128970 0.991648i $$-0.541167\pi$$
−0.128970 + 0.991648i $$0.541167\pi$$
$$284$$ 0 0
$$285$$ 432.000 0.0897876
$$286$$ 0 0
$$287$$ 336.000 0.0691061
$$288$$ 0 0
$$289$$ −4877.00 −0.992673
$$290$$ 0 0
$$291$$ −1842.00 −0.371065
$$292$$ 0 0
$$293$$ −6608.00 −1.31755 −0.658777 0.752338i $$-0.728926\pi$$
−0.658777 + 0.752338i $$0.728926\pi$$
$$294$$ 0 0
$$295$$ 2200.00 0.434200
$$296$$ 0 0
$$297$$ 54.0000 0.0105502
$$298$$ 0 0
$$299$$ −260.000 −0.0502883
$$300$$ 0 0
$$301$$ 752.000 0.144002
$$302$$ 0 0
$$303$$ −4818.00 −0.913488
$$304$$ 0 0
$$305$$ −56.0000 −0.0105133
$$306$$ 0 0
$$307$$ −7664.00 −1.42478 −0.712390 0.701784i $$-0.752387\pi$$
−0.712390 + 0.701784i $$0.752387\pi$$
$$308$$ 0 0
$$309$$ −624.000 −0.114881
$$310$$ 0 0
$$311$$ −2340.00 −0.426653 −0.213327 0.976981i $$-0.568430\pi$$
−0.213327 + 0.976981i $$0.568430\pi$$
$$312$$ 0 0
$$313$$ 6710.00 1.21173 0.605865 0.795567i $$-0.292827\pi$$
0.605865 + 0.795567i $$0.292827\pi$$
$$314$$ 0 0
$$315$$ −144.000 −0.0257571
$$316$$ 0 0
$$317$$ −4164.00 −0.737771 −0.368886 0.929475i $$-0.620261\pi$$
−0.368886 + 0.929475i $$0.620261\pi$$
$$318$$ 0 0
$$319$$ −28.0000 −0.00491442
$$320$$ 0 0
$$321$$ −744.000 −0.129365
$$322$$ 0 0
$$323$$ −216.000 −0.0372092
$$324$$ 0 0
$$325$$ −1417.00 −0.241849
$$326$$ 0 0
$$327$$ −1626.00 −0.274979
$$328$$ 0 0
$$329$$ 1016.00 0.170255
$$330$$ 0 0
$$331$$ 10072.0 1.67253 0.836265 0.548326i $$-0.184735\pi$$
0.836265 + 0.548326i $$0.184735\pi$$
$$332$$ 0 0
$$333$$ 2322.00 0.382117
$$334$$ 0 0
$$335$$ 1792.00 0.292261
$$336$$ 0 0
$$337$$ 2990.00 0.483311 0.241655 0.970362i $$-0.422310\pi$$
0.241655 + 0.970362i $$0.422310\pi$$
$$338$$ 0 0
$$339$$ 6126.00 0.981471
$$340$$ 0 0
$$341$$ 304.000 0.0482772
$$342$$ 0 0
$$343$$ −2680.00 −0.421885
$$344$$ 0 0
$$345$$ −240.000 −0.0374527
$$346$$ 0 0
$$347$$ −6564.00 −1.01549 −0.507743 0.861508i $$-0.669520\pi$$
−0.507743 + 0.861508i $$0.669520\pi$$
$$348$$ 0 0
$$349$$ 674.000 0.103376 0.0516882 0.998663i $$-0.483540\pi$$
0.0516882 + 0.998663i $$0.483540\pi$$
$$350$$ 0 0
$$351$$ −351.000 −0.0533761
$$352$$ 0 0
$$353$$ −10732.0 −1.61815 −0.809075 0.587706i $$-0.800031\pi$$
−0.809075 + 0.587706i $$0.800031\pi$$
$$354$$ 0 0
$$355$$ −3704.00 −0.553769
$$356$$ 0 0
$$357$$ 72.0000 0.0106741
$$358$$ 0 0
$$359$$ −4842.00 −0.711841 −0.355921 0.934516i $$-0.615833\pi$$
−0.355921 + 0.934516i $$0.615833\pi$$
$$360$$ 0 0
$$361$$ −5563.00 −0.811051
$$362$$ 0 0
$$363$$ 3981.00 0.575615
$$364$$ 0 0
$$365$$ −1016.00 −0.145698
$$366$$ 0 0
$$367$$ −6280.00 −0.893224 −0.446612 0.894728i $$-0.647370\pi$$
−0.446612 + 0.894728i $$0.647370\pi$$
$$368$$ 0 0
$$369$$ 756.000 0.106655
$$370$$ 0 0
$$371$$ −1464.00 −0.204871
$$372$$ 0 0
$$373$$ −6434.00 −0.893136 −0.446568 0.894750i $$-0.647354\pi$$
−0.446568 + 0.894750i $$0.647354\pi$$
$$374$$ 0 0
$$375$$ −2808.00 −0.386679
$$376$$ 0 0
$$377$$ 182.000 0.0248633
$$378$$ 0 0
$$379$$ 9068.00 1.22900 0.614501 0.788916i $$-0.289357\pi$$
0.614501 + 0.788916i $$0.289357\pi$$
$$380$$ 0 0
$$381$$ 1464.00 0.196858
$$382$$ 0 0
$$383$$ 3162.00 0.421855 0.210928 0.977502i $$-0.432352\pi$$
0.210928 + 0.977502i $$0.432352\pi$$
$$384$$ 0 0
$$385$$ 32.0000 0.00423603
$$386$$ 0 0
$$387$$ 1692.00 0.222246
$$388$$ 0 0
$$389$$ 3666.00 0.477824 0.238912 0.971041i $$-0.423209\pi$$
0.238912 + 0.971041i $$0.423209\pi$$
$$390$$ 0 0
$$391$$ 120.000 0.0155209
$$392$$ 0 0
$$393$$ 5232.00 0.671551
$$394$$ 0 0
$$395$$ −5312.00 −0.676647
$$396$$ 0 0
$$397$$ −11054.0 −1.39744 −0.698721 0.715394i $$-0.746247\pi$$
−0.698721 + 0.715394i $$0.746247\pi$$
$$398$$ 0 0
$$399$$ −432.000 −0.0542031
$$400$$ 0 0
$$401$$ −5328.00 −0.663510 −0.331755 0.943366i $$-0.607641\pi$$
−0.331755 + 0.943366i $$0.607641\pi$$
$$402$$ 0 0
$$403$$ −1976.00 −0.244247
$$404$$ 0 0
$$405$$ −324.000 −0.0397523
$$406$$ 0 0
$$407$$ −516.000 −0.0628432
$$408$$ 0 0
$$409$$ −12074.0 −1.45971 −0.729854 0.683603i $$-0.760412\pi$$
−0.729854 + 0.683603i $$0.760412\pi$$
$$410$$ 0 0
$$411$$ 2484.00 0.298118
$$412$$ 0 0
$$413$$ −2200.00 −0.262118
$$414$$ 0 0
$$415$$ 744.000 0.0880037
$$416$$ 0 0
$$417$$ −1212.00 −0.142331
$$418$$ 0 0
$$419$$ −13584.0 −1.58382 −0.791911 0.610636i $$-0.790914\pi$$
−0.791911 + 0.610636i $$0.790914\pi$$
$$420$$ 0 0
$$421$$ 7406.00 0.857355 0.428677 0.903458i $$-0.358980\pi$$
0.428677 + 0.903458i $$0.358980\pi$$
$$422$$ 0 0
$$423$$ 2286.00 0.262764
$$424$$ 0 0
$$425$$ 654.000 0.0746439
$$426$$ 0 0
$$427$$ 56.0000 0.00634667
$$428$$ 0 0
$$429$$ 78.0000 0.00877826
$$430$$ 0 0
$$431$$ −10134.0 −1.13257 −0.566285 0.824210i $$-0.691620\pi$$
−0.566285 + 0.824210i $$0.691620\pi$$
$$432$$ 0 0
$$433$$ 9406.00 1.04393 0.521967 0.852966i $$-0.325198\pi$$
0.521967 + 0.852966i $$0.325198\pi$$
$$434$$ 0 0
$$435$$ 168.000 0.0185172
$$436$$ 0 0
$$437$$ −720.000 −0.0788153
$$438$$ 0 0
$$439$$ 4088.00 0.444441 0.222220 0.974996i $$-0.428670\pi$$
0.222220 + 0.974996i $$0.428670\pi$$
$$440$$ 0 0
$$441$$ −2943.00 −0.317784
$$442$$ 0 0
$$443$$ 5328.00 0.571424 0.285712 0.958315i $$-0.407770\pi$$
0.285712 + 0.958315i $$0.407770\pi$$
$$444$$ 0 0
$$445$$ 1344.00 0.143172
$$446$$ 0 0
$$447$$ 8784.00 0.929461
$$448$$ 0 0
$$449$$ 13160.0 1.38320 0.691602 0.722279i $$-0.256905\pi$$
0.691602 + 0.722279i $$0.256905\pi$$
$$450$$ 0 0
$$451$$ −168.000 −0.0175406
$$452$$ 0 0
$$453$$ −5832.00 −0.604881
$$454$$ 0 0
$$455$$ −208.000 −0.0214312
$$456$$ 0 0
$$457$$ −9146.00 −0.936175 −0.468087 0.883682i $$-0.655057\pi$$
−0.468087 + 0.883682i $$0.655057\pi$$
$$458$$ 0 0
$$459$$ 162.000 0.0164739
$$460$$ 0 0
$$461$$ −5580.00 −0.563745 −0.281873 0.959452i $$-0.590956\pi$$
−0.281873 + 0.959452i $$0.590956\pi$$
$$462$$ 0 0
$$463$$ 14788.0 1.48436 0.742178 0.670203i $$-0.233793\pi$$
0.742178 + 0.670203i $$0.233793\pi$$
$$464$$ 0 0
$$465$$ −1824.00 −0.181905
$$466$$ 0 0
$$467$$ −12376.0 −1.22632 −0.613162 0.789957i $$-0.710103\pi$$
−0.613162 + 0.789957i $$0.710103\pi$$
$$468$$ 0 0
$$469$$ −1792.00 −0.176433
$$470$$ 0 0
$$471$$ 10770.0 1.05362
$$472$$ 0 0
$$473$$ −376.000 −0.0365507
$$474$$ 0 0
$$475$$ −3924.00 −0.379043
$$476$$ 0 0
$$477$$ −3294.00 −0.316188
$$478$$ 0 0
$$479$$ 834.000 0.0795541 0.0397771 0.999209i $$-0.487335\pi$$
0.0397771 + 0.999209i $$0.487335\pi$$
$$480$$ 0 0
$$481$$ 3354.00 0.317940
$$482$$ 0 0
$$483$$ 240.000 0.0226095
$$484$$ 0 0
$$485$$ −2456.00 −0.229941
$$486$$ 0 0
$$487$$ −13192.0 −1.22749 −0.613744 0.789505i $$-0.710337\pi$$
−0.613744 + 0.789505i $$0.710337\pi$$
$$488$$ 0 0
$$489$$ −6852.00 −0.633657
$$490$$ 0 0
$$491$$ −16568.0 −1.52282 −0.761409 0.648272i $$-0.775492\pi$$
−0.761409 + 0.648272i $$0.775492\pi$$
$$492$$ 0 0
$$493$$ −84.0000 −0.00767377
$$494$$ 0 0
$$495$$ 72.0000 0.00653770
$$496$$ 0 0
$$497$$ 3704.00 0.334300
$$498$$ 0 0
$$499$$ 10136.0 0.909318 0.454659 0.890666i $$-0.349761\pi$$
0.454659 + 0.890666i $$0.349761\pi$$
$$500$$ 0 0
$$501$$ −9522.00 −0.849125
$$502$$ 0 0
$$503$$ 10412.0 0.922959 0.461479 0.887151i $$-0.347319\pi$$
0.461479 + 0.887151i $$0.347319\pi$$
$$504$$ 0 0
$$505$$ −6424.00 −0.566068
$$506$$ 0 0
$$507$$ −507.000 −0.0444116
$$508$$ 0 0
$$509$$ 4180.00 0.363999 0.181999 0.983299i $$-0.441743\pi$$
0.181999 + 0.983299i $$0.441743\pi$$
$$510$$ 0 0
$$511$$ 1016.00 0.0879554
$$512$$ 0 0
$$513$$ −972.000 −0.0836547
$$514$$ 0 0
$$515$$ −832.000 −0.0711889
$$516$$ 0 0
$$517$$ −508.000 −0.0432143
$$518$$ 0 0
$$519$$ −4074.00 −0.344564
$$520$$ 0 0
$$521$$ −14610.0 −1.22855 −0.614276 0.789091i $$-0.710552\pi$$
−0.614276 + 0.789091i $$0.710552\pi$$
$$522$$ 0 0
$$523$$ 2172.00 0.181596 0.0907982 0.995869i $$-0.471058\pi$$
0.0907982 + 0.995869i $$0.471058\pi$$
$$524$$ 0 0
$$525$$ 1308.00 0.108735
$$526$$ 0 0
$$527$$ 912.000 0.0753840
$$528$$ 0 0
$$529$$ −11767.0 −0.967124
$$530$$ 0 0
$$531$$ −4950.00 −0.404542
$$532$$ 0 0
$$533$$ 1092.00 0.0887425
$$534$$ 0 0
$$535$$ −992.000 −0.0801643
$$536$$ 0 0
$$537$$ 2124.00 0.170684
$$538$$ 0 0
$$539$$ 654.000 0.0522630
$$540$$ 0 0
$$541$$ 11758.0 0.934410 0.467205 0.884149i $$-0.345261\pi$$
0.467205 + 0.884149i $$0.345261\pi$$
$$542$$ 0 0
$$543$$ −1638.00 −0.129454
$$544$$ 0 0
$$545$$ −2168.00 −0.170398
$$546$$ 0 0
$$547$$ −340.000 −0.0265765 −0.0132883 0.999912i $$-0.504230\pi$$
−0.0132883 + 0.999912i $$0.504230\pi$$
$$548$$ 0 0
$$549$$ 126.000 0.00979517
$$550$$ 0 0
$$551$$ 504.000 0.0389676
$$552$$ 0 0
$$553$$ 5312.00 0.408480
$$554$$ 0 0
$$555$$ 3096.00 0.236789
$$556$$ 0 0
$$557$$ 3768.00 0.286634 0.143317 0.989677i $$-0.454223\pi$$
0.143317 + 0.989677i $$0.454223\pi$$
$$558$$ 0 0
$$559$$ 2444.00 0.184920
$$560$$ 0 0
$$561$$ −36.0000 −0.00270931
$$562$$ 0 0
$$563$$ 10172.0 0.761454 0.380727 0.924687i $$-0.375674\pi$$
0.380727 + 0.924687i $$0.375674\pi$$
$$564$$ 0 0
$$565$$ 8168.00 0.608195
$$566$$ 0 0
$$567$$ 324.000 0.0239977
$$568$$ 0 0
$$569$$ −5506.00 −0.405665 −0.202833 0.979213i $$-0.565015\pi$$
−0.202833 + 0.979213i $$0.565015\pi$$
$$570$$ 0 0
$$571$$ −2340.00 −0.171499 −0.0857495 0.996317i $$-0.527328\pi$$
−0.0857495 + 0.996317i $$0.527328\pi$$
$$572$$ 0 0
$$573$$ 10416.0 0.759397
$$574$$ 0 0
$$575$$ 2180.00 0.158108
$$576$$ 0 0
$$577$$ −20094.0 −1.44978 −0.724891 0.688864i $$-0.758110\pi$$
−0.724891 + 0.688864i $$0.758110\pi$$
$$578$$ 0 0
$$579$$ 930.000 0.0667521
$$580$$ 0 0
$$581$$ −744.000 −0.0531262
$$582$$ 0 0
$$583$$ 732.000 0.0520006
$$584$$ 0 0
$$585$$ −468.000 −0.0330759
$$586$$ 0 0
$$587$$ 7118.00 0.500496 0.250248 0.968182i $$-0.419488\pi$$
0.250248 + 0.968182i $$0.419488\pi$$
$$588$$ 0 0
$$589$$ −5472.00 −0.382801
$$590$$ 0 0
$$591$$ 3060.00 0.212981
$$592$$ 0 0
$$593$$ −10328.0 −0.715211 −0.357606 0.933873i $$-0.616407\pi$$
−0.357606 + 0.933873i $$0.616407\pi$$
$$594$$ 0 0
$$595$$ 96.0000 0.00661448
$$596$$ 0 0
$$597$$ 9768.00 0.669644
$$598$$ 0 0
$$599$$ −19732.0 −1.34596 −0.672978 0.739662i $$-0.734985\pi$$
−0.672978 + 0.739662i $$0.734985\pi$$
$$600$$ 0 0
$$601$$ −12026.0 −0.816224 −0.408112 0.912932i $$-0.633813\pi$$
−0.408112 + 0.912932i $$0.633813\pi$$
$$602$$ 0 0
$$603$$ −4032.00 −0.272298
$$604$$ 0 0
$$605$$ 5308.00 0.356696
$$606$$ 0 0
$$607$$ 17016.0 1.13782 0.568911 0.822399i $$-0.307365\pi$$
0.568911 + 0.822399i $$0.307365\pi$$
$$608$$ 0 0
$$609$$ −168.000 −0.0111785
$$610$$ 0 0
$$611$$ 3302.00 0.218633
$$612$$ 0 0
$$613$$ −11654.0 −0.767864 −0.383932 0.923361i $$-0.625430\pi$$
−0.383932 + 0.923361i $$0.625430\pi$$
$$614$$ 0 0
$$615$$ 1008.00 0.0660918
$$616$$ 0 0
$$617$$ 11612.0 0.757669 0.378834 0.925465i $$-0.376325\pi$$
0.378834 + 0.925465i $$0.376325\pi$$
$$618$$ 0 0
$$619$$ −4024.00 −0.261290 −0.130645 0.991429i $$-0.541705\pi$$
−0.130645 + 0.991429i $$0.541705\pi$$
$$620$$ 0 0
$$621$$ 540.000 0.0348945
$$622$$ 0 0
$$623$$ −1344.00 −0.0864305
$$624$$ 0 0
$$625$$ 9881.00 0.632384
$$626$$ 0 0
$$627$$ 216.000 0.0137579
$$628$$ 0 0
$$629$$ −1548.00 −0.0981285
$$630$$ 0 0
$$631$$ −1088.00 −0.0686412 −0.0343206 0.999411i $$-0.510927\pi$$
−0.0343206 + 0.999411i $$0.510927\pi$$
$$632$$ 0 0
$$633$$ −13692.0 −0.859729
$$634$$ 0 0
$$635$$ 1952.00 0.121989
$$636$$ 0 0
$$637$$ −4251.00 −0.264412
$$638$$ 0 0
$$639$$ 8334.00 0.515944
$$640$$ 0 0
$$641$$ −7078.00 −0.436138 −0.218069 0.975933i $$-0.569976\pi$$
−0.218069 + 0.975933i $$0.569976\pi$$
$$642$$ 0 0
$$643$$ −8336.00 −0.511259 −0.255630 0.966775i $$-0.582283\pi$$
−0.255630 + 0.966775i $$0.582283\pi$$
$$644$$ 0 0
$$645$$ 2256.00 0.137721
$$646$$ 0 0
$$647$$ 32.0000 0.00194444 0.000972218 1.00000i $$-0.499691\pi$$
0.000972218 1.00000i $$0.499691\pi$$
$$648$$ 0 0
$$649$$ 1100.00 0.0665312
$$650$$ 0 0
$$651$$ 1824.00 0.109813
$$652$$ 0 0
$$653$$ 15822.0 0.948182 0.474091 0.880476i $$-0.342777\pi$$
0.474091 + 0.880476i $$0.342777\pi$$
$$654$$ 0 0
$$655$$ 6976.00 0.416145
$$656$$ 0 0
$$657$$ 2286.00 0.135746
$$658$$ 0 0
$$659$$ −21540.0 −1.27326 −0.636631 0.771169i $$-0.719672\pi$$
−0.636631 + 0.771169i $$0.719672\pi$$
$$660$$ 0 0
$$661$$ −8270.00 −0.486635 −0.243317 0.969947i $$-0.578236\pi$$
−0.243317 + 0.969947i $$0.578236\pi$$
$$662$$ 0 0
$$663$$ 234.000 0.0137071
$$664$$ 0 0
$$665$$ −576.000 −0.0335885
$$666$$ 0 0
$$667$$ −280.000 −0.0162543
$$668$$ 0 0
$$669$$ 216.000 0.0124829
$$670$$ 0 0
$$671$$ −28.0000 −0.00161092
$$672$$ 0 0
$$673$$ 8482.00 0.485820 0.242910 0.970049i $$-0.421898\pi$$
0.242910 + 0.970049i $$0.421898\pi$$
$$674$$ 0 0
$$675$$ 2943.00 0.167816
$$676$$ 0 0
$$677$$ −2550.00 −0.144763 −0.0723814 0.997377i $$-0.523060\pi$$
−0.0723814 + 0.997377i $$0.523060\pi$$
$$678$$ 0 0
$$679$$ 2456.00 0.138811
$$680$$ 0 0
$$681$$ 8082.00 0.454777
$$682$$ 0 0
$$683$$ 31534.0 1.76664 0.883320 0.468771i $$-0.155303\pi$$
0.883320 + 0.468771i $$0.155303\pi$$
$$684$$ 0 0
$$685$$ 3312.00 0.184737
$$686$$ 0 0
$$687$$ 17766.0 0.986631
$$688$$ 0 0
$$689$$ −4758.00 −0.263085
$$690$$ 0 0
$$691$$ −33832.0 −1.86256 −0.931281 0.364302i $$-0.881307\pi$$
−0.931281 + 0.364302i $$0.881307\pi$$
$$692$$ 0 0
$$693$$ −72.0000 −0.00394669
$$694$$ 0 0
$$695$$ −1616.00 −0.0881991
$$696$$ 0 0
$$697$$ −504.000 −0.0273893
$$698$$ 0 0
$$699$$ 15366.0 0.831467
$$700$$ 0 0
$$701$$ −19422.0 −1.04645 −0.523223 0.852196i $$-0.675271\pi$$
−0.523223 + 0.852196i $$0.675271\pi$$
$$702$$ 0 0
$$703$$ 9288.00 0.498298
$$704$$ 0 0
$$705$$ 3048.00 0.162829
$$706$$ 0 0
$$707$$ 6424.00 0.341725
$$708$$ 0 0
$$709$$ 1894.00 0.100325 0.0501627 0.998741i $$-0.484026\pi$$
0.0501627 + 0.998741i $$0.484026\pi$$
$$710$$ 0 0
$$711$$ 11952.0 0.630429
$$712$$ 0 0
$$713$$ 3040.00 0.159676
$$714$$ 0 0
$$715$$ 104.000 0.00543969
$$716$$ 0 0
$$717$$ −15066.0 −0.784728
$$718$$ 0 0
$$719$$ −20156.0 −1.04547 −0.522734 0.852496i $$-0.675088\pi$$
−0.522734 + 0.852496i $$0.675088\pi$$
$$720$$ 0 0
$$721$$ 832.000 0.0429754
$$722$$ 0 0
$$723$$ 3654.00 0.187958
$$724$$ 0 0
$$725$$ −1526.00 −0.0781713
$$726$$ 0 0
$$727$$ 11128.0 0.567696 0.283848 0.958869i $$-0.408389\pi$$
0.283848 + 0.958869i $$0.408389\pi$$
$$728$$ 0 0
$$729$$ 729.000 0.0370370
$$730$$ 0 0
$$731$$ −1128.00 −0.0570733
$$732$$ 0 0
$$733$$ −16202.0 −0.816418 −0.408209 0.912888i $$-0.633847\pi$$
−0.408209 + 0.912888i $$0.633847\pi$$
$$734$$ 0 0
$$735$$ −3924.00 −0.196924
$$736$$ 0 0
$$737$$ 896.000 0.0447823
$$738$$ 0 0
$$739$$ 5328.00 0.265215 0.132607 0.991169i $$-0.457665\pi$$
0.132607 + 0.991169i $$0.457665\pi$$
$$740$$ 0 0
$$741$$ −1404.00 −0.0696049
$$742$$ 0 0
$$743$$ −20482.0 −1.01132 −0.505661 0.862732i $$-0.668751\pi$$
−0.505661 + 0.862732i $$0.668751\pi$$
$$744$$ 0 0
$$745$$ 11712.0 0.575966
$$746$$ 0 0
$$747$$ −1674.00 −0.0819926
$$748$$ 0 0
$$749$$ 992.000 0.0483937
$$750$$ 0 0
$$751$$ 8040.00 0.390657 0.195329 0.980738i $$-0.437423\pi$$
0.195329 + 0.980738i $$0.437423\pi$$
$$752$$ 0 0
$$753$$ −6336.00 −0.306636
$$754$$ 0 0
$$755$$ −7776.00 −0.374831
$$756$$ 0 0
$$757$$ 15822.0 0.759657 0.379829 0.925057i $$-0.375983\pi$$
0.379829 + 0.925057i $$0.375983\pi$$
$$758$$ 0 0
$$759$$ −120.000 −0.00573877
$$760$$ 0 0
$$761$$ −1452.00 −0.0691655 −0.0345828 0.999402i $$-0.511010\pi$$
−0.0345828 + 0.999402i $$0.511010\pi$$
$$762$$ 0 0
$$763$$ 2168.00 0.102866
$$764$$ 0 0
$$765$$ 216.000 0.0102085
$$766$$ 0 0
$$767$$ −7150.00 −0.336599
$$768$$ 0 0
$$769$$ 32298.0 1.51456 0.757279 0.653091i $$-0.226528\pi$$
0.757279 + 0.653091i $$0.226528\pi$$
$$770$$ 0 0
$$771$$ −8442.00 −0.394334
$$772$$ 0 0
$$773$$ −18736.0 −0.871781 −0.435891 0.900000i $$-0.643567\pi$$
−0.435891 + 0.900000i $$0.643567\pi$$
$$774$$ 0 0
$$775$$ 16568.0 0.767923
$$776$$ 0 0
$$777$$ −3096.00 −0.142945
$$778$$ 0 0
$$779$$ 3024.00 0.139083
$$780$$ 0 0
$$781$$ −1852.00 −0.0848525
$$782$$ 0 0
$$783$$ −378.000 −0.0172524
$$784$$ 0 0
$$785$$ 14360.0 0.652905
$$786$$ 0 0
$$787$$ 40816.0 1.84871 0.924354 0.381536i $$-0.124605\pi$$
0.924354 + 0.381536i $$0.124605\pi$$
$$788$$ 0 0
$$789$$ 12132.0 0.547415
$$790$$ 0 0
$$791$$ −8168.00 −0.367156
$$792$$ 0 0
$$793$$ 182.000 0.00815008
$$794$$ 0 0
$$795$$ −4392.00 −0.195935
$$796$$ 0 0
$$797$$ −4518.00 −0.200798 −0.100399 0.994947i $$-0.532012\pi$$
−0.100399 + 0.994947i $$0.532012\pi$$
$$798$$ 0 0
$$799$$ −1524.00 −0.0674784
$$800$$ 0 0
$$801$$ −3024.00 −0.133393
$$802$$ 0 0
$$803$$ −508.000 −0.0223249
$$804$$ 0 0
$$805$$ 320.000 0.0140106
$$806$$ 0 0
$$807$$ −4410.00 −0.192366
$$808$$ 0 0
$$809$$ −5058.00 −0.219814 −0.109907 0.993942i $$-0.535055\pi$$
−0.109907 + 0.993942i $$0.535055\pi$$
$$810$$ 0 0
$$811$$ 22564.0 0.976978 0.488489 0.872570i $$-0.337548\pi$$
0.488489 + 0.872570i $$0.337548\pi$$
$$812$$ 0 0
$$813$$ 5532.00 0.238642
$$814$$ 0 0
$$815$$ −9136.00 −0.392663
$$816$$ 0 0
$$817$$ 6768.00 0.289819
$$818$$ 0 0
$$819$$ 468.000 0.0199673
$$820$$ 0 0
$$821$$ −32584.0 −1.38513 −0.692564 0.721357i $$-0.743519\pi$$
−0.692564 + 0.721357i $$0.743519\pi$$
$$822$$ 0 0
$$823$$ −9288.00 −0.393389 −0.196695 0.980465i $$-0.563021\pi$$
−0.196695 + 0.980465i $$0.563021\pi$$
$$824$$ 0 0
$$825$$ −654.000 −0.0275992
$$826$$ 0 0
$$827$$ −20586.0 −0.865593 −0.432796 0.901492i $$-0.642473\pi$$
−0.432796 + 0.901492i $$0.642473\pi$$
$$828$$ 0 0
$$829$$ 46118.0 1.93214 0.966070 0.258280i $$-0.0831556\pi$$
0.966070 + 0.258280i $$0.0831556\pi$$
$$830$$ 0 0
$$831$$ 17298.0 0.722095
$$832$$ 0 0
$$833$$ 1962.00 0.0816078
$$834$$ 0 0
$$835$$ −12696.0 −0.526183
$$836$$ 0 0
$$837$$ 4104.00 0.169480
$$838$$ 0 0
$$839$$ −39230.0 −1.61427 −0.807133 0.590369i $$-0.798982\pi$$
−0.807133 + 0.590369i $$0.798982\pi$$
$$840$$ 0 0
$$841$$ −24193.0 −0.991964
$$842$$ 0 0
$$843$$ 22404.0 0.915344
$$844$$ 0 0
$$845$$ −676.000 −0.0275208
$$846$$ 0 0
$$847$$ −5308.00 −0.215331
$$848$$ 0 0
$$849$$ 3684.00 0.148922
$$850$$ 0 0
$$851$$ −5160.00 −0.207853
$$852$$ 0 0
$$853$$ 18674.0 0.749573 0.374786 0.927111i $$-0.377716\pi$$
0.374786 + 0.927111i $$0.377716\pi$$
$$854$$ 0 0
$$855$$ −1296.00 −0.0518389
$$856$$ 0 0
$$857$$ 41678.0 1.66125 0.830626 0.556830i $$-0.187983\pi$$
0.830626 + 0.556830i $$0.187983\pi$$
$$858$$ 0 0
$$859$$ 14740.0 0.585474 0.292737 0.956193i $$-0.405434\pi$$
0.292737 + 0.956193i $$0.405434\pi$$
$$860$$ 0 0
$$861$$ −1008.00 −0.0398984
$$862$$ 0 0
$$863$$ −24982.0 −0.985396 −0.492698 0.870200i $$-0.663989\pi$$
−0.492698 + 0.870200i $$0.663989\pi$$
$$864$$ 0 0
$$865$$ −5432.00 −0.213519
$$866$$ 0 0
$$867$$ 14631.0 0.573120
$$868$$ 0 0
$$869$$ −2656.00 −0.103681
$$870$$ 0 0
$$871$$ −5824.00 −0.226566
$$872$$ 0 0
$$873$$ 5526.00 0.214235
$$874$$ 0 0
$$875$$ 3744.00 0.144652
$$876$$ 0 0
$$877$$ −1134.00 −0.0436630 −0.0218315 0.999762i $$-0.506950\pi$$
−0.0218315 + 0.999762i $$0.506950\pi$$
$$878$$ 0 0
$$879$$ 19824.0 0.760690
$$880$$ 0 0
$$881$$ 34950.0 1.33654 0.668272 0.743917i $$-0.267034\pi$$
0.668272 + 0.743917i $$0.267034\pi$$
$$882$$ 0 0
$$883$$ 3068.00 0.116927 0.0584634 0.998290i $$-0.481380\pi$$
0.0584634 + 0.998290i $$0.481380\pi$$
$$884$$ 0 0
$$885$$ −6600.00 −0.250685
$$886$$ 0 0
$$887$$ −14080.0 −0.532988 −0.266494 0.963837i $$-0.585865\pi$$
−0.266494 + 0.963837i $$0.585865\pi$$
$$888$$ 0 0
$$889$$ −1952.00 −0.0736423
$$890$$ 0 0
$$891$$ −162.000 −0.00609114
$$892$$ 0 0
$$893$$ 9144.00 0.342657
$$894$$ 0 0
$$895$$ 2832.00 0.105769
$$896$$ 0 0
$$897$$ 780.000 0.0290339
$$898$$ 0 0
$$899$$ −2128.00 −0.0789464
$$900$$ 0 0
$$901$$ 2196.00 0.0811980
$$902$$ 0 0
$$903$$ −2256.00 −0.0831395
$$904$$ 0 0
$$905$$ −2184.00 −0.0802195
$$906$$ 0 0
$$907$$ 24876.0 0.910688 0.455344 0.890316i $$-0.349516\pi$$
0.455344 + 0.890316i $$0.349516\pi$$
$$908$$ 0 0
$$909$$ 14454.0 0.527403
$$910$$ 0 0
$$911$$ 51456.0 1.87136 0.935682 0.352843i $$-0.114785\pi$$
0.935682 + 0.352843i $$0.114785\pi$$
$$912$$ 0 0
$$913$$ 372.000 0.0134846
$$914$$ 0 0
$$915$$ 168.000 0.00606985
$$916$$ 0 0
$$917$$ −6976.00 −0.251219
$$918$$ 0 0
$$919$$ −31032.0 −1.11388 −0.556938 0.830554i $$-0.688024\pi$$
−0.556938 + 0.830554i $$0.688024\pi$$
$$920$$ 0 0
$$921$$ 22992.0 0.822597
$$922$$ 0 0
$$923$$ 12038.0 0.429291
$$924$$ 0 0
$$925$$ −28122.0 −0.999617
$$926$$ 0 0
$$927$$ 1872.00 0.0663264
$$928$$ 0 0
$$929$$ 50820.0 1.79478 0.897390 0.441239i $$-0.145461\pi$$
0.897390 + 0.441239i $$0.145461\pi$$
$$930$$ 0 0
$$931$$ −11772.0 −0.414406
$$932$$ 0 0
$$933$$ 7020.00 0.246328
$$934$$ 0 0
$$935$$ −48.0000 −0.00167890
$$936$$ 0 0
$$937$$ 5982.00 0.208563 0.104281 0.994548i $$-0.466746\pi$$
0.104281 + 0.994548i $$0.466746\pi$$
$$938$$ 0 0
$$939$$ −20130.0 −0.699593
$$940$$ 0 0
$$941$$ 20224.0 0.700620 0.350310 0.936634i $$-0.386076\pi$$
0.350310 + 0.936634i $$0.386076\pi$$
$$942$$ 0 0
$$943$$ −1680.00 −0.0580152
$$944$$ 0 0
$$945$$ 432.000 0.0148709
$$946$$ 0 0
$$947$$ −8478.00 −0.290917 −0.145458 0.989364i $$-0.546466\pi$$
−0.145458 + 0.989364i $$0.546466\pi$$
$$948$$ 0 0
$$949$$ 3302.00 0.112948
$$950$$ 0 0
$$951$$ 12492.0 0.425953
$$952$$ 0 0
$$953$$ 40918.0 1.39083 0.695417 0.718607i $$-0.255220\pi$$
0.695417 + 0.718607i $$0.255220\pi$$
$$954$$ 0 0
$$955$$ 13888.0 0.470581
$$956$$ 0 0
$$957$$ 84.0000 0.00283734
$$958$$ 0 0
$$959$$ −3312.00 −0.111522
$$960$$ 0 0
$$961$$ −6687.00 −0.224464
$$962$$ 0 0
$$963$$ 2232.00 0.0746887
$$964$$ 0 0
$$965$$ 1240.00 0.0413648
$$966$$ 0 0
$$967$$ −4624.00 −0.153772 −0.0768862 0.997040i $$-0.524498\pi$$
−0.0768862 + 0.997040i $$0.524498\pi$$
$$968$$ 0 0
$$969$$ 648.000 0.0214827
$$970$$ 0 0
$$971$$ −15300.0 −0.505665 −0.252832 0.967510i $$-0.581362\pi$$
−0.252832 + 0.967510i $$0.581362\pi$$
$$972$$ 0 0
$$973$$ 1616.00 0.0532442
$$974$$ 0 0
$$975$$ 4251.00 0.139632
$$976$$ 0 0
$$977$$ 19584.0 0.641298 0.320649 0.947198i $$-0.396099\pi$$
0.320649 + 0.947198i $$0.396099\pi$$
$$978$$ 0 0
$$979$$ 672.000 0.0219379
$$980$$ 0 0
$$981$$ 4878.00 0.158759
$$982$$ 0 0
$$983$$ −17582.0 −0.570477 −0.285238 0.958457i $$-0.592073\pi$$
−0.285238 + 0.958457i $$0.592073\pi$$
$$984$$ 0 0
$$985$$ 4080.00 0.131979
$$986$$ 0 0
$$987$$ −3048.00 −0.0982968
$$988$$ 0 0
$$989$$ −3760.00 −0.120891
$$990$$ 0 0
$$991$$ 47904.0 1.53554 0.767770 0.640725i $$-0.221366\pi$$
0.767770 + 0.640725i $$0.221366\pi$$
$$992$$ 0 0
$$993$$ −30216.0 −0.965635
$$994$$ 0 0
$$995$$ 13024.0 0.414963
$$996$$ 0 0
$$997$$ 44578.0 1.41605 0.708024 0.706189i $$-0.249587\pi$$
0.708024 + 0.706189i $$0.249587\pi$$
$$998$$ 0 0
$$999$$ −6966.00 −0.220615
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 2496.4.a.c.1.1 1
4.3 odd 2 2496.4.a.l.1.1 1
8.3 odd 2 624.4.a.c.1.1 1
8.5 even 2 78.4.a.f.1.1 1
24.5 odd 2 234.4.a.c.1.1 1
24.11 even 2 1872.4.a.f.1.1 1
40.29 even 2 1950.4.a.a.1.1 1
104.5 odd 4 1014.4.b.g.337.1 2
104.21 odd 4 1014.4.b.g.337.2 2
104.77 even 2 1014.4.a.e.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
78.4.a.f.1.1 1 8.5 even 2
234.4.a.c.1.1 1 24.5 odd 2
624.4.a.c.1.1 1 8.3 odd 2
1014.4.a.e.1.1 1 104.77 even 2
1014.4.b.g.337.1 2 104.5 odd 4
1014.4.b.g.337.2 2 104.21 odd 4
1872.4.a.f.1.1 1 24.11 even 2
1950.4.a.a.1.1 1 40.29 even 2
2496.4.a.c.1.1 1 1.1 even 1 trivial
2496.4.a.l.1.1 1 4.3 odd 2