# Properties

 Label 2368.2.a.k.1.1 Level $2368$ Weight $2$ Character 2368.1 Self dual yes Analytic conductor $18.909$ 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] = [2368,2,Mod(1,2368)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2368, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("2368.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2368 = 2^{6} \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2368.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: $$18.9085751986$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 296) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 2368.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{3} -3.00000 q^{7} -2.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{3} -3.00000 q^{7} -2.00000 q^{9} +3.00000 q^{11} +2.00000 q^{17} +2.00000 q^{19} -3.00000 q^{21} -6.00000 q^{23} -5.00000 q^{25} -5.00000 q^{27} +2.00000 q^{29} -4.00000 q^{31} +3.00000 q^{33} -1.00000 q^{37} +7.00000 q^{41} -4.00000 q^{43} +1.00000 q^{47} +2.00000 q^{49} +2.00000 q^{51} -9.00000 q^{53} +2.00000 q^{57} -8.00000 q^{59} +4.00000 q^{61} +6.00000 q^{63} -12.0000 q^{67} -6.00000 q^{69} -5.00000 q^{71} -13.0000 q^{73} -5.00000 q^{75} -9.00000 q^{77} -10.0000 q^{79} +1.00000 q^{81} +1.00000 q^{83} +2.00000 q^{87} -2.00000 q^{89} -4.00000 q^{93} -12.0000 q^{97} -6.00000 q^{99} +O(q^{100})$$

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 1.00000 0.577350 0.288675 0.957427i $$-0.406785\pi$$
0.288675 + 0.957427i $$0.406785\pi$$
$$4$$ 0 0
$$5$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$6$$ 0 0
$$7$$ −3.00000 −1.13389 −0.566947 0.823754i $$-0.691875\pi$$
−0.566947 + 0.823754i $$0.691875\pi$$
$$8$$ 0 0
$$9$$ −2.00000 −0.666667
$$10$$ 0 0
$$11$$ 3.00000 0.904534 0.452267 0.891883i $$-0.350615\pi$$
0.452267 + 0.891883i $$0.350615\pi$$
$$12$$ 0 0
$$13$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 2.00000 0.485071 0.242536 0.970143i $$-0.422021\pi$$
0.242536 + 0.970143i $$0.422021\pi$$
$$18$$ 0 0
$$19$$ 2.00000 0.458831 0.229416 0.973329i $$-0.426318\pi$$
0.229416 + 0.973329i $$0.426318\pi$$
$$20$$ 0 0
$$21$$ −3.00000 −0.654654
$$22$$ 0 0
$$23$$ −6.00000 −1.25109 −0.625543 0.780189i $$-0.715123\pi$$
−0.625543 + 0.780189i $$0.715123\pi$$
$$24$$ 0 0
$$25$$ −5.00000 −1.00000
$$26$$ 0 0
$$27$$ −5.00000 −0.962250
$$28$$ 0 0
$$29$$ 2.00000 0.371391 0.185695 0.982607i $$-0.440546\pi$$
0.185695 + 0.982607i $$0.440546\pi$$
$$30$$ 0 0
$$31$$ −4.00000 −0.718421 −0.359211 0.933257i $$-0.616954\pi$$
−0.359211 + 0.933257i $$0.616954\pi$$
$$32$$ 0 0
$$33$$ 3.00000 0.522233
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −1.00000 −0.164399
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 7.00000 1.09322 0.546608 0.837389i $$-0.315919\pi$$
0.546608 + 0.837389i $$0.315919\pi$$
$$42$$ 0 0
$$43$$ −4.00000 −0.609994 −0.304997 0.952353i $$-0.598656\pi$$
−0.304997 + 0.952353i $$0.598656\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 1.00000 0.145865 0.0729325 0.997337i $$-0.476764\pi$$
0.0729325 + 0.997337i $$0.476764\pi$$
$$48$$ 0 0
$$49$$ 2.00000 0.285714
$$50$$ 0 0
$$51$$ 2.00000 0.280056
$$52$$ 0 0
$$53$$ −9.00000 −1.23625 −0.618123 0.786082i $$-0.712106\pi$$
−0.618123 + 0.786082i $$0.712106\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 2.00000 0.264906
$$58$$ 0 0
$$59$$ −8.00000 −1.04151 −0.520756 0.853706i $$-0.674350\pi$$
−0.520756 + 0.853706i $$0.674350\pi$$
$$60$$ 0 0
$$61$$ 4.00000 0.512148 0.256074 0.966657i $$-0.417571\pi$$
0.256074 + 0.966657i $$0.417571\pi$$
$$62$$ 0 0
$$63$$ 6.00000 0.755929
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −12.0000 −1.46603 −0.733017 0.680211i $$-0.761888\pi$$
−0.733017 + 0.680211i $$0.761888\pi$$
$$68$$ 0 0
$$69$$ −6.00000 −0.722315
$$70$$ 0 0
$$71$$ −5.00000 −0.593391 −0.296695 0.954972i $$-0.595885\pi$$
−0.296695 + 0.954972i $$0.595885\pi$$
$$72$$ 0 0
$$73$$ −13.0000 −1.52153 −0.760767 0.649025i $$-0.775177\pi$$
−0.760767 + 0.649025i $$0.775177\pi$$
$$74$$ 0 0
$$75$$ −5.00000 −0.577350
$$76$$ 0 0
$$77$$ −9.00000 −1.02565
$$78$$ 0 0
$$79$$ −10.0000 −1.12509 −0.562544 0.826767i $$-0.690177\pi$$
−0.562544 + 0.826767i $$0.690177\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ 1.00000 0.109764 0.0548821 0.998493i $$-0.482522\pi$$
0.0548821 + 0.998493i $$0.482522\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 2.00000 0.214423
$$88$$ 0 0
$$89$$ −2.00000 −0.212000 −0.106000 0.994366i $$-0.533804\pi$$
−0.106000 + 0.994366i $$0.533804\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ −4.00000 −0.414781
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −12.0000 −1.21842 −0.609208 0.793011i $$-0.708512\pi$$
−0.609208 + 0.793011i $$0.708512\pi$$
$$98$$ 0 0
$$99$$ −6.00000 −0.603023
$$100$$ 0 0
$$101$$ 9.00000 0.895533 0.447767 0.894150i $$-0.352219\pi$$
0.447767 + 0.894150i $$0.352219\pi$$
$$102$$ 0 0
$$103$$ −8.00000 −0.788263 −0.394132 0.919054i $$-0.628955\pi$$
−0.394132 + 0.919054i $$0.628955\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −12.0000 −1.16008 −0.580042 0.814587i $$-0.696964\pi$$
−0.580042 + 0.814587i $$0.696964\pi$$
$$108$$ 0 0
$$109$$ 14.0000 1.34096 0.670478 0.741929i $$-0.266089\pi$$
0.670478 + 0.741929i $$0.266089\pi$$
$$110$$ 0 0
$$111$$ −1.00000 −0.0949158
$$112$$ 0 0
$$113$$ 6.00000 0.564433 0.282216 0.959351i $$-0.408930\pi$$
0.282216 + 0.959351i $$0.408930\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ −6.00000 −0.550019
$$120$$ 0 0
$$121$$ −2.00000 −0.181818
$$122$$ 0 0
$$123$$ 7.00000 0.631169
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 11.0000 0.976092 0.488046 0.872818i $$-0.337710\pi$$
0.488046 + 0.872818i $$0.337710\pi$$
$$128$$ 0 0
$$129$$ −4.00000 −0.352180
$$130$$ 0 0
$$131$$ −10.0000 −0.873704 −0.436852 0.899533i $$-0.643907\pi$$
−0.436852 + 0.899533i $$0.643907\pi$$
$$132$$ 0 0
$$133$$ −6.00000 −0.520266
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 10.0000 0.854358 0.427179 0.904167i $$-0.359507\pi$$
0.427179 + 0.904167i $$0.359507\pi$$
$$138$$ 0 0
$$139$$ 20.0000 1.69638 0.848189 0.529694i $$-0.177693\pi$$
0.848189 + 0.529694i $$0.177693\pi$$
$$140$$ 0 0
$$141$$ 1.00000 0.0842152
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 2.00000 0.164957
$$148$$ 0 0
$$149$$ 21.0000 1.72039 0.860194 0.509968i $$-0.170343\pi$$
0.860194 + 0.509968i $$0.170343\pi$$
$$150$$ 0 0
$$151$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$152$$ 0 0
$$153$$ −4.00000 −0.323381
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 17.0000 1.35675 0.678374 0.734717i $$-0.262685\pi$$
0.678374 + 0.734717i $$0.262685\pi$$
$$158$$ 0 0
$$159$$ −9.00000 −0.713746
$$160$$ 0 0
$$161$$ 18.0000 1.41860
$$162$$ 0 0
$$163$$ −8.00000 −0.626608 −0.313304 0.949653i $$-0.601436\pi$$
−0.313304 + 0.949653i $$0.601436\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −2.00000 −0.154765 −0.0773823 0.997001i $$-0.524656\pi$$
−0.0773823 + 0.997001i $$0.524656\pi$$
$$168$$ 0 0
$$169$$ −13.0000 −1.00000
$$170$$ 0 0
$$171$$ −4.00000 −0.305888
$$172$$ 0 0
$$173$$ 19.0000 1.44454 0.722272 0.691609i $$-0.243098\pi$$
0.722272 + 0.691609i $$0.243098\pi$$
$$174$$ 0 0
$$175$$ 15.0000 1.13389
$$176$$ 0 0
$$177$$ −8.00000 −0.601317
$$178$$ 0 0
$$179$$ −10.0000 −0.747435 −0.373718 0.927543i $$-0.621917\pi$$
−0.373718 + 0.927543i $$0.621917\pi$$
$$180$$ 0 0
$$181$$ −5.00000 −0.371647 −0.185824 0.982583i $$-0.559495\pi$$
−0.185824 + 0.982583i $$0.559495\pi$$
$$182$$ 0 0
$$183$$ 4.00000 0.295689
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 6.00000 0.438763
$$188$$ 0 0
$$189$$ 15.0000 1.09109
$$190$$ 0 0
$$191$$ −20.0000 −1.44715 −0.723575 0.690246i $$-0.757502\pi$$
−0.723575 + 0.690246i $$0.757502\pi$$
$$192$$ 0 0
$$193$$ 4.00000 0.287926 0.143963 0.989583i $$-0.454015\pi$$
0.143963 + 0.989583i $$0.454015\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −11.0000 −0.783718 −0.391859 0.920025i $$-0.628168\pi$$
−0.391859 + 0.920025i $$0.628168\pi$$
$$198$$ 0 0
$$199$$ −14.0000 −0.992434 −0.496217 0.868199i $$-0.665278\pi$$
−0.496217 + 0.868199i $$0.665278\pi$$
$$200$$ 0 0
$$201$$ −12.0000 −0.846415
$$202$$ 0 0
$$203$$ −6.00000 −0.421117
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 12.0000 0.834058
$$208$$ 0 0
$$209$$ 6.00000 0.415029
$$210$$ 0 0
$$211$$ −13.0000 −0.894957 −0.447478 0.894295i $$-0.647678\pi$$
−0.447478 + 0.894295i $$0.647678\pi$$
$$212$$ 0 0
$$213$$ −5.00000 −0.342594
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 12.0000 0.814613
$$218$$ 0 0
$$219$$ −13.0000 −0.878459
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 0 0
$$223$$ 5.00000 0.334825 0.167412 0.985887i $$-0.446459\pi$$
0.167412 + 0.985887i $$0.446459\pi$$
$$224$$ 0 0
$$225$$ 10.0000 0.666667
$$226$$ 0 0
$$227$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$228$$ 0 0
$$229$$ 13.0000 0.859064 0.429532 0.903052i $$-0.358679\pi$$
0.429532 + 0.903052i $$0.358679\pi$$
$$230$$ 0 0
$$231$$ −9.00000 −0.592157
$$232$$ 0 0
$$233$$ −18.0000 −1.17922 −0.589610 0.807688i $$-0.700718\pi$$
−0.589610 + 0.807688i $$0.700718\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ −10.0000 −0.649570
$$238$$ 0 0
$$239$$ −6.00000 −0.388108 −0.194054 0.980991i $$-0.562164\pi$$
−0.194054 + 0.980991i $$0.562164\pi$$
$$240$$ 0 0
$$241$$ 18.0000 1.15948 0.579741 0.814801i $$-0.303154\pi$$
0.579741 + 0.814801i $$0.303154\pi$$
$$242$$ 0 0
$$243$$ 16.0000 1.02640
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 0 0
$$249$$ 1.00000 0.0633724
$$250$$ 0 0
$$251$$ 24.0000 1.51487 0.757433 0.652913i $$-0.226453\pi$$
0.757433 + 0.652913i $$0.226453\pi$$
$$252$$ 0 0
$$253$$ −18.0000 −1.13165
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 28.0000 1.74659 0.873296 0.487190i $$-0.161978\pi$$
0.873296 + 0.487190i $$0.161978\pi$$
$$258$$ 0 0
$$259$$ 3.00000 0.186411
$$260$$ 0 0
$$261$$ −4.00000 −0.247594
$$262$$ 0 0
$$263$$ −19.0000 −1.17159 −0.585795 0.810459i $$-0.699218\pi$$
−0.585795 + 0.810459i $$0.699218\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ −2.00000 −0.122398
$$268$$ 0 0
$$269$$ −14.0000 −0.853595 −0.426798 0.904347i $$-0.640358\pi$$
−0.426798 + 0.904347i $$0.640358\pi$$
$$270$$ 0 0
$$271$$ 15.0000 0.911185 0.455593 0.890188i $$-0.349427\pi$$
0.455593 + 0.890188i $$0.349427\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ −15.0000 −0.904534
$$276$$ 0 0
$$277$$ −24.0000 −1.44202 −0.721010 0.692925i $$-0.756322\pi$$
−0.721010 + 0.692925i $$0.756322\pi$$
$$278$$ 0 0
$$279$$ 8.00000 0.478947
$$280$$ 0 0
$$281$$ 32.0000 1.90896 0.954480 0.298275i $$-0.0964112\pi$$
0.954480 + 0.298275i $$0.0964112\pi$$
$$282$$ 0 0
$$283$$ 8.00000 0.475551 0.237775 0.971320i $$-0.423582\pi$$
0.237775 + 0.971320i $$0.423582\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −21.0000 −1.23959
$$288$$ 0 0
$$289$$ −13.0000 −0.764706
$$290$$ 0 0
$$291$$ −12.0000 −0.703452
$$292$$ 0 0
$$293$$ −22.0000 −1.28525 −0.642627 0.766179i $$-0.722155\pi$$
−0.642627 + 0.766179i $$0.722155\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ −15.0000 −0.870388
$$298$$ 0 0
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 12.0000 0.691669
$$302$$ 0 0
$$303$$ 9.00000 0.517036
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 7.00000 0.399511 0.199756 0.979846i $$-0.435985\pi$$
0.199756 + 0.979846i $$0.435985\pi$$
$$308$$ 0 0
$$309$$ −8.00000 −0.455104
$$310$$ 0 0
$$311$$ 30.0000 1.70114 0.850572 0.525859i $$-0.176256\pi$$
0.850572 + 0.525859i $$0.176256\pi$$
$$312$$ 0 0
$$313$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −6.00000 −0.336994 −0.168497 0.985702i $$-0.553891\pi$$
−0.168497 + 0.985702i $$0.553891\pi$$
$$318$$ 0 0
$$319$$ 6.00000 0.335936
$$320$$ 0 0
$$321$$ −12.0000 −0.669775
$$322$$ 0 0
$$323$$ 4.00000 0.222566
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 14.0000 0.774202
$$328$$ 0 0
$$329$$ −3.00000 −0.165395
$$330$$ 0 0
$$331$$ 22.0000 1.20923 0.604615 0.796518i $$-0.293327\pi$$
0.604615 + 0.796518i $$0.293327\pi$$
$$332$$ 0 0
$$333$$ 2.00000 0.109599
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 15.0000 0.817102 0.408551 0.912735i $$-0.366034\pi$$
0.408551 + 0.912735i $$0.366034\pi$$
$$338$$ 0 0
$$339$$ 6.00000 0.325875
$$340$$ 0 0
$$341$$ −12.0000 −0.649836
$$342$$ 0 0
$$343$$ 15.0000 0.809924
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 10.0000 0.536828 0.268414 0.963304i $$-0.413500\pi$$
0.268414 + 0.963304i $$0.413500\pi$$
$$348$$ 0 0
$$349$$ −22.0000 −1.17763 −0.588817 0.808267i $$-0.700406\pi$$
−0.588817 + 0.808267i $$0.700406\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ −30.0000 −1.59674 −0.798369 0.602168i $$-0.794304\pi$$
−0.798369 + 0.602168i $$0.794304\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ −6.00000 −0.317554
$$358$$ 0 0
$$359$$ 15.0000 0.791670 0.395835 0.918322i $$-0.370455\pi$$
0.395835 + 0.918322i $$0.370455\pi$$
$$360$$ 0 0
$$361$$ −15.0000 −0.789474
$$362$$ 0 0
$$363$$ −2.00000 −0.104973
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ −32.0000 −1.67039 −0.835193 0.549957i $$-0.814644\pi$$
−0.835193 + 0.549957i $$0.814644\pi$$
$$368$$ 0 0
$$369$$ −14.0000 −0.728811
$$370$$ 0 0
$$371$$ 27.0000 1.40177
$$372$$ 0 0
$$373$$ −1.00000 −0.0517780 −0.0258890 0.999665i $$-0.508242\pi$$
−0.0258890 + 0.999665i $$0.508242\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 0 0
$$378$$ 0 0
$$379$$ −5.00000 −0.256833 −0.128416 0.991720i $$-0.540989\pi$$
−0.128416 + 0.991720i $$0.540989\pi$$
$$380$$ 0 0
$$381$$ 11.0000 0.563547
$$382$$ 0 0
$$383$$ −20.0000 −1.02195 −0.510976 0.859595i $$-0.670716\pi$$
−0.510976 + 0.859595i $$0.670716\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 8.00000 0.406663
$$388$$ 0 0
$$389$$ −20.0000 −1.01404 −0.507020 0.861934i $$-0.669253\pi$$
−0.507020 + 0.861934i $$0.669253\pi$$
$$390$$ 0 0
$$391$$ −12.0000 −0.606866
$$392$$ 0 0
$$393$$ −10.0000 −0.504433
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 25.0000 1.25471 0.627357 0.778732i $$-0.284137\pi$$
0.627357 + 0.778732i $$0.284137\pi$$
$$398$$ 0 0
$$399$$ −6.00000 −0.300376
$$400$$ 0 0
$$401$$ 28.0000 1.39825 0.699127 0.714998i $$-0.253572\pi$$
0.699127 + 0.714998i $$0.253572\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −3.00000 −0.148704
$$408$$ 0 0
$$409$$ −32.0000 −1.58230 −0.791149 0.611623i $$-0.790517\pi$$
−0.791149 + 0.611623i $$0.790517\pi$$
$$410$$ 0 0
$$411$$ 10.0000 0.493264
$$412$$ 0 0
$$413$$ 24.0000 1.18096
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 20.0000 0.979404
$$418$$ 0 0
$$419$$ 3.00000 0.146560 0.0732798 0.997311i $$-0.476653\pi$$
0.0732798 + 0.997311i $$0.476653\pi$$
$$420$$ 0 0
$$421$$ 10.0000 0.487370 0.243685 0.969854i $$-0.421644\pi$$
0.243685 + 0.969854i $$0.421644\pi$$
$$422$$ 0 0
$$423$$ −2.00000 −0.0972433
$$424$$ 0 0
$$425$$ −10.0000 −0.485071
$$426$$ 0 0
$$427$$ −12.0000 −0.580721
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$432$$ 0 0
$$433$$ 21.0000 1.00920 0.504598 0.863355i $$-0.331641\pi$$
0.504598 + 0.863355i $$0.331641\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ −12.0000 −0.574038
$$438$$ 0 0
$$439$$ 32.0000 1.52728 0.763638 0.645644i $$-0.223411\pi$$
0.763638 + 0.645644i $$0.223411\pi$$
$$440$$ 0 0
$$441$$ −4.00000 −0.190476
$$442$$ 0 0
$$443$$ 41.0000 1.94797 0.973984 0.226615i $$-0.0727659\pi$$
0.973984 + 0.226615i $$0.0727659\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 21.0000 0.993266
$$448$$ 0 0
$$449$$ −30.0000 −1.41579 −0.707894 0.706319i $$-0.750354\pi$$
−0.707894 + 0.706319i $$0.750354\pi$$
$$450$$ 0 0
$$451$$ 21.0000 0.988851
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 28.0000 1.30978 0.654892 0.755722i $$-0.272714\pi$$
0.654892 + 0.755722i $$0.272714\pi$$
$$458$$ 0 0
$$459$$ −10.0000 −0.466760
$$460$$ 0 0
$$461$$ 8.00000 0.372597 0.186299 0.982493i $$-0.440351\pi$$
0.186299 + 0.982493i $$0.440351\pi$$
$$462$$ 0 0
$$463$$ 38.0000 1.76601 0.883005 0.469364i $$-0.155517\pi$$
0.883005 + 0.469364i $$0.155517\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −6.00000 −0.277647 −0.138823 0.990317i $$-0.544332\pi$$
−0.138823 + 0.990317i $$0.544332\pi$$
$$468$$ 0 0
$$469$$ 36.0000 1.66233
$$470$$ 0 0
$$471$$ 17.0000 0.783319
$$472$$ 0 0
$$473$$ −12.0000 −0.551761
$$474$$ 0 0
$$475$$ −10.0000 −0.458831
$$476$$ 0 0
$$477$$ 18.0000 0.824163
$$478$$ 0 0
$$479$$ −38.0000 −1.73626 −0.868132 0.496333i $$-0.834679\pi$$
−0.868132 + 0.496333i $$0.834679\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 0 0
$$483$$ 18.0000 0.819028
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 26.0000 1.17817 0.589086 0.808070i $$-0.299488\pi$$
0.589086 + 0.808070i $$0.299488\pi$$
$$488$$ 0 0
$$489$$ −8.00000 −0.361773
$$490$$ 0 0
$$491$$ −12.0000 −0.541552 −0.270776 0.962642i $$-0.587280\pi$$
−0.270776 + 0.962642i $$0.587280\pi$$
$$492$$ 0 0
$$493$$ 4.00000 0.180151
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 15.0000 0.672842
$$498$$ 0 0
$$499$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$500$$ 0 0
$$501$$ −2.00000 −0.0893534
$$502$$ 0 0
$$503$$ −18.0000 −0.802580 −0.401290 0.915951i $$-0.631438\pi$$
−0.401290 + 0.915951i $$0.631438\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ −13.0000 −0.577350
$$508$$ 0 0
$$509$$ 15.0000 0.664863 0.332432 0.943127i $$-0.392131\pi$$
0.332432 + 0.943127i $$0.392131\pi$$
$$510$$ 0 0
$$511$$ 39.0000 1.72526
$$512$$ 0 0
$$513$$ −10.0000 −0.441511
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 3.00000 0.131940
$$518$$ 0 0
$$519$$ 19.0000 0.834007
$$520$$ 0 0
$$521$$ −29.0000 −1.27051 −0.635257 0.772301i $$-0.719106\pi$$
−0.635257 + 0.772301i $$0.719106\pi$$
$$522$$ 0 0
$$523$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$524$$ 0 0
$$525$$ 15.0000 0.654654
$$526$$ 0 0
$$527$$ −8.00000 −0.348485
$$528$$ 0 0
$$529$$ 13.0000 0.565217
$$530$$ 0 0
$$531$$ 16.0000 0.694341
$$532$$ 0 0
$$533$$ 0 0
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ −10.0000 −0.431532
$$538$$ 0 0
$$539$$ 6.00000 0.258438
$$540$$ 0 0
$$541$$ 36.0000 1.54776 0.773880 0.633332i $$-0.218313\pi$$
0.773880 + 0.633332i $$0.218313\pi$$
$$542$$ 0 0
$$543$$ −5.00000 −0.214571
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −12.0000 −0.513083 −0.256541 0.966533i $$-0.582583\pi$$
−0.256541 + 0.966533i $$0.582583\pi$$
$$548$$ 0 0
$$549$$ −8.00000 −0.341432
$$550$$ 0 0
$$551$$ 4.00000 0.170406
$$552$$ 0 0
$$553$$ 30.0000 1.27573
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ −18.0000 −0.762684 −0.381342 0.924434i $$-0.624538\pi$$
−0.381342 + 0.924434i $$0.624538\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ 0 0
$$561$$ 6.00000 0.253320
$$562$$ 0 0
$$563$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ −3.00000 −0.125988
$$568$$ 0 0
$$569$$ −24.0000 −1.00613 −0.503066 0.864248i $$-0.667795\pi$$
−0.503066 + 0.864248i $$0.667795\pi$$
$$570$$ 0 0
$$571$$ 15.0000 0.627730 0.313865 0.949468i $$-0.398376\pi$$
0.313865 + 0.949468i $$0.398376\pi$$
$$572$$ 0 0
$$573$$ −20.0000 −0.835512
$$574$$ 0 0
$$575$$ 30.0000 1.25109
$$576$$ 0 0
$$577$$ 38.0000 1.58196 0.790980 0.611842i $$-0.209571\pi$$
0.790980 + 0.611842i $$0.209571\pi$$
$$578$$ 0 0
$$579$$ 4.00000 0.166234
$$580$$ 0 0
$$581$$ −3.00000 −0.124461
$$582$$ 0 0
$$583$$ −27.0000 −1.11823
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −42.0000 −1.73353 −0.866763 0.498721i $$-0.833803\pi$$
−0.866763 + 0.498721i $$0.833803\pi$$
$$588$$ 0 0
$$589$$ −8.00000 −0.329634
$$590$$ 0 0
$$591$$ −11.0000 −0.452480
$$592$$ 0 0
$$593$$ 11.0000 0.451716 0.225858 0.974160i $$-0.427481\pi$$
0.225858 + 0.974160i $$0.427481\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ −14.0000 −0.572982
$$598$$ 0 0
$$599$$ 23.0000 0.939755 0.469877 0.882732i $$-0.344298\pi$$
0.469877 + 0.882732i $$0.344298\pi$$
$$600$$ 0 0
$$601$$ 14.0000 0.571072 0.285536 0.958368i $$-0.407828\pi$$
0.285536 + 0.958368i $$0.407828\pi$$
$$602$$ 0 0
$$603$$ 24.0000 0.977356
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 34.0000 1.38002 0.690009 0.723801i $$-0.257607\pi$$
0.690009 + 0.723801i $$0.257607\pi$$
$$608$$ 0 0
$$609$$ −6.00000 −0.243132
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 0 0
$$613$$ −43.0000 −1.73675 −0.868377 0.495905i $$-0.834836\pi$$
−0.868377 + 0.495905i $$0.834836\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −23.0000 −0.925945 −0.462973 0.886373i $$-0.653217\pi$$
−0.462973 + 0.886373i $$0.653217\pi$$
$$618$$ 0 0
$$619$$ −37.0000 −1.48716 −0.743578 0.668649i $$-0.766873\pi$$
−0.743578 + 0.668649i $$0.766873\pi$$
$$620$$ 0 0
$$621$$ 30.0000 1.20386
$$622$$ 0 0
$$623$$ 6.00000 0.240385
$$624$$ 0 0
$$625$$ 25.0000 1.00000
$$626$$ 0 0
$$627$$ 6.00000 0.239617
$$628$$ 0 0
$$629$$ −2.00000 −0.0797452
$$630$$ 0 0
$$631$$ 2.00000 0.0796187 0.0398094 0.999207i $$-0.487325\pi$$
0.0398094 + 0.999207i $$0.487325\pi$$
$$632$$ 0 0
$$633$$ −13.0000 −0.516704
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ 10.0000 0.395594
$$640$$ 0 0
$$641$$ −33.0000 −1.30342 −0.651711 0.758468i $$-0.725948\pi$$
−0.651711 + 0.758468i $$0.725948\pi$$
$$642$$ 0 0
$$643$$ 34.0000 1.34083 0.670415 0.741987i $$-0.266116\pi$$
0.670415 + 0.741987i $$0.266116\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −46.0000 −1.80845 −0.904223 0.427060i $$-0.859549\pi$$
−0.904223 + 0.427060i $$0.859549\pi$$
$$648$$ 0 0
$$649$$ −24.0000 −0.942082
$$650$$ 0 0
$$651$$ 12.0000 0.470317
$$652$$ 0 0
$$653$$ −4.00000 −0.156532 −0.0782660 0.996933i $$-0.524938\pi$$
−0.0782660 + 0.996933i $$0.524938\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 26.0000 1.01436
$$658$$ 0 0
$$659$$ −35.0000 −1.36341 −0.681703 0.731629i $$-0.738760\pi$$
−0.681703 + 0.731629i $$0.738760\pi$$
$$660$$ 0 0
$$661$$ −42.0000 −1.63361 −0.816805 0.576913i $$-0.804257\pi$$
−0.816805 + 0.576913i $$0.804257\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −12.0000 −0.464642
$$668$$ 0 0
$$669$$ 5.00000 0.193311
$$670$$ 0 0
$$671$$ 12.0000 0.463255
$$672$$ 0 0
$$673$$ −41.0000 −1.58043 −0.790217 0.612827i $$-0.790032\pi$$
−0.790217 + 0.612827i $$0.790032\pi$$
$$674$$ 0 0
$$675$$ 25.0000 0.962250
$$676$$ 0 0
$$677$$ 31.0000 1.19143 0.595713 0.803197i $$-0.296869\pi$$
0.595713 + 0.803197i $$0.296869\pi$$
$$678$$ 0 0
$$679$$ 36.0000 1.38155
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ −12.0000 −0.459167 −0.229584 0.973289i $$-0.573736\pi$$
−0.229584 + 0.973289i $$0.573736\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 13.0000 0.495981
$$688$$ 0 0
$$689$$ 0 0
$$690$$ 0 0
$$691$$ 28.0000 1.06517 0.532585 0.846376i $$-0.321221\pi$$
0.532585 + 0.846376i $$0.321221\pi$$
$$692$$ 0 0
$$693$$ 18.0000 0.683763
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 14.0000 0.530288
$$698$$ 0 0
$$699$$ −18.0000 −0.680823
$$700$$ 0 0
$$701$$ −28.0000 −1.05755 −0.528773 0.848763i $$-0.677348\pi$$
−0.528773 + 0.848763i $$0.677348\pi$$
$$702$$ 0 0
$$703$$ −2.00000 −0.0754314
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ −27.0000 −1.01544
$$708$$ 0 0
$$709$$ −6.00000 −0.225335 −0.112667 0.993633i $$-0.535939\pi$$
−0.112667 + 0.993633i $$0.535939\pi$$
$$710$$ 0 0
$$711$$ 20.0000 0.750059
$$712$$ 0 0
$$713$$ 24.0000 0.898807
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ −6.00000 −0.224074
$$718$$ 0 0
$$719$$ 13.0000 0.484818 0.242409 0.970174i $$-0.422062\pi$$
0.242409 + 0.970174i $$0.422062\pi$$
$$720$$ 0 0
$$721$$ 24.0000 0.893807
$$722$$ 0 0
$$723$$ 18.0000 0.669427
$$724$$ 0 0
$$725$$ −10.0000 −0.371391
$$726$$ 0 0
$$727$$ −26.0000 −0.964287 −0.482143 0.876092i $$-0.660142\pi$$
−0.482143 + 0.876092i $$0.660142\pi$$
$$728$$ 0 0
$$729$$ 13.0000 0.481481
$$730$$ 0 0
$$731$$ −8.00000 −0.295891
$$732$$ 0 0
$$733$$ −31.0000 −1.14501 −0.572506 0.819901i $$-0.694029\pi$$
−0.572506 + 0.819901i $$0.694029\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −36.0000 −1.32608
$$738$$ 0 0
$$739$$ 3.00000 0.110357 0.0551784 0.998477i $$-0.482427\pi$$
0.0551784 + 0.998477i $$0.482427\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 31.0000 1.13728 0.568640 0.822587i $$-0.307470\pi$$
0.568640 + 0.822587i $$0.307470\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ −2.00000 −0.0731762
$$748$$ 0 0
$$749$$ 36.0000 1.31541
$$750$$ 0 0
$$751$$ 3.00000 0.109472 0.0547358 0.998501i $$-0.482568\pi$$
0.0547358 + 0.998501i $$0.482568\pi$$
$$752$$ 0 0
$$753$$ 24.0000 0.874609
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 26.0000 0.944986 0.472493 0.881334i $$-0.343354\pi$$
0.472493 + 0.881334i $$0.343354\pi$$
$$758$$ 0 0
$$759$$ −18.0000 −0.653359
$$760$$ 0 0
$$761$$ 17.0000 0.616250 0.308125 0.951346i $$-0.400299\pi$$
0.308125 + 0.951346i $$0.400299\pi$$
$$762$$ 0 0
$$763$$ −42.0000 −1.52050
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 0 0
$$768$$ 0 0
$$769$$ 14.0000 0.504853 0.252426 0.967616i $$-0.418771\pi$$
0.252426 + 0.967616i $$0.418771\pi$$
$$770$$ 0 0
$$771$$ 28.0000 1.00840
$$772$$ 0 0
$$773$$ 33.0000 1.18693 0.593464 0.804861i $$-0.297760\pi$$
0.593464 + 0.804861i $$0.297760\pi$$
$$774$$ 0 0
$$775$$ 20.0000 0.718421
$$776$$ 0 0
$$777$$ 3.00000 0.107624
$$778$$ 0 0
$$779$$ 14.0000 0.501602
$$780$$ 0 0
$$781$$ −15.0000 −0.536742
$$782$$ 0 0
$$783$$ −10.0000 −0.357371
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 39.0000 1.39020 0.695100 0.718913i $$-0.255360\pi$$
0.695100 + 0.718913i $$0.255360\pi$$
$$788$$ 0 0
$$789$$ −19.0000 −0.676418
$$790$$ 0 0
$$791$$ −18.0000 −0.640006
$$792$$ 0 0
$$793$$ 0 0
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ −16.0000 −0.566749 −0.283375 0.959009i $$-0.591454\pi$$
−0.283375 + 0.959009i $$0.591454\pi$$
$$798$$ 0 0
$$799$$ 2.00000 0.0707549
$$800$$ 0 0
$$801$$ 4.00000 0.141333
$$802$$ 0 0
$$803$$ −39.0000 −1.37628
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ −14.0000 −0.492823
$$808$$ 0 0
$$809$$ −10.0000 −0.351581 −0.175791 0.984428i $$-0.556248\pi$$
−0.175791 + 0.984428i $$0.556248\pi$$
$$810$$ 0 0
$$811$$ 27.0000 0.948098 0.474049 0.880498i $$-0.342792\pi$$
0.474049 + 0.880498i $$0.342792\pi$$
$$812$$ 0 0
$$813$$ 15.0000 0.526073
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ −8.00000 −0.279885
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 15.0000 0.523504 0.261752 0.965135i $$-0.415700\pi$$
0.261752 + 0.965135i $$0.415700\pi$$
$$822$$ 0 0
$$823$$ −48.0000 −1.67317 −0.836587 0.547833i $$-0.815453\pi$$
−0.836587 + 0.547833i $$0.815453\pi$$
$$824$$ 0 0
$$825$$ −15.0000 −0.522233
$$826$$ 0 0
$$827$$ 38.0000 1.32139 0.660695 0.750655i $$-0.270262\pi$$
0.660695 + 0.750655i $$0.270262\pi$$
$$828$$ 0 0
$$829$$ 56.0000 1.94496 0.972480 0.232986i $$-0.0748495\pi$$
0.972480 + 0.232986i $$0.0748495\pi$$
$$830$$ 0 0
$$831$$ −24.0000 −0.832551
$$832$$ 0 0
$$833$$ 4.00000 0.138592
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 20.0000 0.691301
$$838$$ 0 0
$$839$$ 8.00000 0.276191 0.138095 0.990419i $$-0.455902\pi$$
0.138095 + 0.990419i $$0.455902\pi$$
$$840$$ 0 0
$$841$$ −25.0000 −0.862069
$$842$$ 0 0
$$843$$ 32.0000 1.10214
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 6.00000 0.206162
$$848$$ 0 0
$$849$$ 8.00000 0.274559
$$850$$ 0 0
$$851$$ 6.00000 0.205677
$$852$$ 0 0
$$853$$ −34.0000 −1.16414 −0.582069 0.813139i $$-0.697757\pi$$
−0.582069 + 0.813139i $$0.697757\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 6.00000 0.204956 0.102478 0.994735i $$-0.467323\pi$$
0.102478 + 0.994735i $$0.467323\pi$$
$$858$$ 0 0
$$859$$ 34.0000 1.16007 0.580033 0.814593i $$-0.303040\pi$$
0.580033 + 0.814593i $$0.303040\pi$$
$$860$$ 0 0
$$861$$ −21.0000 −0.715678
$$862$$ 0 0
$$863$$ −48.0000 −1.63394 −0.816970 0.576681i $$-0.804348\pi$$
−0.816970 + 0.576681i $$0.804348\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ −13.0000 −0.441503
$$868$$ 0 0
$$869$$ −30.0000 −1.01768
$$870$$ 0 0
$$871$$ 0 0
$$872$$ 0 0
$$873$$ 24.0000 0.812277
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 22.0000 0.742887 0.371444 0.928456i $$-0.378863\pi$$
0.371444 + 0.928456i $$0.378863\pi$$
$$878$$ 0 0
$$879$$ −22.0000 −0.742042
$$880$$ 0 0
$$881$$ 18.0000 0.606435 0.303218 0.952921i $$-0.401939\pi$$
0.303218 + 0.952921i $$0.401939\pi$$
$$882$$ 0 0
$$883$$ −32.0000 −1.07689 −0.538443 0.842662i $$-0.680987\pi$$
−0.538443 + 0.842662i $$0.680987\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −9.00000 −0.302190 −0.151095 0.988519i $$-0.548280\pi$$
−0.151095 + 0.988519i $$0.548280\pi$$
$$888$$ 0 0
$$889$$ −33.0000 −1.10678
$$890$$ 0 0
$$891$$ 3.00000 0.100504
$$892$$ 0 0
$$893$$ 2.00000 0.0669274
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ −8.00000 −0.266815
$$900$$ 0 0
$$901$$ −18.0000 −0.599667
$$902$$ 0 0
$$903$$ 12.0000 0.399335
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 28.0000 0.929725 0.464862 0.885383i $$-0.346104\pi$$
0.464862 + 0.885383i $$0.346104\pi$$
$$908$$ 0 0
$$909$$ −18.0000 −0.597022
$$910$$ 0 0
$$911$$ −4.00000 −0.132526 −0.0662630 0.997802i $$-0.521108\pi$$
−0.0662630 + 0.997802i $$0.521108\pi$$
$$912$$ 0 0
$$913$$ 3.00000 0.0992855
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 30.0000 0.990687
$$918$$ 0 0
$$919$$ −4.00000 −0.131948 −0.0659739 0.997821i $$-0.521015\pi$$
−0.0659739 + 0.997821i $$0.521015\pi$$
$$920$$ 0 0
$$921$$ 7.00000 0.230658
$$922$$ 0 0
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 5.00000 0.164399
$$926$$ 0 0
$$927$$ 16.0000 0.525509
$$928$$ 0 0
$$929$$ −10.0000 −0.328089 −0.164045 0.986453i $$-0.552454\pi$$
−0.164045 + 0.986453i $$0.552454\pi$$
$$930$$ 0 0
$$931$$ 4.00000 0.131095
$$932$$ 0 0
$$933$$ 30.0000 0.982156
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 17.0000 0.555366 0.277683 0.960673i $$-0.410434\pi$$
0.277683 + 0.960673i $$0.410434\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 2.00000 0.0651981 0.0325991 0.999469i $$-0.489622\pi$$
0.0325991 + 0.999469i $$0.489622\pi$$
$$942$$ 0 0
$$943$$ −42.0000 −1.36771
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 2.00000 0.0649913 0.0324956 0.999472i $$-0.489654\pi$$
0.0324956 + 0.999472i $$0.489654\pi$$
$$948$$ 0 0
$$949$$ 0 0
$$950$$ 0 0
$$951$$ −6.00000 −0.194563
$$952$$ 0 0
$$953$$ 45.0000 1.45769 0.728846 0.684677i $$-0.240057\pi$$
0.728846 + 0.684677i $$0.240057\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 6.00000 0.193952
$$958$$ 0 0
$$959$$ −30.0000 −0.968751
$$960$$ 0 0
$$961$$ −15.0000 −0.483871
$$962$$ 0 0
$$963$$ 24.0000 0.773389
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −22.0000 −0.707472 −0.353736 0.935345i $$-0.615089\pi$$
−0.353736 + 0.935345i $$0.615089\pi$$
$$968$$ 0 0
$$969$$ 4.00000 0.128499
$$970$$ 0 0
$$971$$ −28.0000 −0.898563 −0.449281 0.893390i $$-0.648320\pi$$
−0.449281 + 0.893390i $$0.648320\pi$$
$$972$$ 0 0
$$973$$ −60.0000 −1.92351
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 4.00000 0.127971 0.0639857 0.997951i $$-0.479619\pi$$
0.0639857 + 0.997951i $$0.479619\pi$$
$$978$$ 0 0
$$979$$ −6.00000 −0.191761
$$980$$ 0 0
$$981$$ −28.0000 −0.893971
$$982$$ 0 0
$$983$$ −33.0000 −1.05254 −0.526268 0.850319i $$-0.676409\pi$$
−0.526268 + 0.850319i $$0.676409\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ −3.00000 −0.0954911
$$988$$ 0 0
$$989$$ 24.0000 0.763156
$$990$$ 0 0
$$991$$ 44.0000 1.39771 0.698853 0.715265i $$-0.253694\pi$$
0.698853 + 0.715265i $$0.253694\pi$$
$$992$$ 0 0
$$993$$ 22.0000 0.698149
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ −18.0000 −0.570066 −0.285033 0.958518i $$-0.592005\pi$$
−0.285033 + 0.958518i $$0.592005\pi$$
$$998$$ 0 0
$$999$$ 5.00000 0.158193
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 2368.2.a.k.1.1 1
4.3 odd 2 2368.2.a.f.1.1 1
8.3 odd 2 592.2.a.d.1.1 1
8.5 even 2 296.2.a.b.1.1 1
24.5 odd 2 2664.2.a.c.1.1 1
24.11 even 2 5328.2.a.m.1.1 1
40.29 even 2 7400.2.a.g.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
296.2.a.b.1.1 1 8.5 even 2
592.2.a.d.1.1 1 8.3 odd 2
2368.2.a.f.1.1 1 4.3 odd 2
2368.2.a.k.1.1 1 1.1 even 1 trivial
2664.2.a.c.1.1 1 24.5 odd 2
5328.2.a.m.1.1 1 24.11 even 2
7400.2.a.g.1.1 1 40.29 even 2