# Properties

 Label 48.4.a.b.1.1 Level $48$ Weight $4$ Character 48.1 Self dual yes Analytic conductor $2.832$ Analytic rank $0$ 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] = [48,4,Mod(1,48)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(48, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("48.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$48 = 2^{4} \cdot 3$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 48.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: $$2.83209168028$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 24) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 48.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-3.00000 q^{3} +14.0000 q^{5} +24.0000 q^{7} +9.00000 q^{9} +O(q^{10})$$ $$q-3.00000 q^{3} +14.0000 q^{5} +24.0000 q^{7} +9.00000 q^{9} +28.0000 q^{11} -74.0000 q^{13} -42.0000 q^{15} +82.0000 q^{17} -92.0000 q^{19} -72.0000 q^{21} -8.00000 q^{23} +71.0000 q^{25} -27.0000 q^{27} -138.000 q^{29} -80.0000 q^{31} -84.0000 q^{33} +336.000 q^{35} +30.0000 q^{37} +222.000 q^{39} +282.000 q^{41} -4.00000 q^{43} +126.000 q^{45} -240.000 q^{47} +233.000 q^{49} -246.000 q^{51} -130.000 q^{53} +392.000 q^{55} +276.000 q^{57} -596.000 q^{59} -218.000 q^{61} +216.000 q^{63} -1036.00 q^{65} +436.000 q^{67} +24.0000 q^{69} -856.000 q^{71} -998.000 q^{73} -213.000 q^{75} +672.000 q^{77} +32.0000 q^{79} +81.0000 q^{81} +1508.00 q^{83} +1148.00 q^{85} +414.000 q^{87} -246.000 q^{89} -1776.00 q^{91} +240.000 q^{93} -1288.00 q^{95} +866.000 q^{97} +252.000 q^{99} +O(q^{100})$$

## 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
$$5$$ 14.0000 1.25220 0.626099 0.779744i $$-0.284651\pi$$
0.626099 + 0.779744i $$0.284651\pi$$
$$6$$ 0 0
$$7$$ 24.0000 1.29588 0.647939 0.761692i $$-0.275631\pi$$
0.647939 + 0.761692i $$0.275631\pi$$
$$8$$ 0 0
$$9$$ 9.00000 0.333333
$$10$$ 0 0
$$11$$ 28.0000 0.767483 0.383742 0.923440i $$-0.374635\pi$$
0.383742 + 0.923440i $$0.374635\pi$$
$$12$$ 0 0
$$13$$ −74.0000 −1.57876 −0.789381 0.613904i $$-0.789598\pi$$
−0.789381 + 0.613904i $$0.789598\pi$$
$$14$$ 0 0
$$15$$ −42.0000 −0.722957
$$16$$ 0 0
$$17$$ 82.0000 1.16988 0.584939 0.811077i $$-0.301118\pi$$
0.584939 + 0.811077i $$0.301118\pi$$
$$18$$ 0 0
$$19$$ −92.0000 −1.11086 −0.555428 0.831565i $$-0.687445\pi$$
−0.555428 + 0.831565i $$0.687445\pi$$
$$20$$ 0 0
$$21$$ −72.0000 −0.748176
$$22$$ 0 0
$$23$$ −8.00000 −0.0725268 −0.0362634 0.999342i $$-0.511546\pi$$
−0.0362634 + 0.999342i $$0.511546\pi$$
$$24$$ 0 0
$$25$$ 71.0000 0.568000
$$26$$ 0 0
$$27$$ −27.0000 −0.192450
$$28$$ 0 0
$$29$$ −138.000 −0.883654 −0.441827 0.897100i $$-0.645669\pi$$
−0.441827 + 0.897100i $$0.645669\pi$$
$$30$$ 0 0
$$31$$ −80.0000 −0.463498 −0.231749 0.972776i $$-0.574445\pi$$
−0.231749 + 0.972776i $$0.574445\pi$$
$$32$$ 0 0
$$33$$ −84.0000 −0.443107
$$34$$ 0 0
$$35$$ 336.000 1.62270
$$36$$ 0 0
$$37$$ 30.0000 0.133296 0.0666482 0.997777i $$-0.478769\pi$$
0.0666482 + 0.997777i $$0.478769\pi$$
$$38$$ 0 0
$$39$$ 222.000 0.911499
$$40$$ 0 0
$$41$$ 282.000 1.07417 0.537085 0.843528i $$-0.319525\pi$$
0.537085 + 0.843528i $$0.319525\pi$$
$$42$$ 0 0
$$43$$ −4.00000 −0.0141859 −0.00709296 0.999975i $$-0.502258\pi$$
−0.00709296 + 0.999975i $$0.502258\pi$$
$$44$$ 0 0
$$45$$ 126.000 0.417399
$$46$$ 0 0
$$47$$ −240.000 −0.744843 −0.372421 0.928064i $$-0.621472\pi$$
−0.372421 + 0.928064i $$0.621472\pi$$
$$48$$ 0 0
$$49$$ 233.000 0.679300
$$50$$ 0 0
$$51$$ −246.000 −0.675429
$$52$$ 0 0
$$53$$ −130.000 −0.336922 −0.168461 0.985708i $$-0.553880\pi$$
−0.168461 + 0.985708i $$0.553880\pi$$
$$54$$ 0 0
$$55$$ 392.000 0.961041
$$56$$ 0 0
$$57$$ 276.000 0.641353
$$58$$ 0 0
$$59$$ −596.000 −1.31513 −0.657564 0.753398i $$-0.728413\pi$$
−0.657564 + 0.753398i $$0.728413\pi$$
$$60$$ 0 0
$$61$$ −218.000 −0.457574 −0.228787 0.973476i $$-0.573476\pi$$
−0.228787 + 0.973476i $$0.573476\pi$$
$$62$$ 0 0
$$63$$ 216.000 0.431959
$$64$$ 0 0
$$65$$ −1036.00 −1.97692
$$66$$ 0 0
$$67$$ 436.000 0.795013 0.397507 0.917599i $$-0.369876\pi$$
0.397507 + 0.917599i $$0.369876\pi$$
$$68$$ 0 0
$$69$$ 24.0000 0.0418733
$$70$$ 0 0
$$71$$ −856.000 −1.43082 −0.715412 0.698703i $$-0.753761\pi$$
−0.715412 + 0.698703i $$0.753761\pi$$
$$72$$ 0 0
$$73$$ −998.000 −1.60010 −0.800048 0.599935i $$-0.795193\pi$$
−0.800048 + 0.599935i $$0.795193\pi$$
$$74$$ 0 0
$$75$$ −213.000 −0.327935
$$76$$ 0 0
$$77$$ 672.000 0.994565
$$78$$ 0 0
$$79$$ 32.0000 0.0455732 0.0227866 0.999740i $$-0.492746\pi$$
0.0227866 + 0.999740i $$0.492746\pi$$
$$80$$ 0 0
$$81$$ 81.0000 0.111111
$$82$$ 0 0
$$83$$ 1508.00 1.99427 0.997136 0.0756351i $$-0.0240984\pi$$
0.997136 + 0.0756351i $$0.0240984\pi$$
$$84$$ 0 0
$$85$$ 1148.00 1.46492
$$86$$ 0 0
$$87$$ 414.000 0.510178
$$88$$ 0 0
$$89$$ −246.000 −0.292988 −0.146494 0.989212i $$-0.546799\pi$$
−0.146494 + 0.989212i $$0.546799\pi$$
$$90$$ 0 0
$$91$$ −1776.00 −2.04588
$$92$$ 0 0
$$93$$ 240.000 0.267600
$$94$$ 0 0
$$95$$ −1288.00 −1.39101
$$96$$ 0 0
$$97$$ 866.000 0.906484 0.453242 0.891387i $$-0.350267\pi$$
0.453242 + 0.891387i $$0.350267\pi$$
$$98$$ 0 0
$$99$$ 252.000 0.255828
$$100$$ 0 0
$$101$$ 270.000 0.266000 0.133000 0.991116i $$-0.457539\pi$$
0.133000 + 0.991116i $$0.457539\pi$$
$$102$$ 0 0
$$103$$ 1496.00 1.43112 0.715560 0.698552i $$-0.246172\pi$$
0.715560 + 0.698552i $$0.246172\pi$$
$$104$$ 0 0
$$105$$ −1008.00 −0.936864
$$106$$ 0 0
$$107$$ 1692.00 1.52871 0.764354 0.644797i $$-0.223058\pi$$
0.764354 + 0.644797i $$0.223058\pi$$
$$108$$ 0 0
$$109$$ 406.000 0.356768 0.178384 0.983961i $$-0.442913\pi$$
0.178384 + 0.983961i $$0.442913\pi$$
$$110$$ 0 0
$$111$$ −90.0000 −0.0769588
$$112$$ 0 0
$$113$$ 786.000 0.654342 0.327171 0.944965i $$-0.393905\pi$$
0.327171 + 0.944965i $$0.393905\pi$$
$$114$$ 0 0
$$115$$ −112.000 −0.0908179
$$116$$ 0 0
$$117$$ −666.000 −0.526254
$$118$$ 0 0
$$119$$ 1968.00 1.51602
$$120$$ 0 0
$$121$$ −547.000 −0.410969
$$122$$ 0 0
$$123$$ −846.000 −0.620173
$$124$$ 0 0
$$125$$ −756.000 −0.540950
$$126$$ 0 0
$$127$$ −1744.00 −1.21854 −0.609272 0.792962i $$-0.708538\pi$$
−0.609272 + 0.792962i $$0.708538\pi$$
$$128$$ 0 0
$$129$$ 12.0000 0.00819024
$$130$$ 0 0
$$131$$ −652.000 −0.434851 −0.217426 0.976077i $$-0.569766\pi$$
−0.217426 + 0.976077i $$0.569766\pi$$
$$132$$ 0 0
$$133$$ −2208.00 −1.43953
$$134$$ 0 0
$$135$$ −378.000 −0.240986
$$136$$ 0 0
$$137$$ 1530.00 0.954137 0.477068 0.878866i $$-0.341699\pi$$
0.477068 + 0.878866i $$0.341699\pi$$
$$138$$ 0 0
$$139$$ −516.000 −0.314867 −0.157434 0.987530i $$-0.550322\pi$$
−0.157434 + 0.987530i $$0.550322\pi$$
$$140$$ 0 0
$$141$$ 720.000 0.430035
$$142$$ 0 0
$$143$$ −2072.00 −1.21167
$$144$$ 0 0
$$145$$ −1932.00 −1.10651
$$146$$ 0 0
$$147$$ −699.000 −0.392194
$$148$$ 0 0
$$149$$ 1342.00 0.737859 0.368929 0.929457i $$-0.379724\pi$$
0.368929 + 0.929457i $$0.379724\pi$$
$$150$$ 0 0
$$151$$ 424.000 0.228507 0.114254 0.993452i $$-0.463552\pi$$
0.114254 + 0.993452i $$0.463552\pi$$
$$152$$ 0 0
$$153$$ 738.000 0.389959
$$154$$ 0 0
$$155$$ −1120.00 −0.580391
$$156$$ 0 0
$$157$$ 262.000 0.133184 0.0665920 0.997780i $$-0.478787\pi$$
0.0665920 + 0.997780i $$0.478787\pi$$
$$158$$ 0 0
$$159$$ 390.000 0.194522
$$160$$ 0 0
$$161$$ −192.000 −0.0939858
$$162$$ 0 0
$$163$$ 2292.00 1.10137 0.550685 0.834713i $$-0.314367\pi$$
0.550685 + 0.834713i $$0.314367\pi$$
$$164$$ 0 0
$$165$$ −1176.00 −0.554857
$$166$$ 0 0
$$167$$ 1896.00 0.878544 0.439272 0.898354i $$-0.355236\pi$$
0.439272 + 0.898354i $$0.355236\pi$$
$$168$$ 0 0
$$169$$ 3279.00 1.49249
$$170$$ 0 0
$$171$$ −828.000 −0.370285
$$172$$ 0 0
$$173$$ −2874.00 −1.26304 −0.631521 0.775359i $$-0.717569\pi$$
−0.631521 + 0.775359i $$0.717569\pi$$
$$174$$ 0 0
$$175$$ 1704.00 0.736059
$$176$$ 0 0
$$177$$ 1788.00 0.759290
$$178$$ 0 0
$$179$$ 1188.00 0.496063 0.248032 0.968752i $$-0.420216\pi$$
0.248032 + 0.968752i $$0.420216\pi$$
$$180$$ 0 0
$$181$$ −3474.00 −1.42663 −0.713316 0.700843i $$-0.752808\pi$$
−0.713316 + 0.700843i $$0.752808\pi$$
$$182$$ 0 0
$$183$$ 654.000 0.264181
$$184$$ 0 0
$$185$$ 420.000 0.166914
$$186$$ 0 0
$$187$$ 2296.00 0.897862
$$188$$ 0 0
$$189$$ −648.000 −0.249392
$$190$$ 0 0
$$191$$ −192.000 −0.0727363 −0.0363681 0.999338i $$-0.511579\pi$$
−0.0363681 + 0.999338i $$0.511579\pi$$
$$192$$ 0 0
$$193$$ 4802.00 1.79096 0.895481 0.445100i $$-0.146832\pi$$
0.895481 + 0.445100i $$0.146832\pi$$
$$194$$ 0 0
$$195$$ 3108.00 1.14138
$$196$$ 0 0
$$197$$ 1518.00 0.549000 0.274500 0.961587i $$-0.411488\pi$$
0.274500 + 0.961587i $$0.411488\pi$$
$$198$$ 0 0
$$199$$ −5128.00 −1.82670 −0.913352 0.407170i $$-0.866516\pi$$
−0.913352 + 0.407170i $$0.866516\pi$$
$$200$$ 0 0
$$201$$ −1308.00 −0.459001
$$202$$ 0 0
$$203$$ −3312.00 −1.14511
$$204$$ 0 0
$$205$$ 3948.00 1.34507
$$206$$ 0 0
$$207$$ −72.0000 −0.0241756
$$208$$ 0 0
$$209$$ −2576.00 −0.852563
$$210$$ 0 0
$$211$$ −1084.00 −0.353676 −0.176838 0.984240i $$-0.556587\pi$$
−0.176838 + 0.984240i $$0.556587\pi$$
$$212$$ 0 0
$$213$$ 2568.00 0.826087
$$214$$ 0 0
$$215$$ −56.0000 −0.0177636
$$216$$ 0 0
$$217$$ −1920.00 −0.600636
$$218$$ 0 0
$$219$$ 2994.00 0.923816
$$220$$ 0 0
$$221$$ −6068.00 −1.84696
$$222$$ 0 0
$$223$$ −688.000 −0.206600 −0.103300 0.994650i $$-0.532940\pi$$
−0.103300 + 0.994650i $$0.532940\pi$$
$$224$$ 0 0
$$225$$ 639.000 0.189333
$$226$$ 0 0
$$227$$ −4812.00 −1.40698 −0.703488 0.710707i $$-0.748375\pi$$
−0.703488 + 0.710707i $$0.748375\pi$$
$$228$$ 0 0
$$229$$ 2494.00 0.719686 0.359843 0.933013i $$-0.382830\pi$$
0.359843 + 0.933013i $$0.382830\pi$$
$$230$$ 0 0
$$231$$ −2016.00 −0.574212
$$232$$ 0 0
$$233$$ 698.000 0.196255 0.0981277 0.995174i $$-0.468715\pi$$
0.0981277 + 0.995174i $$0.468715\pi$$
$$234$$ 0 0
$$235$$ −3360.00 −0.932690
$$236$$ 0 0
$$237$$ −96.0000 −0.0263117
$$238$$ 0 0
$$239$$ 6320.00 1.71049 0.855244 0.518225i $$-0.173407\pi$$
0.855244 + 0.518225i $$0.173407\pi$$
$$240$$ 0 0
$$241$$ −6510.00 −1.74002 −0.870012 0.493030i $$-0.835889\pi$$
−0.870012 + 0.493030i $$0.835889\pi$$
$$242$$ 0 0
$$243$$ −243.000 −0.0641500
$$244$$ 0 0
$$245$$ 3262.00 0.850619
$$246$$ 0 0
$$247$$ 6808.00 1.75378
$$248$$ 0 0
$$249$$ −4524.00 −1.15139
$$250$$ 0 0
$$251$$ −628.000 −0.157924 −0.0789622 0.996878i $$-0.525161\pi$$
−0.0789622 + 0.996878i $$0.525161\pi$$
$$252$$ 0 0
$$253$$ −224.000 −0.0556631
$$254$$ 0 0
$$255$$ −3444.00 −0.845771
$$256$$ 0 0
$$257$$ −4862.00 −1.18009 −0.590045 0.807370i $$-0.700890\pi$$
−0.590045 + 0.807370i $$0.700890\pi$$
$$258$$ 0 0
$$259$$ 720.000 0.172736
$$260$$ 0 0
$$261$$ −1242.00 −0.294551
$$262$$ 0 0
$$263$$ −5816.00 −1.36361 −0.681806 0.731533i $$-0.738805\pi$$
−0.681806 + 0.731533i $$0.738805\pi$$
$$264$$ 0 0
$$265$$ −1820.00 −0.421893
$$266$$ 0 0
$$267$$ 738.000 0.169157
$$268$$ 0 0
$$269$$ 3526.00 0.799197 0.399599 0.916690i $$-0.369150\pi$$
0.399599 + 0.916690i $$0.369150\pi$$
$$270$$ 0 0
$$271$$ 256.000 0.0573834 0.0286917 0.999588i $$-0.490866\pi$$
0.0286917 + 0.999588i $$0.490866\pi$$
$$272$$ 0 0
$$273$$ 5328.00 1.18119
$$274$$ 0 0
$$275$$ 1988.00 0.435931
$$276$$ 0 0
$$277$$ 142.000 0.0308013 0.0154006 0.999881i $$-0.495098\pi$$
0.0154006 + 0.999881i $$0.495098\pi$$
$$278$$ 0 0
$$279$$ −720.000 −0.154499
$$280$$ 0 0
$$281$$ 8842.00 1.87712 0.938558 0.345122i $$-0.112162\pi$$
0.938558 + 0.345122i $$0.112162\pi$$
$$282$$ 0 0
$$283$$ 7180.00 1.50815 0.754075 0.656788i $$-0.228085\pi$$
0.754075 + 0.656788i $$0.228085\pi$$
$$284$$ 0 0
$$285$$ 3864.00 0.803100
$$286$$ 0 0
$$287$$ 6768.00 1.39199
$$288$$ 0 0
$$289$$ 1811.00 0.368614
$$290$$ 0 0
$$291$$ −2598.00 −0.523359
$$292$$ 0 0
$$293$$ 7374.00 1.47029 0.735143 0.677912i $$-0.237115\pi$$
0.735143 + 0.677912i $$0.237115\pi$$
$$294$$ 0 0
$$295$$ −8344.00 −1.64680
$$296$$ 0 0
$$297$$ −756.000 −0.147702
$$298$$ 0 0
$$299$$ 592.000 0.114502
$$300$$ 0 0
$$301$$ −96.0000 −0.0183832
$$302$$ 0 0
$$303$$ −810.000 −0.153575
$$304$$ 0 0
$$305$$ −3052.00 −0.572974
$$306$$ 0 0
$$307$$ −1500.00 −0.278858 −0.139429 0.990232i $$-0.544527\pi$$
−0.139429 + 0.990232i $$0.544527\pi$$
$$308$$ 0 0
$$309$$ −4488.00 −0.826257
$$310$$ 0 0
$$311$$ 7608.00 1.38717 0.693585 0.720374i $$-0.256030\pi$$
0.693585 + 0.720374i $$0.256030\pi$$
$$312$$ 0 0
$$313$$ −4758.00 −0.859227 −0.429614 0.903013i $$-0.641350\pi$$
−0.429614 + 0.903013i $$0.641350\pi$$
$$314$$ 0 0
$$315$$ 3024.00 0.540899
$$316$$ 0 0
$$317$$ 4374.00 0.774979 0.387489 0.921874i $$-0.373342\pi$$
0.387489 + 0.921874i $$0.373342\pi$$
$$318$$ 0 0
$$319$$ −3864.00 −0.678190
$$320$$ 0 0
$$321$$ −5076.00 −0.882600
$$322$$ 0 0
$$323$$ −7544.00 −1.29956
$$324$$ 0 0
$$325$$ −5254.00 −0.896737
$$326$$ 0 0
$$327$$ −1218.00 −0.205980
$$328$$ 0 0
$$329$$ −5760.00 −0.965225
$$330$$ 0 0
$$331$$ 7804.00 1.29591 0.647956 0.761678i $$-0.275624\pi$$
0.647956 + 0.761678i $$0.275624\pi$$
$$332$$ 0 0
$$333$$ 270.000 0.0444322
$$334$$ 0 0
$$335$$ 6104.00 0.995514
$$336$$ 0 0
$$337$$ 5106.00 0.825346 0.412673 0.910879i $$-0.364595\pi$$
0.412673 + 0.910879i $$0.364595\pi$$
$$338$$ 0 0
$$339$$ −2358.00 −0.377785
$$340$$ 0 0
$$341$$ −2240.00 −0.355727
$$342$$ 0 0
$$343$$ −2640.00 −0.415588
$$344$$ 0 0
$$345$$ 336.000 0.0524337
$$346$$ 0 0
$$347$$ 4716.00 0.729591 0.364796 0.931088i $$-0.381139\pi$$
0.364796 + 0.931088i $$0.381139\pi$$
$$348$$ 0 0
$$349$$ 7302.00 1.11996 0.559982 0.828505i $$-0.310808\pi$$
0.559982 + 0.828505i $$0.310808\pi$$
$$350$$ 0 0
$$351$$ 1998.00 0.303833
$$352$$ 0 0
$$353$$ −4382.00 −0.660709 −0.330355 0.943857i $$-0.607168\pi$$
−0.330355 + 0.943857i $$0.607168\pi$$
$$354$$ 0 0
$$355$$ −11984.0 −1.79168
$$356$$ 0 0
$$357$$ −5904.00 −0.875274
$$358$$ 0 0
$$359$$ −7224.00 −1.06203 −0.531014 0.847363i $$-0.678189\pi$$
−0.531014 + 0.847363i $$0.678189\pi$$
$$360$$ 0 0
$$361$$ 1605.00 0.233999
$$362$$ 0 0
$$363$$ 1641.00 0.237273
$$364$$ 0 0
$$365$$ −13972.0 −2.00364
$$366$$ 0 0
$$367$$ −1408.00 −0.200264 −0.100132 0.994974i $$-0.531927\pi$$
−0.100132 + 0.994974i $$0.531927\pi$$
$$368$$ 0 0
$$369$$ 2538.00 0.358057
$$370$$ 0 0
$$371$$ −3120.00 −0.436610
$$372$$ 0 0
$$373$$ −1714.00 −0.237929 −0.118965 0.992899i $$-0.537957\pi$$
−0.118965 + 0.992899i $$0.537957\pi$$
$$374$$ 0 0
$$375$$ 2268.00 0.312317
$$376$$ 0 0
$$377$$ 10212.0 1.39508
$$378$$ 0 0
$$379$$ −884.000 −0.119810 −0.0599051 0.998204i $$-0.519080\pi$$
−0.0599051 + 0.998204i $$0.519080\pi$$
$$380$$ 0 0
$$381$$ 5232.00 0.703526
$$382$$ 0 0
$$383$$ −10368.0 −1.38324 −0.691619 0.722263i $$-0.743102\pi$$
−0.691619 + 0.722263i $$0.743102\pi$$
$$384$$ 0 0
$$385$$ 9408.00 1.24539
$$386$$ 0 0
$$387$$ −36.0000 −0.00472864
$$388$$ 0 0
$$389$$ 398.000 0.0518751 0.0259375 0.999664i $$-0.491743\pi$$
0.0259375 + 0.999664i $$0.491743\pi$$
$$390$$ 0 0
$$391$$ −656.000 −0.0848474
$$392$$ 0 0
$$393$$ 1956.00 0.251061
$$394$$ 0 0
$$395$$ 448.000 0.0570666
$$396$$ 0 0
$$397$$ −5098.00 −0.644487 −0.322243 0.946657i $$-0.604437\pi$$
−0.322243 + 0.946657i $$0.604437\pi$$
$$398$$ 0 0
$$399$$ 6624.00 0.831115
$$400$$ 0 0
$$401$$ 10002.0 1.24558 0.622788 0.782391i $$-0.286000\pi$$
0.622788 + 0.782391i $$0.286000\pi$$
$$402$$ 0 0
$$403$$ 5920.00 0.731752
$$404$$ 0 0
$$405$$ 1134.00 0.139133
$$406$$ 0 0
$$407$$ 840.000 0.102303
$$408$$ 0 0
$$409$$ −9270.00 −1.12071 −0.560357 0.828251i $$-0.689336\pi$$
−0.560357 + 0.828251i $$0.689336\pi$$
$$410$$ 0 0
$$411$$ −4590.00 −0.550871
$$412$$ 0 0
$$413$$ −14304.0 −1.70425
$$414$$ 0 0
$$415$$ 21112.0 2.49722
$$416$$ 0 0
$$417$$ 1548.00 0.181789
$$418$$ 0 0
$$419$$ 6516.00 0.759731 0.379866 0.925042i $$-0.375970\pi$$
0.379866 + 0.925042i $$0.375970\pi$$
$$420$$ 0 0
$$421$$ −2626.00 −0.303999 −0.151999 0.988381i $$-0.548571\pi$$
−0.151999 + 0.988381i $$0.548571\pi$$
$$422$$ 0 0
$$423$$ −2160.00 −0.248281
$$424$$ 0 0
$$425$$ 5822.00 0.664491
$$426$$ 0 0
$$427$$ −5232.00 −0.592961
$$428$$ 0 0
$$429$$ 6216.00 0.699560
$$430$$ 0 0
$$431$$ 4304.00 0.481012 0.240506 0.970648i $$-0.422687\pi$$
0.240506 + 0.970648i $$0.422687\pi$$
$$432$$ 0 0
$$433$$ 11794.0 1.30897 0.654484 0.756076i $$-0.272886\pi$$
0.654484 + 0.756076i $$0.272886\pi$$
$$434$$ 0 0
$$435$$ 5796.00 0.638844
$$436$$ 0 0
$$437$$ 736.000 0.0805667
$$438$$ 0 0
$$439$$ 5544.00 0.602735 0.301368 0.953508i $$-0.402557\pi$$
0.301368 + 0.953508i $$0.402557\pi$$
$$440$$ 0 0
$$441$$ 2097.00 0.226433
$$442$$ 0 0
$$443$$ 3788.00 0.406260 0.203130 0.979152i $$-0.434889\pi$$
0.203130 + 0.979152i $$0.434889\pi$$
$$444$$ 0 0
$$445$$ −3444.00 −0.366879
$$446$$ 0 0
$$447$$ −4026.00 −0.426003
$$448$$ 0 0
$$449$$ −13342.0 −1.40233 −0.701167 0.712997i $$-0.747337\pi$$
−0.701167 + 0.712997i $$0.747337\pi$$
$$450$$ 0 0
$$451$$ 7896.00 0.824408
$$452$$ 0 0
$$453$$ −1272.00 −0.131929
$$454$$ 0 0
$$455$$ −24864.0 −2.56185
$$456$$ 0 0
$$457$$ −4390.00 −0.449356 −0.224678 0.974433i $$-0.572133\pi$$
−0.224678 + 0.974433i $$0.572133\pi$$
$$458$$ 0 0
$$459$$ −2214.00 −0.225143
$$460$$ 0 0
$$461$$ 5798.00 0.585770 0.292885 0.956148i $$-0.405385\pi$$
0.292885 + 0.956148i $$0.405385\pi$$
$$462$$ 0 0
$$463$$ 14656.0 1.47111 0.735553 0.677467i $$-0.236922\pi$$
0.735553 + 0.677467i $$0.236922\pi$$
$$464$$ 0 0
$$465$$ 3360.00 0.335089
$$466$$ 0 0
$$467$$ −8412.00 −0.833535 −0.416768 0.909013i $$-0.636837\pi$$
−0.416768 + 0.909013i $$0.636837\pi$$
$$468$$ 0 0
$$469$$ 10464.0 1.03024
$$470$$ 0 0
$$471$$ −786.000 −0.0768938
$$472$$ 0 0
$$473$$ −112.000 −0.0108875
$$474$$ 0 0
$$475$$ −6532.00 −0.630966
$$476$$ 0 0
$$477$$ −1170.00 −0.112307
$$478$$ 0 0
$$479$$ −14848.0 −1.41633 −0.708165 0.706047i $$-0.750477\pi$$
−0.708165 + 0.706047i $$0.750477\pi$$
$$480$$ 0 0
$$481$$ −2220.00 −0.210443
$$482$$ 0 0
$$483$$ 576.000 0.0542627
$$484$$ 0 0
$$485$$ 12124.0 1.13510
$$486$$ 0 0
$$487$$ −18568.0 −1.72771 −0.863857 0.503738i $$-0.831958\pi$$
−0.863857 + 0.503738i $$0.831958\pi$$
$$488$$ 0 0
$$489$$ −6876.00 −0.635876
$$490$$ 0 0
$$491$$ 14364.0 1.32024 0.660120 0.751160i $$-0.270505\pi$$
0.660120 + 0.751160i $$0.270505\pi$$
$$492$$ 0 0
$$493$$ −11316.0 −1.03377
$$494$$ 0 0
$$495$$ 3528.00 0.320347
$$496$$ 0 0
$$497$$ −20544.0 −1.85417
$$498$$ 0 0
$$499$$ −21660.0 −1.94316 −0.971578 0.236720i $$-0.923928\pi$$
−0.971578 + 0.236720i $$0.923928\pi$$
$$500$$ 0 0
$$501$$ −5688.00 −0.507228
$$502$$ 0 0
$$503$$ 17112.0 1.51687 0.758436 0.651748i $$-0.225964\pi$$
0.758436 + 0.651748i $$0.225964\pi$$
$$504$$ 0 0
$$505$$ 3780.00 0.333085
$$506$$ 0 0
$$507$$ −9837.00 −0.861689
$$508$$ 0 0
$$509$$ 11478.0 0.999516 0.499758 0.866165i $$-0.333422\pi$$
0.499758 + 0.866165i $$0.333422\pi$$
$$510$$ 0 0
$$511$$ −23952.0 −2.07353
$$512$$ 0 0
$$513$$ 2484.00 0.213784
$$514$$ 0 0
$$515$$ 20944.0 1.79204
$$516$$ 0 0
$$517$$ −6720.00 −0.571654
$$518$$ 0 0
$$519$$ 8622.00 0.729217
$$520$$ 0 0
$$521$$ 13114.0 1.10275 0.551377 0.834256i $$-0.314103\pi$$
0.551377 + 0.834256i $$0.314103\pi$$
$$522$$ 0 0
$$523$$ 4508.00 0.376905 0.188452 0.982082i $$-0.439653\pi$$
0.188452 + 0.982082i $$0.439653\pi$$
$$524$$ 0 0
$$525$$ −5112.00 −0.424964
$$526$$ 0 0
$$527$$ −6560.00 −0.542235
$$528$$ 0 0
$$529$$ −12103.0 −0.994740
$$530$$ 0 0
$$531$$ −5364.00 −0.438376
$$532$$ 0 0
$$533$$ −20868.0 −1.69586
$$534$$ 0 0
$$535$$ 23688.0 1.91425
$$536$$ 0 0
$$537$$ −3564.00 −0.286402
$$538$$ 0 0
$$539$$ 6524.00 0.521352
$$540$$ 0 0
$$541$$ 22950.0 1.82384 0.911920 0.410368i $$-0.134600\pi$$
0.911920 + 0.410368i $$0.134600\pi$$
$$542$$ 0 0
$$543$$ 10422.0 0.823666
$$544$$ 0 0
$$545$$ 5684.00 0.446745
$$546$$ 0 0
$$547$$ 6580.00 0.514334 0.257167 0.966367i $$-0.417211\pi$$
0.257167 + 0.966367i $$0.417211\pi$$
$$548$$ 0 0
$$549$$ −1962.00 −0.152525
$$550$$ 0 0
$$551$$ 12696.0 0.981611
$$552$$ 0 0
$$553$$ 768.000 0.0590573
$$554$$ 0 0
$$555$$ −1260.00 −0.0963676
$$556$$ 0 0
$$557$$ 7046.00 0.535994 0.267997 0.963420i $$-0.413638\pi$$
0.267997 + 0.963420i $$0.413638\pi$$
$$558$$ 0 0
$$559$$ 296.000 0.0223962
$$560$$ 0 0
$$561$$ −6888.00 −0.518381
$$562$$ 0 0
$$563$$ −8252.00 −0.617727 −0.308864 0.951106i $$-0.599949\pi$$
−0.308864 + 0.951106i $$0.599949\pi$$
$$564$$ 0 0
$$565$$ 11004.0 0.819366
$$566$$ 0 0
$$567$$ 1944.00 0.143986
$$568$$ 0 0
$$569$$ −6838.00 −0.503803 −0.251901 0.967753i $$-0.581056\pi$$
−0.251901 + 0.967753i $$0.581056\pi$$
$$570$$ 0 0
$$571$$ −23316.0 −1.70883 −0.854417 0.519588i $$-0.826085\pi$$
−0.854417 + 0.519588i $$0.826085\pi$$
$$572$$ 0 0
$$573$$ 576.000 0.0419943
$$574$$ 0 0
$$575$$ −568.000 −0.0411952
$$576$$ 0 0
$$577$$ −10558.0 −0.761760 −0.380880 0.924625i $$-0.624379\pi$$
−0.380880 + 0.924625i $$0.624379\pi$$
$$578$$ 0 0
$$579$$ −14406.0 −1.03401
$$580$$ 0 0
$$581$$ 36192.0 2.58433
$$582$$ 0 0
$$583$$ −3640.00 −0.258582
$$584$$ 0 0
$$585$$ −9324.00 −0.658974
$$586$$ 0 0
$$587$$ −1028.00 −0.0722830 −0.0361415 0.999347i $$-0.511507\pi$$
−0.0361415 + 0.999347i $$0.511507\pi$$
$$588$$ 0 0
$$589$$ 7360.00 0.514879
$$590$$ 0 0
$$591$$ −4554.00 −0.316965
$$592$$ 0 0
$$593$$ 1202.00 0.0832382 0.0416191 0.999134i $$-0.486748\pi$$
0.0416191 + 0.999134i $$0.486748\pi$$
$$594$$ 0 0
$$595$$ 27552.0 1.89836
$$596$$ 0 0
$$597$$ 15384.0 1.05465
$$598$$ 0 0
$$599$$ 3576.00 0.243926 0.121963 0.992535i $$-0.461081\pi$$
0.121963 + 0.992535i $$0.461081\pi$$
$$600$$ 0 0
$$601$$ 8650.00 0.587090 0.293545 0.955945i $$-0.405165\pi$$
0.293545 + 0.955945i $$0.405165\pi$$
$$602$$ 0 0
$$603$$ 3924.00 0.265004
$$604$$ 0 0
$$605$$ −7658.00 −0.514615
$$606$$ 0 0
$$607$$ −12656.0 −0.846279 −0.423139 0.906065i $$-0.639072\pi$$
−0.423139 + 0.906065i $$0.639072\pi$$
$$608$$ 0 0
$$609$$ 9936.00 0.661128
$$610$$ 0 0
$$611$$ 17760.0 1.17593
$$612$$ 0 0
$$613$$ −3298.00 −0.217300 −0.108650 0.994080i $$-0.534653\pi$$
−0.108650 + 0.994080i $$0.534653\pi$$
$$614$$ 0 0
$$615$$ −11844.0 −0.776579
$$616$$ 0 0
$$617$$ 5370.00 0.350386 0.175193 0.984534i $$-0.443945\pi$$
0.175193 + 0.984534i $$0.443945\pi$$
$$618$$ 0 0
$$619$$ 16220.0 1.05321 0.526605 0.850110i $$-0.323465\pi$$
0.526605 + 0.850110i $$0.323465\pi$$
$$620$$ 0 0
$$621$$ 216.000 0.0139578
$$622$$ 0 0
$$623$$ −5904.00 −0.379677
$$624$$ 0 0
$$625$$ −19459.0 −1.24538
$$626$$ 0 0
$$627$$ 7728.00 0.492227
$$628$$ 0 0
$$629$$ 2460.00 0.155941
$$630$$ 0 0
$$631$$ 20360.0 1.28450 0.642249 0.766496i $$-0.278001\pi$$
0.642249 + 0.766496i $$0.278001\pi$$
$$632$$ 0 0
$$633$$ 3252.00 0.204195
$$634$$ 0 0
$$635$$ −24416.0 −1.52586
$$636$$ 0 0
$$637$$ −17242.0 −1.07245
$$638$$ 0 0
$$639$$ −7704.00 −0.476941
$$640$$ 0 0
$$641$$ 14498.0 0.893349 0.446674 0.894697i $$-0.352608\pi$$
0.446674 + 0.894697i $$0.352608\pi$$
$$642$$ 0 0
$$643$$ −21612.0 −1.32550 −0.662748 0.748842i $$-0.730610\pi$$
−0.662748 + 0.748842i $$0.730610\pi$$
$$644$$ 0 0
$$645$$ 168.000 0.0102558
$$646$$ 0 0
$$647$$ −12184.0 −0.740344 −0.370172 0.928963i $$-0.620701\pi$$
−0.370172 + 0.928963i $$0.620701\pi$$
$$648$$ 0 0
$$649$$ −16688.0 −1.00934
$$650$$ 0 0
$$651$$ 5760.00 0.346778
$$652$$ 0 0
$$653$$ −28122.0 −1.68530 −0.842648 0.538464i $$-0.819005\pi$$
−0.842648 + 0.538464i $$0.819005\pi$$
$$654$$ 0 0
$$655$$ −9128.00 −0.544520
$$656$$ 0 0
$$657$$ −8982.00 −0.533366
$$658$$ 0 0
$$659$$ 5700.00 0.336935 0.168468 0.985707i $$-0.446118\pi$$
0.168468 + 0.985707i $$0.446118\pi$$
$$660$$ 0 0
$$661$$ −29458.0 −1.73341 −0.866705 0.498822i $$-0.833766\pi$$
−0.866705 + 0.498822i $$0.833766\pi$$
$$662$$ 0 0
$$663$$ 18204.0 1.06634
$$664$$ 0 0
$$665$$ −30912.0 −1.80258
$$666$$ 0 0
$$667$$ 1104.00 0.0640885
$$668$$ 0 0
$$669$$ 2064.00 0.119281
$$670$$ 0 0
$$671$$ −6104.00 −0.351181
$$672$$ 0 0
$$673$$ 19810.0 1.13465 0.567325 0.823494i $$-0.307978\pi$$
0.567325 + 0.823494i $$0.307978\pi$$
$$674$$ 0 0
$$675$$ −1917.00 −0.109312
$$676$$ 0 0
$$677$$ −10450.0 −0.593244 −0.296622 0.954995i $$-0.595860\pi$$
−0.296622 + 0.954995i $$0.595860\pi$$
$$678$$ 0 0
$$679$$ 20784.0 1.17469
$$680$$ 0 0
$$681$$ 14436.0 0.812318
$$682$$ 0 0
$$683$$ −23300.0 −1.30534 −0.652672 0.757641i $$-0.726352\pi$$
−0.652672 + 0.757641i $$0.726352\pi$$
$$684$$ 0 0
$$685$$ 21420.0 1.19477
$$686$$ 0 0
$$687$$ −7482.00 −0.415511
$$688$$ 0 0
$$689$$ 9620.00 0.531920
$$690$$ 0 0
$$691$$ 14212.0 0.782417 0.391208 0.920302i $$-0.372057\pi$$
0.391208 + 0.920302i $$0.372057\pi$$
$$692$$ 0 0
$$693$$ 6048.00 0.331522
$$694$$ 0 0
$$695$$ −7224.00 −0.394276
$$696$$ 0 0
$$697$$ 23124.0 1.25665
$$698$$ 0 0
$$699$$ −2094.00 −0.113308
$$700$$ 0 0
$$701$$ −15978.0 −0.860885 −0.430443 0.902618i $$-0.641643\pi$$
−0.430443 + 0.902618i $$0.641643\pi$$
$$702$$ 0 0
$$703$$ −2760.00 −0.148073
$$704$$ 0 0
$$705$$ 10080.0 0.538489
$$706$$ 0 0
$$707$$ 6480.00 0.344704
$$708$$ 0 0
$$709$$ −8866.00 −0.469633 −0.234816 0.972040i $$-0.575449\pi$$
−0.234816 + 0.972040i $$0.575449\pi$$
$$710$$ 0 0
$$711$$ 288.000 0.0151911
$$712$$ 0 0
$$713$$ 640.000 0.0336160
$$714$$ 0 0
$$715$$ −29008.0 −1.51726
$$716$$ 0 0
$$717$$ −18960.0 −0.987551
$$718$$ 0 0
$$719$$ −7760.00 −0.402502 −0.201251 0.979540i $$-0.564501\pi$$
−0.201251 + 0.979540i $$0.564501\pi$$
$$720$$ 0 0
$$721$$ 35904.0 1.85456
$$722$$ 0 0
$$723$$ 19530.0 1.00460
$$724$$ 0 0
$$725$$ −9798.00 −0.501915
$$726$$ 0 0
$$727$$ −13080.0 −0.667277 −0.333638 0.942701i $$-0.608276\pi$$
−0.333638 + 0.942701i $$0.608276\pi$$
$$728$$ 0 0
$$729$$ 729.000 0.0370370
$$730$$ 0 0
$$731$$ −328.000 −0.0165958
$$732$$ 0 0
$$733$$ 16934.0 0.853304 0.426652 0.904416i $$-0.359693\pi$$
0.426652 + 0.904416i $$0.359693\pi$$
$$734$$ 0 0
$$735$$ −9786.00 −0.491105
$$736$$ 0 0
$$737$$ 12208.0 0.610159
$$738$$ 0 0
$$739$$ 7060.00 0.351429 0.175715 0.984441i $$-0.443776\pi$$
0.175715 + 0.984441i $$0.443776\pi$$
$$740$$ 0 0
$$741$$ −20424.0 −1.01254
$$742$$ 0 0
$$743$$ 12520.0 0.618189 0.309094 0.951031i $$-0.399974\pi$$
0.309094 + 0.951031i $$0.399974\pi$$
$$744$$ 0 0
$$745$$ 18788.0 0.923945
$$746$$ 0 0
$$747$$ 13572.0 0.664757
$$748$$ 0 0
$$749$$ 40608.0 1.98102
$$750$$ 0 0
$$751$$ 9792.00 0.475786 0.237893 0.971291i $$-0.423543\pi$$
0.237893 + 0.971291i $$0.423543\pi$$
$$752$$ 0 0
$$753$$ 1884.00 0.0911777
$$754$$ 0 0
$$755$$ 5936.00 0.286137
$$756$$ 0 0
$$757$$ 13166.0 0.632135 0.316068 0.948737i $$-0.397637\pi$$
0.316068 + 0.948737i $$0.397637\pi$$
$$758$$ 0 0
$$759$$ 672.000 0.0321371
$$760$$ 0 0
$$761$$ −23222.0 −1.10617 −0.553086 0.833124i $$-0.686550\pi$$
−0.553086 + 0.833124i $$0.686550\pi$$
$$762$$ 0 0
$$763$$ 9744.00 0.462328
$$764$$ 0 0
$$765$$ 10332.0 0.488306
$$766$$ 0 0
$$767$$ 44104.0 2.07628
$$768$$ 0 0
$$769$$ −39934.0 −1.87264 −0.936318 0.351154i $$-0.885789\pi$$
−0.936318 + 0.351154i $$0.885789\pi$$
$$770$$ 0 0
$$771$$ 14586.0 0.681325
$$772$$ 0 0
$$773$$ −17106.0 −0.795938 −0.397969 0.917399i $$-0.630285\pi$$
−0.397969 + 0.917399i $$0.630285\pi$$
$$774$$ 0 0
$$775$$ −5680.00 −0.263267
$$776$$ 0 0
$$777$$ −2160.00 −0.0997292
$$778$$ 0 0
$$779$$ −25944.0 −1.19325
$$780$$ 0 0
$$781$$ −23968.0 −1.09813
$$782$$ 0 0
$$783$$ 3726.00 0.170059
$$784$$ 0 0
$$785$$ 3668.00 0.166773
$$786$$ 0 0
$$787$$ 9956.00 0.450944 0.225472 0.974250i $$-0.427608\pi$$
0.225472 + 0.974250i $$0.427608\pi$$
$$788$$ 0 0
$$789$$ 17448.0 0.787282
$$790$$ 0 0
$$791$$ 18864.0 0.847948
$$792$$ 0 0
$$793$$ 16132.0 0.722401
$$794$$ 0 0
$$795$$ 5460.00 0.243580
$$796$$ 0 0
$$797$$ −9130.00 −0.405773 −0.202887 0.979202i $$-0.565032\pi$$
−0.202887 + 0.979202i $$0.565032\pi$$
$$798$$ 0 0
$$799$$ −19680.0 −0.871375
$$800$$ 0 0
$$801$$ −2214.00 −0.0976627
$$802$$ 0 0
$$803$$ −27944.0 −1.22805
$$804$$ 0 0
$$805$$ −2688.00 −0.117689
$$806$$ 0 0
$$807$$ −10578.0 −0.461417
$$808$$ 0 0
$$809$$ 11482.0 0.498993 0.249497 0.968376i $$-0.419735\pi$$
0.249497 + 0.968376i $$0.419735\pi$$
$$810$$ 0 0
$$811$$ −4612.00 −0.199691 −0.0998454 0.995003i $$-0.531835\pi$$
−0.0998454 + 0.995003i $$0.531835\pi$$
$$812$$ 0 0
$$813$$ −768.000 −0.0331303
$$814$$ 0 0
$$815$$ 32088.0 1.37913
$$816$$ 0 0
$$817$$ 368.000 0.0157585
$$818$$ 0 0
$$819$$ −15984.0 −0.681961
$$820$$ 0 0
$$821$$ −35010.0 −1.48826 −0.744128 0.668038i $$-0.767135\pi$$
−0.744128 + 0.668038i $$0.767135\pi$$
$$822$$ 0 0
$$823$$ −13688.0 −0.579749 −0.289875 0.957065i $$-0.593614\pi$$
−0.289875 + 0.957065i $$0.593614\pi$$
$$824$$ 0 0
$$825$$ −5964.00 −0.251685
$$826$$ 0 0
$$827$$ −11668.0 −0.490612 −0.245306 0.969446i $$-0.578888\pi$$
−0.245306 + 0.969446i $$0.578888\pi$$
$$828$$ 0 0
$$829$$ −29306.0 −1.22779 −0.613896 0.789387i $$-0.710399\pi$$
−0.613896 + 0.789387i $$0.710399\pi$$
$$830$$ 0 0
$$831$$ −426.000 −0.0177831
$$832$$ 0 0
$$833$$ 19106.0 0.794698
$$834$$ 0 0
$$835$$ 26544.0 1.10011
$$836$$ 0 0
$$837$$ 2160.00 0.0892001
$$838$$ 0 0
$$839$$ 2664.00 0.109620 0.0548102 0.998497i $$-0.482545\pi$$
0.0548102 + 0.998497i $$0.482545\pi$$
$$840$$ 0 0
$$841$$ −5345.00 −0.219156
$$842$$ 0 0
$$843$$ −26526.0 −1.08375
$$844$$ 0 0
$$845$$ 45906.0 1.86889
$$846$$ 0 0
$$847$$ −13128.0 −0.532566
$$848$$ 0 0
$$849$$ −21540.0 −0.870731
$$850$$ 0 0
$$851$$ −240.000 −0.00966756
$$852$$ 0 0
$$853$$ 26030.0 1.04484 0.522421 0.852688i $$-0.325029\pi$$
0.522421 + 0.852688i $$0.325029\pi$$
$$854$$ 0 0
$$855$$ −11592.0 −0.463670
$$856$$ 0 0
$$857$$ 44202.0 1.76186 0.880929 0.473249i $$-0.156919\pi$$
0.880929 + 0.473249i $$0.156919\pi$$
$$858$$ 0 0
$$859$$ 32748.0 1.30075 0.650377 0.759612i $$-0.274611\pi$$
0.650377 + 0.759612i $$0.274611\pi$$
$$860$$ 0 0
$$861$$ −20304.0 −0.803668
$$862$$ 0 0
$$863$$ −45344.0 −1.78856 −0.894280 0.447507i $$-0.852312\pi$$
−0.894280 + 0.447507i $$0.852312\pi$$
$$864$$ 0 0
$$865$$ −40236.0 −1.58158
$$866$$ 0 0
$$867$$ −5433.00 −0.212819
$$868$$ 0 0
$$869$$ 896.000 0.0349767
$$870$$ 0 0
$$871$$ −32264.0 −1.25514
$$872$$ 0 0
$$873$$ 7794.00 0.302161
$$874$$ 0 0
$$875$$ −18144.0 −0.701005
$$876$$ 0 0
$$877$$ −8778.00 −0.337984 −0.168992 0.985617i $$-0.554051\pi$$
−0.168992 + 0.985617i $$0.554051\pi$$
$$878$$ 0 0
$$879$$ −22122.0 −0.848870
$$880$$ 0 0
$$881$$ −4142.00 −0.158397 −0.0791984 0.996859i $$-0.525236\pi$$
−0.0791984 + 0.996859i $$0.525236\pi$$
$$882$$ 0 0
$$883$$ −22076.0 −0.841355 −0.420678 0.907210i $$-0.638208\pi$$
−0.420678 + 0.907210i $$0.638208\pi$$
$$884$$ 0 0
$$885$$ 25032.0 0.950781
$$886$$ 0 0
$$887$$ 40376.0 1.52840 0.764201 0.644978i $$-0.223133\pi$$
0.764201 + 0.644978i $$0.223133\pi$$
$$888$$ 0 0
$$889$$ −41856.0 −1.57908
$$890$$ 0 0
$$891$$ 2268.00 0.0852759
$$892$$ 0 0
$$893$$ 22080.0 0.827412
$$894$$ 0 0
$$895$$ 16632.0 0.621169
$$896$$ 0 0
$$897$$ −1776.00 −0.0661080
$$898$$ 0 0
$$899$$ 11040.0 0.409571
$$900$$ 0 0
$$901$$ −10660.0 −0.394158
$$902$$ 0 0
$$903$$ 288.000 0.0106136
$$904$$ 0 0
$$905$$ −48636.0 −1.78643
$$906$$ 0 0
$$907$$ 26396.0 0.966334 0.483167 0.875528i $$-0.339486\pi$$
0.483167 + 0.875528i $$0.339486\pi$$
$$908$$ 0 0
$$909$$ 2430.00 0.0886667
$$910$$ 0 0
$$911$$ −24368.0 −0.886222 −0.443111 0.896467i $$-0.646125\pi$$
−0.443111 + 0.896467i $$0.646125\pi$$
$$912$$ 0 0
$$913$$ 42224.0 1.53057
$$914$$ 0 0
$$915$$ 9156.00 0.330807
$$916$$ 0 0
$$917$$ −15648.0 −0.563514
$$918$$ 0 0
$$919$$ 5096.00 0.182918 0.0914589 0.995809i $$-0.470847\pi$$
0.0914589 + 0.995809i $$0.470847\pi$$
$$920$$ 0 0
$$921$$ 4500.00 0.160999
$$922$$ 0 0
$$923$$ 63344.0 2.25893
$$924$$ 0 0
$$925$$ 2130.00 0.0757124
$$926$$ 0 0
$$927$$ 13464.0 0.477040
$$928$$ 0 0
$$929$$ −18494.0 −0.653142 −0.326571 0.945173i $$-0.605893\pi$$
−0.326571 + 0.945173i $$0.605893\pi$$
$$930$$ 0 0
$$931$$ −21436.0 −0.754604
$$932$$ 0 0
$$933$$ −22824.0 −0.800883
$$934$$ 0 0
$$935$$ 32144.0 1.12430
$$936$$ 0 0
$$937$$ −33222.0 −1.15829 −0.579144 0.815225i $$-0.696613\pi$$
−0.579144 + 0.815225i $$0.696613\pi$$
$$938$$ 0 0
$$939$$ 14274.0 0.496075
$$940$$ 0 0
$$941$$ 27846.0 0.964669 0.482335 0.875987i $$-0.339789\pi$$
0.482335 + 0.875987i $$0.339789\pi$$
$$942$$ 0 0
$$943$$ −2256.00 −0.0779061
$$944$$ 0 0
$$945$$ −9072.00 −0.312288
$$946$$ 0 0
$$947$$ −41052.0 −1.40867 −0.704335 0.709868i $$-0.748755\pi$$
−0.704335 + 0.709868i $$0.748755\pi$$
$$948$$ 0 0
$$949$$ 73852.0 2.52617
$$950$$ 0 0
$$951$$ −13122.0 −0.447434
$$952$$ 0 0
$$953$$ 5706.00 0.193951 0.0969756 0.995287i $$-0.469083\pi$$
0.0969756 + 0.995287i $$0.469083\pi$$
$$954$$ 0 0
$$955$$ −2688.00 −0.0910802
$$956$$ 0 0
$$957$$ 11592.0 0.391553
$$958$$ 0 0
$$959$$ 36720.0 1.23644
$$960$$ 0 0
$$961$$ −23391.0 −0.785170
$$962$$ 0 0
$$963$$ 15228.0 0.509570
$$964$$ 0 0
$$965$$ 67228.0 2.24264
$$966$$ 0 0
$$967$$ 39352.0 1.30866 0.654330 0.756209i $$-0.272951\pi$$
0.654330 + 0.756209i $$0.272951\pi$$
$$968$$ 0 0
$$969$$ 22632.0 0.750304
$$970$$ 0 0
$$971$$ 33180.0 1.09660 0.548299 0.836282i $$-0.315276\pi$$
0.548299 + 0.836282i $$0.315276\pi$$
$$972$$ 0 0
$$973$$ −12384.0 −0.408030
$$974$$ 0 0
$$975$$ 15762.0 0.517731
$$976$$ 0 0
$$977$$ −4014.00 −0.131442 −0.0657212 0.997838i $$-0.520935\pi$$
−0.0657212 + 0.997838i $$0.520935\pi$$
$$978$$ 0 0
$$979$$ −6888.00 −0.224864
$$980$$ 0 0
$$981$$ 3654.00 0.118923
$$982$$ 0 0
$$983$$ −20328.0 −0.659575 −0.329788 0.944055i $$-0.606977\pi$$
−0.329788 + 0.944055i $$0.606977\pi$$
$$984$$ 0 0
$$985$$ 21252.0 0.687457
$$986$$ 0 0
$$987$$ 17280.0 0.557273
$$988$$ 0 0
$$989$$ 32.0000 0.00102886
$$990$$ 0 0
$$991$$ −11728.0 −0.375936 −0.187968 0.982175i $$-0.560190\pi$$
−0.187968 + 0.982175i $$0.560190\pi$$
$$992$$ 0 0
$$993$$ −23412.0 −0.748195
$$994$$ 0 0
$$995$$ −71792.0 −2.28740
$$996$$ 0 0
$$997$$ 50974.0 1.61922 0.809610 0.586968i $$-0.199679\pi$$
0.809610 + 0.586968i $$0.199679\pi$$
$$998$$ 0 0
$$999$$ −810.000 −0.0256529
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 48.4.a.b.1.1 1
3.2 odd 2 144.4.a.b.1.1 1
4.3 odd 2 24.4.a.a.1.1 1
5.2 odd 4 1200.4.f.p.49.2 2
5.3 odd 4 1200.4.f.p.49.1 2
5.4 even 2 1200.4.a.u.1.1 1
7.6 odd 2 2352.4.a.w.1.1 1
8.3 odd 2 192.4.a.a.1.1 1
8.5 even 2 192.4.a.g.1.1 1
12.11 even 2 72.4.a.b.1.1 1
16.3 odd 4 768.4.d.o.385.2 2
16.5 even 4 768.4.d.b.385.2 2
16.11 odd 4 768.4.d.o.385.1 2
16.13 even 4 768.4.d.b.385.1 2
20.3 even 4 600.4.f.b.49.2 2
20.7 even 4 600.4.f.b.49.1 2
20.19 odd 2 600.4.a.h.1.1 1
24.5 odd 2 576.4.a.v.1.1 1
24.11 even 2 576.4.a.u.1.1 1
28.27 even 2 1176.4.a.a.1.1 1
36.7 odd 6 648.4.i.b.433.1 2
36.11 even 6 648.4.i.k.433.1 2
36.23 even 6 648.4.i.k.217.1 2
36.31 odd 6 648.4.i.b.217.1 2
60.23 odd 4 1800.4.f.q.649.2 2
60.47 odd 4 1800.4.f.q.649.1 2
60.59 even 2 1800.4.a.bg.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
24.4.a.a.1.1 1 4.3 odd 2
48.4.a.b.1.1 1 1.1 even 1 trivial
72.4.a.b.1.1 1 12.11 even 2
144.4.a.b.1.1 1 3.2 odd 2
192.4.a.a.1.1 1 8.3 odd 2
192.4.a.g.1.1 1 8.5 even 2
576.4.a.u.1.1 1 24.11 even 2
576.4.a.v.1.1 1 24.5 odd 2
600.4.a.h.1.1 1 20.19 odd 2
600.4.f.b.49.1 2 20.7 even 4
600.4.f.b.49.2 2 20.3 even 4
648.4.i.b.217.1 2 36.31 odd 6
648.4.i.b.433.1 2 36.7 odd 6
648.4.i.k.217.1 2 36.23 even 6
648.4.i.k.433.1 2 36.11 even 6
768.4.d.b.385.1 2 16.13 even 4
768.4.d.b.385.2 2 16.5 even 4
768.4.d.o.385.1 2 16.11 odd 4
768.4.d.o.385.2 2 16.3 odd 4
1176.4.a.a.1.1 1 28.27 even 2
1200.4.a.u.1.1 1 5.4 even 2
1200.4.f.p.49.1 2 5.3 odd 4
1200.4.f.p.49.2 2 5.2 odd 4
1800.4.a.bg.1.1 1 60.59 even 2
1800.4.f.q.649.1 2 60.47 odd 4
1800.4.f.q.649.2 2 60.23 odd 4
2352.4.a.w.1.1 1 7.6 odd 2