# Properties

 Label 1638.2.a.j.1.1 Level $1638$ Weight $2$ Character 1638.1 Self dual yes Analytic conductor $13.079$ 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] = [1638,2,Mod(1,1638)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1638, 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("1638.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1638 = 2 \cdot 3^{2} \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1638.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: $$13.0794958511$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 182) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 1638.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} +1.00000 q^{4} +4.00000 q^{5} -1.00000 q^{7} -1.00000 q^{8} +O(q^{10})$$ $$q-1.00000 q^{2} +1.00000 q^{4} +4.00000 q^{5} -1.00000 q^{7} -1.00000 q^{8} -4.00000 q^{10} -1.00000 q^{11} -1.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} -6.00000 q^{19} +4.00000 q^{20} +1.00000 q^{22} +7.00000 q^{23} +11.0000 q^{25} +1.00000 q^{26} -1.00000 q^{28} +4.00000 q^{29} +7.00000 q^{31} -1.00000 q^{32} -4.00000 q^{35} +9.00000 q^{37} +6.00000 q^{38} -4.00000 q^{40} +3.00000 q^{41} +4.00000 q^{43} -1.00000 q^{44} -7.00000 q^{46} -7.00000 q^{47} +1.00000 q^{49} -11.0000 q^{50} -1.00000 q^{52} -4.00000 q^{55} +1.00000 q^{56} -4.00000 q^{58} +10.0000 q^{59} +1.00000 q^{61} -7.00000 q^{62} +1.00000 q^{64} -4.00000 q^{65} +1.00000 q^{67} +4.00000 q^{70} -16.0000 q^{71} +5.00000 q^{73} -9.00000 q^{74} -6.00000 q^{76} +1.00000 q^{77} +11.0000 q^{79} +4.00000 q^{80} -3.00000 q^{82} -4.00000 q^{86} +1.00000 q^{88} +6.00000 q^{89} +1.00000 q^{91} +7.00000 q^{92} +7.00000 q^{94} -24.0000 q^{95} -1.00000 q^{97} -1.00000 q^{98} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.00000 −0.707107
$$3$$ 0 0
$$4$$ 1.00000 0.500000
$$5$$ 4.00000 1.78885 0.894427 0.447214i $$-0.147584\pi$$
0.894427 + 0.447214i $$0.147584\pi$$
$$6$$ 0 0
$$7$$ −1.00000 −0.377964
$$8$$ −1.00000 −0.353553
$$9$$ 0 0
$$10$$ −4.00000 −1.26491
$$11$$ −1.00000 −0.301511 −0.150756 0.988571i $$-0.548171\pi$$
−0.150756 + 0.988571i $$0.548171\pi$$
$$12$$ 0 0
$$13$$ −1.00000 −0.277350
$$14$$ 1.00000 0.267261
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$18$$ 0 0
$$19$$ −6.00000 −1.37649 −0.688247 0.725476i $$-0.741620\pi$$
−0.688247 + 0.725476i $$0.741620\pi$$
$$20$$ 4.00000 0.894427
$$21$$ 0 0
$$22$$ 1.00000 0.213201
$$23$$ 7.00000 1.45960 0.729800 0.683660i $$-0.239613\pi$$
0.729800 + 0.683660i $$0.239613\pi$$
$$24$$ 0 0
$$25$$ 11.0000 2.20000
$$26$$ 1.00000 0.196116
$$27$$ 0 0
$$28$$ −1.00000 −0.188982
$$29$$ 4.00000 0.742781 0.371391 0.928477i $$-0.378881\pi$$
0.371391 + 0.928477i $$0.378881\pi$$
$$30$$ 0 0
$$31$$ 7.00000 1.25724 0.628619 0.777714i $$-0.283621\pi$$
0.628619 + 0.777714i $$0.283621\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ 0 0
$$34$$ 0 0
$$35$$ −4.00000 −0.676123
$$36$$ 0 0
$$37$$ 9.00000 1.47959 0.739795 0.672832i $$-0.234922\pi$$
0.739795 + 0.672832i $$0.234922\pi$$
$$38$$ 6.00000 0.973329
$$39$$ 0 0
$$40$$ −4.00000 −0.632456
$$41$$ 3.00000 0.468521 0.234261 0.972174i $$-0.424733\pi$$
0.234261 + 0.972174i $$0.424733\pi$$
$$42$$ 0 0
$$43$$ 4.00000 0.609994 0.304997 0.952353i $$-0.401344\pi$$
0.304997 + 0.952353i $$0.401344\pi$$
$$44$$ −1.00000 −0.150756
$$45$$ 0 0
$$46$$ −7.00000 −1.03209
$$47$$ −7.00000 −1.02105 −0.510527 0.859861i $$-0.670550\pi$$
−0.510527 + 0.859861i $$0.670550\pi$$
$$48$$ 0 0
$$49$$ 1.00000 0.142857
$$50$$ −11.0000 −1.55563
$$51$$ 0 0
$$52$$ −1.00000 −0.138675
$$53$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$54$$ 0 0
$$55$$ −4.00000 −0.539360
$$56$$ 1.00000 0.133631
$$57$$ 0 0
$$58$$ −4.00000 −0.525226
$$59$$ 10.0000 1.30189 0.650945 0.759125i $$-0.274373\pi$$
0.650945 + 0.759125i $$0.274373\pi$$
$$60$$ 0 0
$$61$$ 1.00000 0.128037 0.0640184 0.997949i $$-0.479608\pi$$
0.0640184 + 0.997949i $$0.479608\pi$$
$$62$$ −7.00000 −0.889001
$$63$$ 0 0
$$64$$ 1.00000 0.125000
$$65$$ −4.00000 −0.496139
$$66$$ 0 0
$$67$$ 1.00000 0.122169 0.0610847 0.998133i $$-0.480544\pi$$
0.0610847 + 0.998133i $$0.480544\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 4.00000 0.478091
$$71$$ −16.0000 −1.89885 −0.949425 0.313993i $$-0.898333\pi$$
−0.949425 + 0.313993i $$0.898333\pi$$
$$72$$ 0 0
$$73$$ 5.00000 0.585206 0.292603 0.956234i $$-0.405479\pi$$
0.292603 + 0.956234i $$0.405479\pi$$
$$74$$ −9.00000 −1.04623
$$75$$ 0 0
$$76$$ −6.00000 −0.688247
$$77$$ 1.00000 0.113961
$$78$$ 0 0
$$79$$ 11.0000 1.23760 0.618798 0.785550i $$-0.287620\pi$$
0.618798 + 0.785550i $$0.287620\pi$$
$$80$$ 4.00000 0.447214
$$81$$ 0 0
$$82$$ −3.00000 −0.331295
$$83$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ −4.00000 −0.431331
$$87$$ 0 0
$$88$$ 1.00000 0.106600
$$89$$ 6.00000 0.635999 0.317999 0.948091i $$-0.396989\pi$$
0.317999 + 0.948091i $$0.396989\pi$$
$$90$$ 0 0
$$91$$ 1.00000 0.104828
$$92$$ 7.00000 0.729800
$$93$$ 0 0
$$94$$ 7.00000 0.721995
$$95$$ −24.0000 −2.46235
$$96$$ 0 0
$$97$$ −1.00000 −0.101535 −0.0507673 0.998711i $$-0.516167\pi$$
−0.0507673 + 0.998711i $$0.516167\pi$$
$$98$$ −1.00000 −0.101015
$$99$$ 0 0
$$100$$ 11.0000 1.10000
$$101$$ 5.00000 0.497519 0.248759 0.968565i $$-0.419977\pi$$
0.248759 + 0.968565i $$0.419977\pi$$
$$102$$ 0 0
$$103$$ −14.0000 −1.37946 −0.689730 0.724066i $$-0.742271\pi$$
−0.689730 + 0.724066i $$0.742271\pi$$
$$104$$ 1.00000 0.0980581
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 4.00000 0.386695 0.193347 0.981130i $$-0.438066\pi$$
0.193347 + 0.981130i $$0.438066\pi$$
$$108$$ 0 0
$$109$$ −14.0000 −1.34096 −0.670478 0.741929i $$-0.733911\pi$$
−0.670478 + 0.741929i $$0.733911\pi$$
$$110$$ 4.00000 0.381385
$$111$$ 0 0
$$112$$ −1.00000 −0.0944911
$$113$$ 7.00000 0.658505 0.329252 0.944242i $$-0.393203\pi$$
0.329252 + 0.944242i $$0.393203\pi$$
$$114$$ 0 0
$$115$$ 28.0000 2.61101
$$116$$ 4.00000 0.371391
$$117$$ 0 0
$$118$$ −10.0000 −0.920575
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −10.0000 −0.909091
$$122$$ −1.00000 −0.0905357
$$123$$ 0 0
$$124$$ 7.00000 0.628619
$$125$$ 24.0000 2.14663
$$126$$ 0 0
$$127$$ −11.0000 −0.976092 −0.488046 0.872818i $$-0.662290\pi$$
−0.488046 + 0.872818i $$0.662290\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ 0 0
$$130$$ 4.00000 0.350823
$$131$$ −8.00000 −0.698963 −0.349482 0.936943i $$-0.613642\pi$$
−0.349482 + 0.936943i $$0.613642\pi$$
$$132$$ 0 0
$$133$$ 6.00000 0.520266
$$134$$ −1.00000 −0.0863868
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 6.00000 0.512615 0.256307 0.966595i $$-0.417494\pi$$
0.256307 + 0.966595i $$0.417494\pi$$
$$138$$ 0 0
$$139$$ 4.00000 0.339276 0.169638 0.985506i $$-0.445740\pi$$
0.169638 + 0.985506i $$0.445740\pi$$
$$140$$ −4.00000 −0.338062
$$141$$ 0 0
$$142$$ 16.0000 1.34269
$$143$$ 1.00000 0.0836242
$$144$$ 0 0
$$145$$ 16.0000 1.32873
$$146$$ −5.00000 −0.413803
$$147$$ 0 0
$$148$$ 9.00000 0.739795
$$149$$ −9.00000 −0.737309 −0.368654 0.929567i $$-0.620181\pi$$
−0.368654 + 0.929567i $$0.620181\pi$$
$$150$$ 0 0
$$151$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$152$$ 6.00000 0.486664
$$153$$ 0 0
$$154$$ −1.00000 −0.0805823
$$155$$ 28.0000 2.24901
$$156$$ 0 0
$$157$$ 5.00000 0.399043 0.199522 0.979893i $$-0.436061\pi$$
0.199522 + 0.979893i $$0.436061\pi$$
$$158$$ −11.0000 −0.875113
$$159$$ 0 0
$$160$$ −4.00000 −0.316228
$$161$$ −7.00000 −0.551677
$$162$$ 0 0
$$163$$ −4.00000 −0.313304 −0.156652 0.987654i $$-0.550070\pi$$
−0.156652 + 0.987654i $$0.550070\pi$$
$$164$$ 3.00000 0.234261
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$168$$ 0 0
$$169$$ 1.00000 0.0769231
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 4.00000 0.304997
$$173$$ −2.00000 −0.152057 −0.0760286 0.997106i $$-0.524224\pi$$
−0.0760286 + 0.997106i $$0.524224\pi$$
$$174$$ 0 0
$$175$$ −11.0000 −0.831522
$$176$$ −1.00000 −0.0753778
$$177$$ 0 0
$$178$$ −6.00000 −0.449719
$$179$$ 6.00000 0.448461 0.224231 0.974536i $$-0.428013\pi$$
0.224231 + 0.974536i $$0.428013\pi$$
$$180$$ 0 0
$$181$$ 15.0000 1.11494 0.557471 0.830197i $$-0.311772\pi$$
0.557471 + 0.830197i $$0.311772\pi$$
$$182$$ −1.00000 −0.0741249
$$183$$ 0 0
$$184$$ −7.00000 −0.516047
$$185$$ 36.0000 2.64677
$$186$$ 0 0
$$187$$ 0 0
$$188$$ −7.00000 −0.510527
$$189$$ 0 0
$$190$$ 24.0000 1.74114
$$191$$ 8.00000 0.578860 0.289430 0.957199i $$-0.406534\pi$$
0.289430 + 0.957199i $$0.406534\pi$$
$$192$$ 0 0
$$193$$ 20.0000 1.43963 0.719816 0.694165i $$-0.244226\pi$$
0.719816 + 0.694165i $$0.244226\pi$$
$$194$$ 1.00000 0.0717958
$$195$$ 0 0
$$196$$ 1.00000 0.0714286
$$197$$ 27.0000 1.92367 0.961835 0.273629i $$-0.0882242\pi$$
0.961835 + 0.273629i $$0.0882242\pi$$
$$198$$ 0 0
$$199$$ −4.00000 −0.283552 −0.141776 0.989899i $$-0.545281\pi$$
−0.141776 + 0.989899i $$0.545281\pi$$
$$200$$ −11.0000 −0.777817
$$201$$ 0 0
$$202$$ −5.00000 −0.351799
$$203$$ −4.00000 −0.280745
$$204$$ 0 0
$$205$$ 12.0000 0.838116
$$206$$ 14.0000 0.975426
$$207$$ 0 0
$$208$$ −1.00000 −0.0693375
$$209$$ 6.00000 0.415029
$$210$$ 0 0
$$211$$ −10.0000 −0.688428 −0.344214 0.938891i $$-0.611855\pi$$
−0.344214 + 0.938891i $$0.611855\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ −4.00000 −0.273434
$$215$$ 16.0000 1.09119
$$216$$ 0 0
$$217$$ −7.00000 −0.475191
$$218$$ 14.0000 0.948200
$$219$$ 0 0
$$220$$ −4.00000 −0.269680
$$221$$ 0 0
$$222$$ 0 0
$$223$$ 21.0000 1.40626 0.703132 0.711059i $$-0.251784\pi$$
0.703132 + 0.711059i $$0.251784\pi$$
$$224$$ 1.00000 0.0668153
$$225$$ 0 0
$$226$$ −7.00000 −0.465633
$$227$$ −24.0000 −1.59294 −0.796468 0.604681i $$-0.793301\pi$$
−0.796468 + 0.604681i $$0.793301\pi$$
$$228$$ 0 0
$$229$$ −24.0000 −1.58596 −0.792982 0.609245i $$-0.791473\pi$$
−0.792982 + 0.609245i $$0.791473\pi$$
$$230$$ −28.0000 −1.84627
$$231$$ 0 0
$$232$$ −4.00000 −0.262613
$$233$$ 5.00000 0.327561 0.163780 0.986497i $$-0.447631\pi$$
0.163780 + 0.986497i $$0.447631\pi$$
$$234$$ 0 0
$$235$$ −28.0000 −1.82652
$$236$$ 10.0000 0.650945
$$237$$ 0 0
$$238$$ 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 $$-0.696846\pi$$
−0.579741 + 0.814801i $$0.696846\pi$$
$$242$$ 10.0000 0.642824
$$243$$ 0 0
$$244$$ 1.00000 0.0640184
$$245$$ 4.00000 0.255551
$$246$$ 0 0
$$247$$ 6.00000 0.381771
$$248$$ −7.00000 −0.444500
$$249$$ 0 0
$$250$$ −24.0000 −1.51789
$$251$$ −3.00000 −0.189358 −0.0946792 0.995508i $$-0.530183\pi$$
−0.0946792 + 0.995508i $$0.530183\pi$$
$$252$$ 0 0
$$253$$ −7.00000 −0.440086
$$254$$ 11.0000 0.690201
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ −24.0000 −1.49708 −0.748539 0.663090i $$-0.769245\pi$$
−0.748539 + 0.663090i $$0.769245\pi$$
$$258$$ 0 0
$$259$$ −9.00000 −0.559233
$$260$$ −4.00000 −0.248069
$$261$$ 0 0
$$262$$ 8.00000 0.494242
$$263$$ −24.0000 −1.47990 −0.739952 0.672660i $$-0.765152\pi$$
−0.739952 + 0.672660i $$0.765152\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ −6.00000 −0.367884
$$267$$ 0 0
$$268$$ 1.00000 0.0610847
$$269$$ −9.00000 −0.548740 −0.274370 0.961624i $$-0.588469\pi$$
−0.274370 + 0.961624i $$0.588469\pi$$
$$270$$ 0 0
$$271$$ −23.0000 −1.39715 −0.698575 0.715537i $$-0.746182\pi$$
−0.698575 + 0.715537i $$0.746182\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ −6.00000 −0.362473
$$275$$ −11.0000 −0.663325
$$276$$ 0 0
$$277$$ −18.0000 −1.08152 −0.540758 0.841178i $$-0.681862\pi$$
−0.540758 + 0.841178i $$0.681862\pi$$
$$278$$ −4.00000 −0.239904
$$279$$ 0 0
$$280$$ 4.00000 0.239046
$$281$$ 8.00000 0.477240 0.238620 0.971113i $$-0.423305\pi$$
0.238620 + 0.971113i $$0.423305\pi$$
$$282$$ 0 0
$$283$$ 19.0000 1.12943 0.564716 0.825285i $$-0.308986\pi$$
0.564716 + 0.825285i $$0.308986\pi$$
$$284$$ −16.0000 −0.949425
$$285$$ 0 0
$$286$$ −1.00000 −0.0591312
$$287$$ −3.00000 −0.177084
$$288$$ 0 0
$$289$$ −17.0000 −1.00000
$$290$$ −16.0000 −0.939552
$$291$$ 0 0
$$292$$ 5.00000 0.292603
$$293$$ 26.0000 1.51894 0.759468 0.650545i $$-0.225459\pi$$
0.759468 + 0.650545i $$0.225459\pi$$
$$294$$ 0 0
$$295$$ 40.0000 2.32889
$$296$$ −9.00000 −0.523114
$$297$$ 0 0
$$298$$ 9.00000 0.521356
$$299$$ −7.00000 −0.404820
$$300$$ 0 0
$$301$$ −4.00000 −0.230556
$$302$$ 0 0
$$303$$ 0 0
$$304$$ −6.00000 −0.344124
$$305$$ 4.00000 0.229039
$$306$$ 0 0
$$307$$ −4.00000 −0.228292 −0.114146 0.993464i $$-0.536413\pi$$
−0.114146 + 0.993464i $$0.536413\pi$$
$$308$$ 1.00000 0.0569803
$$309$$ 0 0
$$310$$ −28.0000 −1.59029
$$311$$ −30.0000 −1.70114 −0.850572 0.525859i $$-0.823744\pi$$
−0.850572 + 0.525859i $$0.823744\pi$$
$$312$$ 0 0
$$313$$ 14.0000 0.791327 0.395663 0.918396i $$-0.370515\pi$$
0.395663 + 0.918396i $$0.370515\pi$$
$$314$$ −5.00000 −0.282166
$$315$$ 0 0
$$316$$ 11.0000 0.618798
$$317$$ −21.0000 −1.17948 −0.589739 0.807594i $$-0.700769\pi$$
−0.589739 + 0.807594i $$0.700769\pi$$
$$318$$ 0 0
$$319$$ −4.00000 −0.223957
$$320$$ 4.00000 0.223607
$$321$$ 0 0
$$322$$ 7.00000 0.390095
$$323$$ 0 0
$$324$$ 0 0
$$325$$ −11.0000 −0.610170
$$326$$ 4.00000 0.221540
$$327$$ 0 0
$$328$$ −3.00000 −0.165647
$$329$$ 7.00000 0.385922
$$330$$ 0 0
$$331$$ −7.00000 −0.384755 −0.192377 0.981321i $$-0.561620\pi$$
−0.192377 + 0.981321i $$0.561620\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ 4.00000 0.218543
$$336$$ 0 0
$$337$$ 17.0000 0.926049 0.463025 0.886345i $$-0.346764\pi$$
0.463025 + 0.886345i $$0.346764\pi$$
$$338$$ −1.00000 −0.0543928
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −7.00000 −0.379071
$$342$$ 0 0
$$343$$ −1.00000 −0.0539949
$$344$$ −4.00000 −0.215666
$$345$$ 0 0
$$346$$ 2.00000 0.107521
$$347$$ 24.0000 1.28839 0.644194 0.764862i $$-0.277193\pi$$
0.644194 + 0.764862i $$0.277193\pi$$
$$348$$ 0 0
$$349$$ −26.0000 −1.39175 −0.695874 0.718164i $$-0.744983\pi$$
−0.695874 + 0.718164i $$0.744983\pi$$
$$350$$ 11.0000 0.587975
$$351$$ 0 0
$$352$$ 1.00000 0.0533002
$$353$$ −25.0000 −1.33062 −0.665308 0.746569i $$-0.731700\pi$$
−0.665308 + 0.746569i $$0.731700\pi$$
$$354$$ 0 0
$$355$$ −64.0000 −3.39677
$$356$$ 6.00000 0.317999
$$357$$ 0 0
$$358$$ −6.00000 −0.317110
$$359$$ −12.0000 −0.633336 −0.316668 0.948536i $$-0.602564\pi$$
−0.316668 + 0.948536i $$0.602564\pi$$
$$360$$ 0 0
$$361$$ 17.0000 0.894737
$$362$$ −15.0000 −0.788382
$$363$$ 0 0
$$364$$ 1.00000 0.0524142
$$365$$ 20.0000 1.04685
$$366$$ 0 0
$$367$$ 32.0000 1.67039 0.835193 0.549957i $$-0.185356\pi$$
0.835193 + 0.549957i $$0.185356\pi$$
$$368$$ 7.00000 0.364900
$$369$$ 0 0
$$370$$ −36.0000 −1.87155
$$371$$ 0 0
$$372$$ 0 0
$$373$$ −16.0000 −0.828449 −0.414224 0.910175i $$-0.635947\pi$$
−0.414224 + 0.910175i $$0.635947\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 7.00000 0.360997
$$377$$ −4.00000 −0.206010
$$378$$ 0 0
$$379$$ −8.00000 −0.410932 −0.205466 0.978664i $$-0.565871\pi$$
−0.205466 + 0.978664i $$0.565871\pi$$
$$380$$ −24.0000 −1.23117
$$381$$ 0 0
$$382$$ −8.00000 −0.409316
$$383$$ 15.0000 0.766464 0.383232 0.923652i $$-0.374811\pi$$
0.383232 + 0.923652i $$0.374811\pi$$
$$384$$ 0 0
$$385$$ 4.00000 0.203859
$$386$$ −20.0000 −1.01797
$$387$$ 0 0
$$388$$ −1.00000 −0.0507673
$$389$$ 36.0000 1.82527 0.912636 0.408773i $$-0.134043\pi$$
0.912636 + 0.408773i $$0.134043\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ −1.00000 −0.0505076
$$393$$ 0 0
$$394$$ −27.0000 −1.36024
$$395$$ 44.0000 2.21388
$$396$$ 0 0
$$397$$ −28.0000 −1.40528 −0.702640 0.711546i $$-0.747995\pi$$
−0.702640 + 0.711546i $$0.747995\pi$$
$$398$$ 4.00000 0.200502
$$399$$ 0 0
$$400$$ 11.0000 0.550000
$$401$$ −12.0000 −0.599251 −0.299626 0.954057i $$-0.596862\pi$$
−0.299626 + 0.954057i $$0.596862\pi$$
$$402$$ 0 0
$$403$$ −7.00000 −0.348695
$$404$$ 5.00000 0.248759
$$405$$ 0 0
$$406$$ 4.00000 0.198517
$$407$$ −9.00000 −0.446113
$$408$$ 0 0
$$409$$ −26.0000 −1.28562 −0.642809 0.766027i $$-0.722231\pi$$
−0.642809 + 0.766027i $$0.722231\pi$$
$$410$$ −12.0000 −0.592638
$$411$$ 0 0
$$412$$ −14.0000 −0.689730
$$413$$ −10.0000 −0.492068
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 1.00000 0.0490290
$$417$$ 0 0
$$418$$ −6.00000 −0.293470
$$419$$ −15.0000 −0.732798 −0.366399 0.930458i $$-0.619409\pi$$
−0.366399 + 0.930458i $$0.619409\pi$$
$$420$$ 0 0
$$421$$ 19.0000 0.926003 0.463002 0.886357i $$-0.346772\pi$$
0.463002 + 0.886357i $$0.346772\pi$$
$$422$$ 10.0000 0.486792
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ −1.00000 −0.0483934
$$428$$ 4.00000 0.193347
$$429$$ 0 0
$$430$$ −16.0000 −0.771589
$$431$$ −18.0000 −0.867029 −0.433515 0.901146i $$-0.642727\pi$$
−0.433515 + 0.901146i $$0.642727\pi$$
$$432$$ 0 0
$$433$$ 34.0000 1.63394 0.816968 0.576683i $$-0.195653\pi$$
0.816968 + 0.576683i $$0.195653\pi$$
$$434$$ 7.00000 0.336011
$$435$$ 0 0
$$436$$ −14.0000 −0.670478
$$437$$ −42.0000 −2.00913
$$438$$ 0 0
$$439$$ 2.00000 0.0954548 0.0477274 0.998860i $$-0.484802\pi$$
0.0477274 + 0.998860i $$0.484802\pi$$
$$440$$ 4.00000 0.190693
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −12.0000 −0.570137 −0.285069 0.958507i $$-0.592016\pi$$
−0.285069 + 0.958507i $$0.592016\pi$$
$$444$$ 0 0
$$445$$ 24.0000 1.13771
$$446$$ −21.0000 −0.994379
$$447$$ 0 0
$$448$$ −1.00000 −0.0472456
$$449$$ 2.00000 0.0943858 0.0471929 0.998886i $$-0.484972\pi$$
0.0471929 + 0.998886i $$0.484972\pi$$
$$450$$ 0 0
$$451$$ −3.00000 −0.141264
$$452$$ 7.00000 0.329252
$$453$$ 0 0
$$454$$ 24.0000 1.12638
$$455$$ 4.00000 0.187523
$$456$$ 0 0
$$457$$ −26.0000 −1.21623 −0.608114 0.793849i $$-0.708074\pi$$
−0.608114 + 0.793849i $$0.708074\pi$$
$$458$$ 24.0000 1.12145
$$459$$ 0 0
$$460$$ 28.0000 1.30551
$$461$$ −12.0000 −0.558896 −0.279448 0.960161i $$-0.590151\pi$$
−0.279448 + 0.960161i $$0.590151\pi$$
$$462$$ 0 0
$$463$$ 16.0000 0.743583 0.371792 0.928316i $$-0.378744\pi$$
0.371792 + 0.928316i $$0.378744\pi$$
$$464$$ 4.00000 0.185695
$$465$$ 0 0
$$466$$ −5.00000 −0.231621
$$467$$ 4.00000 0.185098 0.0925490 0.995708i $$-0.470499\pi$$
0.0925490 + 0.995708i $$0.470499\pi$$
$$468$$ 0 0
$$469$$ −1.00000 −0.0461757
$$470$$ 28.0000 1.29154
$$471$$ 0 0
$$472$$ −10.0000 −0.460287
$$473$$ −4.00000 −0.183920
$$474$$ 0 0
$$475$$ −66.0000 −3.02829
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 6.00000 0.274434
$$479$$ −24.0000 −1.09659 −0.548294 0.836286i $$-0.684723\pi$$
−0.548294 + 0.836286i $$0.684723\pi$$
$$480$$ 0 0
$$481$$ −9.00000 −0.410365
$$482$$ 18.0000 0.819878
$$483$$ 0 0
$$484$$ −10.0000 −0.454545
$$485$$ −4.00000 −0.181631
$$486$$ 0 0
$$487$$ 38.0000 1.72194 0.860972 0.508652i $$-0.169856\pi$$
0.860972 + 0.508652i $$0.169856\pi$$
$$488$$ −1.00000 −0.0452679
$$489$$ 0 0
$$490$$ −4.00000 −0.180702
$$491$$ −16.0000 −0.722070 −0.361035 0.932552i $$-0.617576\pi$$
−0.361035 + 0.932552i $$0.617576\pi$$
$$492$$ 0 0
$$493$$ 0 0
$$494$$ −6.00000 −0.269953
$$495$$ 0 0
$$496$$ 7.00000 0.314309
$$497$$ 16.0000 0.717698
$$498$$ 0 0
$$499$$ −5.00000 −0.223831 −0.111915 0.993718i $$-0.535699\pi$$
−0.111915 + 0.993718i $$0.535699\pi$$
$$500$$ 24.0000 1.07331
$$501$$ 0 0
$$502$$ 3.00000 0.133897
$$503$$ 20.0000 0.891756 0.445878 0.895094i $$-0.352892\pi$$
0.445878 + 0.895094i $$0.352892\pi$$
$$504$$ 0 0
$$505$$ 20.0000 0.889988
$$506$$ 7.00000 0.311188
$$507$$ 0 0
$$508$$ −11.0000 −0.488046
$$509$$ −10.0000 −0.443242 −0.221621 0.975133i $$-0.571135\pi$$
−0.221621 + 0.975133i $$0.571135\pi$$
$$510$$ 0 0
$$511$$ −5.00000 −0.221187
$$512$$ −1.00000 −0.0441942
$$513$$ 0 0
$$514$$ 24.0000 1.05859
$$515$$ −56.0000 −2.46765
$$516$$ 0 0
$$517$$ 7.00000 0.307860
$$518$$ 9.00000 0.395437
$$519$$ 0 0
$$520$$ 4.00000 0.175412
$$521$$ 18.0000 0.788594 0.394297 0.918983i $$-0.370988\pi$$
0.394297 + 0.918983i $$0.370988\pi$$
$$522$$ 0 0
$$523$$ 27.0000 1.18063 0.590314 0.807174i $$-0.299004\pi$$
0.590314 + 0.807174i $$0.299004\pi$$
$$524$$ −8.00000 −0.349482
$$525$$ 0 0
$$526$$ 24.0000 1.04645
$$527$$ 0 0
$$528$$ 0 0
$$529$$ 26.0000 1.13043
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 6.00000 0.260133
$$533$$ −3.00000 −0.129944
$$534$$ 0 0
$$535$$ 16.0000 0.691740
$$536$$ −1.00000 −0.0431934
$$537$$ 0 0
$$538$$ 9.00000 0.388018
$$539$$ −1.00000 −0.0430730
$$540$$ 0 0
$$541$$ −10.0000 −0.429934 −0.214967 0.976621i $$-0.568964\pi$$
−0.214967 + 0.976621i $$0.568964\pi$$
$$542$$ 23.0000 0.987935
$$543$$ 0 0
$$544$$ 0 0
$$545$$ −56.0000 −2.39878
$$546$$ 0 0
$$547$$ 28.0000 1.19719 0.598597 0.801050i $$-0.295725\pi$$
0.598597 + 0.801050i $$0.295725\pi$$
$$548$$ 6.00000 0.256307
$$549$$ 0 0
$$550$$ 11.0000 0.469042
$$551$$ −24.0000 −1.02243
$$552$$ 0 0
$$553$$ −11.0000 −0.467768
$$554$$ 18.0000 0.764747
$$555$$ 0 0
$$556$$ 4.00000 0.169638
$$557$$ −9.00000 −0.381342 −0.190671 0.981654i $$-0.561066\pi$$
−0.190671 + 0.981654i $$0.561066\pi$$
$$558$$ 0 0
$$559$$ −4.00000 −0.169182
$$560$$ −4.00000 −0.169031
$$561$$ 0 0
$$562$$ −8.00000 −0.337460
$$563$$ −11.0000 −0.463595 −0.231797 0.972764i $$-0.574461\pi$$
−0.231797 + 0.972764i $$0.574461\pi$$
$$564$$ 0 0
$$565$$ 28.0000 1.17797
$$566$$ −19.0000 −0.798630
$$567$$ 0 0
$$568$$ 16.0000 0.671345
$$569$$ 35.0000 1.46728 0.733638 0.679540i $$-0.237821\pi$$
0.733638 + 0.679540i $$0.237821\pi$$
$$570$$ 0 0
$$571$$ 6.00000 0.251092 0.125546 0.992088i $$-0.459932\pi$$
0.125546 + 0.992088i $$0.459932\pi$$
$$572$$ 1.00000 0.0418121
$$573$$ 0 0
$$574$$ 3.00000 0.125218
$$575$$ 77.0000 3.21112
$$576$$ 0 0
$$577$$ 30.0000 1.24892 0.624458 0.781058i $$-0.285320\pi$$
0.624458 + 0.781058i $$0.285320\pi$$
$$578$$ 17.0000 0.707107
$$579$$ 0 0
$$580$$ 16.0000 0.664364
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 0 0
$$584$$ −5.00000 −0.206901
$$585$$ 0 0
$$586$$ −26.0000 −1.07405
$$587$$ −14.0000 −0.577842 −0.288921 0.957353i $$-0.593296\pi$$
−0.288921 + 0.957353i $$0.593296\pi$$
$$588$$ 0 0
$$589$$ −42.0000 −1.73058
$$590$$ −40.0000 −1.64677
$$591$$ 0 0
$$592$$ 9.00000 0.369898
$$593$$ −14.0000 −0.574911 −0.287456 0.957794i $$-0.592809\pi$$
−0.287456 + 0.957794i $$0.592809\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −9.00000 −0.368654
$$597$$ 0 0
$$598$$ 7.00000 0.286251
$$599$$ 29.0000 1.18491 0.592454 0.805604i $$-0.298159\pi$$
0.592454 + 0.805604i $$0.298159\pi$$
$$600$$ 0 0
$$601$$ −26.0000 −1.06056 −0.530281 0.847822i $$-0.677914\pi$$
−0.530281 + 0.847822i $$0.677914\pi$$
$$602$$ 4.00000 0.163028
$$603$$ 0 0
$$604$$ 0 0
$$605$$ −40.0000 −1.62623
$$606$$ 0 0
$$607$$ 6.00000 0.243532 0.121766 0.992559i $$-0.461144\pi$$
0.121766 + 0.992559i $$0.461144\pi$$
$$608$$ 6.00000 0.243332
$$609$$ 0 0
$$610$$ −4.00000 −0.161955
$$611$$ 7.00000 0.283190
$$612$$ 0 0
$$613$$ −49.0000 −1.97909 −0.989546 0.144220i $$-0.953933\pi$$
−0.989546 + 0.144220i $$0.953933\pi$$
$$614$$ 4.00000 0.161427
$$615$$ 0 0
$$616$$ −1.00000 −0.0402911
$$617$$ −22.0000 −0.885687 −0.442843 0.896599i $$-0.646030\pi$$
−0.442843 + 0.896599i $$0.646030\pi$$
$$618$$ 0 0
$$619$$ 22.0000 0.884255 0.442127 0.896952i $$-0.354224\pi$$
0.442127 + 0.896952i $$0.354224\pi$$
$$620$$ 28.0000 1.12451
$$621$$ 0 0
$$622$$ 30.0000 1.20289
$$623$$ −6.00000 −0.240385
$$624$$ 0 0
$$625$$ 41.0000 1.64000
$$626$$ −14.0000 −0.559553
$$627$$ 0 0
$$628$$ 5.00000 0.199522
$$629$$ 0 0
$$630$$ 0 0
$$631$$ −2.00000 −0.0796187 −0.0398094 0.999207i $$-0.512675\pi$$
−0.0398094 + 0.999207i $$0.512675\pi$$
$$632$$ −11.0000 −0.437557
$$633$$ 0 0
$$634$$ 21.0000 0.834017
$$635$$ −44.0000 −1.74609
$$636$$ 0 0
$$637$$ −1.00000 −0.0396214
$$638$$ 4.00000 0.158362
$$639$$ 0 0
$$640$$ −4.00000 −0.158114
$$641$$ −7.00000 −0.276483 −0.138242 0.990399i $$-0.544145\pi$$
−0.138242 + 0.990399i $$0.544145\pi$$
$$642$$ 0 0
$$643$$ −16.0000 −0.630978 −0.315489 0.948929i $$-0.602169\pi$$
−0.315489 + 0.948929i $$0.602169\pi$$
$$644$$ −7.00000 −0.275839
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 8.00000 0.314512 0.157256 0.987558i $$-0.449735\pi$$
0.157256 + 0.987558i $$0.449735\pi$$
$$648$$ 0 0
$$649$$ −10.0000 −0.392534
$$650$$ 11.0000 0.431455
$$651$$ 0 0
$$652$$ −4.00000 −0.156652
$$653$$ 36.0000 1.40879 0.704394 0.709809i $$-0.251219\pi$$
0.704394 + 0.709809i $$0.251219\pi$$
$$654$$ 0 0
$$655$$ −32.0000 −1.25034
$$656$$ 3.00000 0.117130
$$657$$ 0 0
$$658$$ −7.00000 −0.272888
$$659$$ 18.0000 0.701180 0.350590 0.936529i $$-0.385981\pi$$
0.350590 + 0.936529i $$0.385981\pi$$
$$660$$ 0 0
$$661$$ −4.00000 −0.155582 −0.0777910 0.996970i $$-0.524787\pi$$
−0.0777910 + 0.996970i $$0.524787\pi$$
$$662$$ 7.00000 0.272063
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 24.0000 0.930680
$$666$$ 0 0
$$667$$ 28.0000 1.08416
$$668$$ 0 0
$$669$$ 0 0
$$670$$ −4.00000 −0.154533
$$671$$ −1.00000 −0.0386046
$$672$$ 0 0
$$673$$ 7.00000 0.269830 0.134915 0.990857i $$-0.456924\pi$$
0.134915 + 0.990857i $$0.456924\pi$$
$$674$$ −17.0000 −0.654816
$$675$$ 0 0
$$676$$ 1.00000 0.0384615
$$677$$ −39.0000 −1.49889 −0.749446 0.662066i $$-0.769680\pi$$
−0.749446 + 0.662066i $$0.769680\pi$$
$$678$$ 0 0
$$679$$ 1.00000 0.0383765
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 7.00000 0.268044
$$683$$ −1.00000 −0.0382639 −0.0191320 0.999817i $$-0.506090\pi$$
−0.0191320 + 0.999817i $$0.506090\pi$$
$$684$$ 0 0
$$685$$ 24.0000 0.916993
$$686$$ 1.00000 0.0381802
$$687$$ 0 0
$$688$$ 4.00000 0.152499
$$689$$ 0 0
$$690$$ 0 0
$$691$$ −32.0000 −1.21734 −0.608669 0.793424i $$-0.708296\pi$$
−0.608669 + 0.793424i $$0.708296\pi$$
$$692$$ −2.00000 −0.0760286
$$693$$ 0 0
$$694$$ −24.0000 −0.911028
$$695$$ 16.0000 0.606915
$$696$$ 0 0
$$697$$ 0 0
$$698$$ 26.0000 0.984115
$$699$$ 0 0
$$700$$ −11.0000 −0.415761
$$701$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$702$$ 0 0
$$703$$ −54.0000 −2.03665
$$704$$ −1.00000 −0.0376889
$$705$$ 0 0
$$706$$ 25.0000 0.940887
$$707$$ −5.00000 −0.188044
$$708$$ 0 0
$$709$$ −13.0000 −0.488225 −0.244113 0.969747i $$-0.578497\pi$$
−0.244113 + 0.969747i $$0.578497\pi$$
$$710$$ 64.0000 2.40188
$$711$$ 0 0
$$712$$ −6.00000 −0.224860
$$713$$ 49.0000 1.83506
$$714$$ 0 0
$$715$$ 4.00000 0.149592
$$716$$ 6.00000 0.224231
$$717$$ 0 0
$$718$$ 12.0000 0.447836
$$719$$ −2.00000 −0.0745874 −0.0372937 0.999304i $$-0.511874\pi$$
−0.0372937 + 0.999304i $$0.511874\pi$$
$$720$$ 0 0
$$721$$ 14.0000 0.521387
$$722$$ −17.0000 −0.632674
$$723$$ 0 0
$$724$$ 15.0000 0.557471
$$725$$ 44.0000 1.63412
$$726$$ 0 0
$$727$$ −26.0000 −0.964287 −0.482143 0.876092i $$-0.660142\pi$$
−0.482143 + 0.876092i $$0.660142\pi$$
$$728$$ −1.00000 −0.0370625
$$729$$ 0 0
$$730$$ −20.0000 −0.740233
$$731$$ 0 0
$$732$$ 0 0
$$733$$ −4.00000 −0.147743 −0.0738717 0.997268i $$-0.523536\pi$$
−0.0738717 + 0.997268i $$0.523536\pi$$
$$734$$ −32.0000 −1.18114
$$735$$ 0 0
$$736$$ −7.00000 −0.258023
$$737$$ −1.00000 −0.0368355
$$738$$ 0 0
$$739$$ 24.0000 0.882854 0.441427 0.897297i $$-0.354472\pi$$
0.441427 + 0.897297i $$0.354472\pi$$
$$740$$ 36.0000 1.32339
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −16.0000 −0.586983 −0.293492 0.955962i $$-0.594817\pi$$
−0.293492 + 0.955962i $$0.594817\pi$$
$$744$$ 0 0
$$745$$ −36.0000 −1.31894
$$746$$ 16.0000 0.585802
$$747$$ 0 0
$$748$$ 0 0
$$749$$ −4.00000 −0.146157
$$750$$ 0 0
$$751$$ −27.0000 −0.985244 −0.492622 0.870243i $$-0.663961\pi$$
−0.492622 + 0.870243i $$0.663961\pi$$
$$752$$ −7.00000 −0.255264
$$753$$ 0 0
$$754$$ 4.00000 0.145671
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 54.0000 1.96266 0.981332 0.192323i $$-0.0616021\pi$$
0.981332 + 0.192323i $$0.0616021\pi$$
$$758$$ 8.00000 0.290573
$$759$$ 0 0
$$760$$ 24.0000 0.870572
$$761$$ −45.0000 −1.63125 −0.815624 0.578582i $$-0.803606\pi$$
−0.815624 + 0.578582i $$0.803606\pi$$
$$762$$ 0 0
$$763$$ 14.0000 0.506834
$$764$$ 8.00000 0.289430
$$765$$ 0 0
$$766$$ −15.0000 −0.541972
$$767$$ −10.0000 −0.361079
$$768$$ 0 0
$$769$$ 21.0000 0.757279 0.378640 0.925544i $$-0.376392\pi$$
0.378640 + 0.925544i $$0.376392\pi$$
$$770$$ −4.00000 −0.144150
$$771$$ 0 0
$$772$$ 20.0000 0.719816
$$773$$ −14.0000 −0.503545 −0.251773 0.967786i $$-0.581013\pi$$
−0.251773 + 0.967786i $$0.581013\pi$$
$$774$$ 0 0
$$775$$ 77.0000 2.76592
$$776$$ 1.00000 0.0358979
$$777$$ 0 0
$$778$$ −36.0000 −1.29066
$$779$$ −18.0000 −0.644917
$$780$$ 0 0
$$781$$ 16.0000 0.572525
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 1.00000 0.0357143
$$785$$ 20.0000 0.713831
$$786$$ 0 0
$$787$$ 32.0000 1.14068 0.570338 0.821410i $$-0.306812\pi$$
0.570338 + 0.821410i $$0.306812\pi$$
$$788$$ 27.0000 0.961835
$$789$$ 0 0
$$790$$ −44.0000 −1.56545
$$791$$ −7.00000 −0.248891
$$792$$ 0 0
$$793$$ −1.00000 −0.0355110
$$794$$ 28.0000 0.993683
$$795$$ 0 0
$$796$$ −4.00000 −0.141776
$$797$$ −15.0000 −0.531327 −0.265664 0.964066i $$-0.585591\pi$$
−0.265664 + 0.964066i $$0.585591\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ −11.0000 −0.388909
$$801$$ 0 0
$$802$$ 12.0000 0.423735
$$803$$ −5.00000 −0.176446
$$804$$ 0 0
$$805$$ −28.0000 −0.986870
$$806$$ 7.00000 0.246564
$$807$$ 0 0
$$808$$ −5.00000 −0.175899
$$809$$ −6.00000 −0.210949 −0.105474 0.994422i $$-0.533636\pi$$
−0.105474 + 0.994422i $$0.533636\pi$$
$$810$$ 0 0
$$811$$ −28.0000 −0.983213 −0.491606 0.870817i $$-0.663590\pi$$
−0.491606 + 0.870817i $$0.663590\pi$$
$$812$$ −4.00000 −0.140372
$$813$$ 0 0
$$814$$ 9.00000 0.315450
$$815$$ −16.0000 −0.560456
$$816$$ 0 0
$$817$$ −24.0000 −0.839654
$$818$$ 26.0000 0.909069
$$819$$ 0 0
$$820$$ 12.0000 0.419058
$$821$$ 46.0000 1.60541 0.802706 0.596376i $$-0.203393\pi$$
0.802706 + 0.596376i $$0.203393\pi$$
$$822$$ 0 0
$$823$$ −3.00000 −0.104573 −0.0522867 0.998632i $$-0.516651\pi$$
−0.0522867 + 0.998632i $$0.516651\pi$$
$$824$$ 14.0000 0.487713
$$825$$ 0 0
$$826$$ 10.0000 0.347945
$$827$$ 12.0000 0.417281 0.208640 0.977992i $$-0.433096\pi$$
0.208640 + 0.977992i $$0.433096\pi$$
$$828$$ 0 0
$$829$$ −14.0000 −0.486240 −0.243120 0.969996i $$-0.578171\pi$$
−0.243120 + 0.969996i $$0.578171\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ −1.00000 −0.0346688
$$833$$ 0 0
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 6.00000 0.207514
$$837$$ 0 0
$$838$$ 15.0000 0.518166
$$839$$ −17.0000 −0.586905 −0.293453 0.955974i $$-0.594804\pi$$
−0.293453 + 0.955974i $$0.594804\pi$$
$$840$$ 0 0
$$841$$ −13.0000 −0.448276
$$842$$ −19.0000 −0.654783
$$843$$ 0 0
$$844$$ −10.0000 −0.344214
$$845$$ 4.00000 0.137604
$$846$$ 0 0
$$847$$ 10.0000 0.343604
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 63.0000 2.15961
$$852$$ 0 0
$$853$$ 16.0000 0.547830 0.273915 0.961754i $$-0.411681\pi$$
0.273915 + 0.961754i $$0.411681\pi$$
$$854$$ 1.00000 0.0342193
$$855$$ 0 0
$$856$$ −4.00000 −0.136717
$$857$$ −22.0000 −0.751506 −0.375753 0.926720i $$-0.622616\pi$$
−0.375753 + 0.926720i $$0.622616\pi$$
$$858$$ 0 0
$$859$$ 5.00000 0.170598 0.0852989 0.996355i $$-0.472815\pi$$
0.0852989 + 0.996355i $$0.472815\pi$$
$$860$$ 16.0000 0.545595
$$861$$ 0 0
$$862$$ 18.0000 0.613082
$$863$$ −28.0000 −0.953131 −0.476566 0.879139i $$-0.658119\pi$$
−0.476566 + 0.879139i $$0.658119\pi$$
$$864$$ 0 0
$$865$$ −8.00000 −0.272008
$$866$$ −34.0000 −1.15537
$$867$$ 0 0
$$868$$ −7.00000 −0.237595
$$869$$ −11.0000 −0.373149
$$870$$ 0 0
$$871$$ −1.00000 −0.0338837
$$872$$ 14.0000 0.474100
$$873$$ 0 0
$$874$$ 42.0000 1.42067
$$875$$ −24.0000 −0.811348
$$876$$ 0 0
$$877$$ −45.0000 −1.51954 −0.759771 0.650191i $$-0.774689\pi$$
−0.759771 + 0.650191i $$0.774689\pi$$
$$878$$ −2.00000 −0.0674967
$$879$$ 0 0
$$880$$ −4.00000 −0.134840
$$881$$ −18.0000 −0.606435 −0.303218 0.952921i $$-0.598061\pi$$
−0.303218 + 0.952921i $$0.598061\pi$$
$$882$$ 0 0
$$883$$ −14.0000 −0.471138 −0.235569 0.971858i $$-0.575695\pi$$
−0.235569 + 0.971858i $$0.575695\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 12.0000 0.403148
$$887$$ 38.0000 1.27592 0.637958 0.770072i $$-0.279780\pi$$
0.637958 + 0.770072i $$0.279780\pi$$
$$888$$ 0 0
$$889$$ 11.0000 0.368928
$$890$$ −24.0000 −0.804482
$$891$$ 0 0
$$892$$ 21.0000 0.703132
$$893$$ 42.0000 1.40548
$$894$$ 0 0
$$895$$ 24.0000 0.802232
$$896$$ 1.00000 0.0334077
$$897$$ 0 0
$$898$$ −2.00000 −0.0667409
$$899$$ 28.0000 0.933852
$$900$$ 0 0
$$901$$ 0 0
$$902$$ 3.00000 0.0998891
$$903$$ 0 0
$$904$$ −7.00000 −0.232817
$$905$$ 60.0000 1.99447
$$906$$ 0 0
$$907$$ −30.0000 −0.996134 −0.498067 0.867139i $$-0.665957\pi$$
−0.498067 + 0.867139i $$0.665957\pi$$
$$908$$ −24.0000 −0.796468
$$909$$ 0 0
$$910$$ −4.00000 −0.132599
$$911$$ −36.0000 −1.19273 −0.596367 0.802712i $$-0.703390\pi$$
−0.596367 + 0.802712i $$0.703390\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ 26.0000 0.860004
$$915$$ 0 0
$$916$$ −24.0000 −0.792982
$$917$$ 8.00000 0.264183
$$918$$ 0 0
$$919$$ 27.0000 0.890648 0.445324 0.895370i $$-0.353089\pi$$
0.445324 + 0.895370i $$0.353089\pi$$
$$920$$ −28.0000 −0.923133
$$921$$ 0 0
$$922$$ 12.0000 0.395199
$$923$$ 16.0000 0.526646
$$924$$ 0 0
$$925$$ 99.0000 3.25510
$$926$$ −16.0000 −0.525793
$$927$$ 0 0
$$928$$ −4.00000 −0.131306
$$929$$ 15.0000 0.492134 0.246067 0.969253i $$-0.420862\pi$$
0.246067 + 0.969253i $$0.420862\pi$$
$$930$$ 0 0
$$931$$ −6.00000 −0.196642
$$932$$ 5.00000 0.163780
$$933$$ 0 0
$$934$$ −4.00000 −0.130884
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −36.0000 −1.17607 −0.588034 0.808836i $$-0.700098\pi$$
−0.588034 + 0.808836i $$0.700098\pi$$
$$938$$ 1.00000 0.0326512
$$939$$ 0 0
$$940$$ −28.0000 −0.913259
$$941$$ 50.0000 1.62995 0.814977 0.579494i $$-0.196750\pi$$
0.814977 + 0.579494i $$0.196750\pi$$
$$942$$ 0 0
$$943$$ 21.0000 0.683854
$$944$$ 10.0000 0.325472
$$945$$ 0 0
$$946$$ 4.00000 0.130051
$$947$$ −24.0000 −0.779895 −0.389948 0.920837i $$-0.627507\pi$$
−0.389948 + 0.920837i $$0.627507\pi$$
$$948$$ 0 0
$$949$$ −5.00000 −0.162307
$$950$$ 66.0000 2.14132
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 34.0000 1.10137 0.550684 0.834714i $$-0.314367\pi$$
0.550684 + 0.834714i $$0.314367\pi$$
$$954$$ 0 0
$$955$$ 32.0000 1.03550
$$956$$ −6.00000 −0.194054
$$957$$ 0 0
$$958$$ 24.0000 0.775405
$$959$$ −6.00000 −0.193750
$$960$$ 0 0
$$961$$ 18.0000 0.580645
$$962$$ 9.00000 0.290172
$$963$$ 0 0
$$964$$ −18.0000 −0.579741
$$965$$ 80.0000 2.57529
$$966$$ 0 0
$$967$$ −22.0000 −0.707472 −0.353736 0.935345i $$-0.615089\pi$$
−0.353736 + 0.935345i $$0.615089\pi$$
$$968$$ 10.0000 0.321412
$$969$$ 0 0
$$970$$ 4.00000 0.128432
$$971$$ 15.0000 0.481373 0.240686 0.970603i $$-0.422627\pi$$
0.240686 + 0.970603i $$0.422627\pi$$
$$972$$ 0 0
$$973$$ −4.00000 −0.128234
$$974$$ −38.0000 −1.21760
$$975$$ 0 0
$$976$$ 1.00000 0.0320092
$$977$$ −42.0000 −1.34370 −0.671850 0.740688i $$-0.734500\pi$$
−0.671850 + 0.740688i $$0.734500\pi$$
$$978$$ 0 0
$$979$$ −6.00000 −0.191761
$$980$$ 4.00000 0.127775
$$981$$ 0 0
$$982$$ 16.0000 0.510581
$$983$$ 40.0000 1.27580 0.637901 0.770118i $$-0.279803\pi$$
0.637901 + 0.770118i $$0.279803\pi$$
$$984$$ 0 0
$$985$$ 108.000 3.44117
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 6.00000 0.190885
$$989$$ 28.0000 0.890348
$$990$$ 0 0
$$991$$ −57.0000 −1.81066 −0.905332 0.424704i $$-0.860378\pi$$
−0.905332 + 0.424704i $$0.860378\pi$$
$$992$$ −7.00000 −0.222250
$$993$$ 0 0
$$994$$ −16.0000 −0.507489
$$995$$ −16.0000 −0.507234
$$996$$ 0 0
$$997$$ 5.00000 0.158352 0.0791758 0.996861i $$-0.474771\pi$$
0.0791758 + 0.996861i $$0.474771\pi$$
$$998$$ 5.00000 0.158272
$$999$$ 0 0
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1638.2.a.j.1.1 1
3.2 odd 2 182.2.a.e.1.1 1
12.11 even 2 1456.2.a.a.1.1 1
15.14 odd 2 4550.2.a.a.1.1 1
21.2 odd 6 1274.2.f.b.1145.1 2
21.5 even 6 1274.2.f.k.1145.1 2
21.11 odd 6 1274.2.f.b.79.1 2
21.17 even 6 1274.2.f.k.79.1 2
21.20 even 2 1274.2.a.h.1.1 1
24.5 odd 2 5824.2.a.b.1.1 1
24.11 even 2 5824.2.a.bf.1.1 1
39.5 even 4 2366.2.d.j.337.1 2
39.8 even 4 2366.2.d.j.337.2 2
39.38 odd 2 2366.2.a.h.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
182.2.a.e.1.1 1 3.2 odd 2
1274.2.a.h.1.1 1 21.20 even 2
1274.2.f.b.79.1 2 21.11 odd 6
1274.2.f.b.1145.1 2 21.2 odd 6
1274.2.f.k.79.1 2 21.17 even 6
1274.2.f.k.1145.1 2 21.5 even 6
1456.2.a.a.1.1 1 12.11 even 2
1638.2.a.j.1.1 1 1.1 even 1 trivial
2366.2.a.h.1.1 1 39.38 odd 2
2366.2.d.j.337.1 2 39.5 even 4
2366.2.d.j.337.2 2 39.8 even 4
4550.2.a.a.1.1 1 15.14 odd 2
5824.2.a.b.1.1 1 24.5 odd 2
5824.2.a.bf.1.1 1 24.11 even 2