# Properties

 Label 1521.4.a.y.1.2 Level $1521$ Weight $4$ Character 1521.1 Self dual yes Analytic conductor $89.742$ Analytic rank $1$ Dimension $4$ CM discriminant -39 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1521,4,Mod(1,1521)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1521, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1521.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1521 = 3^{2} \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1521.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: $$89.7419051187$$ Analytic rank: $$1$$ Dimension: $$4$$ Coefficient field: 4.4.8112.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 5x^{2} + 3$$ x^4 - 5*x^2 + 3 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2^{2}\cdot 3^{2}$$ Twist minimal: no (minimal twist has level 117) Fricke sign: $$-1$$ Sato-Tate group: $N(\mathrm{U}(1))$

## Embedding invariants

 Embedding label 1.2 Root $$2.07431$$ of defining polynomial Character $$\chi$$ $$=$$ 1521.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-3.52058 q^{2} +4.39445 q^{4} -9.17304 q^{5} +12.6936 q^{8} +O(q^{10})$$ $$q-3.52058 q^{2} +4.39445 q^{4} -9.17304 q^{5} +12.6936 q^{8} +32.2944 q^{10} +20.4780 q^{11} -79.8444 q^{16} -40.3104 q^{20} -72.0942 q^{22} -40.8554 q^{25} +179.549 q^{32} -116.439 q^{40} -196.898 q^{41} -452.000 q^{43} +89.9894 q^{44} +640.490 q^{47} -343.000 q^{49} +143.834 q^{50} -187.845 q^{55} +579.506 q^{59} +944.654 q^{61} +6.63840 q^{64} +1190.09 q^{71} -418.244 q^{79} +732.416 q^{80} +693.193 q^{82} +94.6440 q^{83} +1591.30 q^{86} +259.939 q^{88} -1672.06 q^{89} -2254.89 q^{94} +1207.56 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 32 q^{4}+O(q^{10})$$ 4 * q + 32 * q^4 $$4 q + 32 q^{4} - 116 q^{10} - 204 q^{16} + 476 q^{22} + 500 q^{25} - 884 q^{40} - 1808 q^{43} - 1372 q^{49} - 3232 q^{55} - 1632 q^{64} - 2924 q^{82} + 2756 q^{88} - 5140 q^{94}+O(q^{100})$$ 4 * q + 32 * q^4 - 116 * q^10 - 204 * q^16 + 476 * q^22 + 500 * q^25 - 884 * q^40 - 1808 * q^43 - 1372 * q^49 - 3232 * q^55 - 1632 * q^64 - 2924 * q^82 + 2756 * q^88 - 5140 * q^94

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −3.52058 −1.24471 −0.622356 0.782735i $$-0.713824\pi$$
−0.622356 + 0.782735i $$0.713824\pi$$
$$3$$ 0 0
$$4$$ 4.39445 0.549306
$$5$$ −9.17304 −0.820462 −0.410231 0.911982i $$-0.634552\pi$$
−0.410231 + 0.911982i $$0.634552\pi$$
$$6$$ 0 0
$$7$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$8$$ 12.6936 0.560984
$$9$$ 0 0
$$10$$ 32.2944 1.02124
$$11$$ 20.4780 0.561304 0.280652 0.959810i $$-0.409449\pi$$
0.280652 + 0.959810i $$0.409449\pi$$
$$12$$ 0 0
$$13$$ 0 0
$$14$$ 0 0
$$15$$ 0 0
$$16$$ −79.8444 −1.24757
$$17$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$18$$ 0 0
$$19$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$20$$ −40.3104 −0.450685
$$21$$ 0 0
$$22$$ −72.0942 −0.698661
$$23$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$24$$ 0 0
$$25$$ −40.8554 −0.326843
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$30$$ 0 0
$$31$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$32$$ 179.549 0.991879
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ −116.439 −0.460266
$$41$$ −196.898 −0.750006 −0.375003 0.927024i $$-0.622358\pi$$
−0.375003 + 0.927024i $$0.622358\pi$$
$$42$$ 0 0
$$43$$ −452.000 −1.60301 −0.801504 0.597989i $$-0.795967\pi$$
−0.801504 + 0.597989i $$0.795967\pi$$
$$44$$ 89.9894 0.308327
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 640.490 1.98777 0.993884 0.110431i $$-0.0352231\pi$$
0.993884 + 0.110431i $$0.0352231\pi$$
$$48$$ 0 0
$$49$$ −343.000 −1.00000
$$50$$ 143.834 0.406825
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$54$$ 0 0
$$55$$ −187.845 −0.460528
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 579.506 1.27873 0.639367 0.768902i $$-0.279197\pi$$
0.639367 + 0.768902i $$0.279197\pi$$
$$60$$ 0 0
$$61$$ 944.654 1.98280 0.991398 0.130879i $$-0.0417798\pi$$
0.991398 + 0.130879i $$0.0417798\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 6.63840 0.0129656
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 1190.09 1.98926 0.994632 0.103474i $$-0.0329957\pi$$
0.994632 + 0.103474i $$0.0329957\pi$$
$$72$$ 0 0
$$73$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −418.244 −0.595647 −0.297824 0.954621i $$-0.596261\pi$$
−0.297824 + 0.954621i $$0.596261\pi$$
$$80$$ 732.416 1.02358
$$81$$ 0 0
$$82$$ 693.193 0.933541
$$83$$ 94.6440 0.125163 0.0625815 0.998040i $$-0.480067\pi$$
0.0625815 + 0.998040i $$0.480067\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 1591.30 1.99528
$$87$$ 0 0
$$88$$ 259.939 0.314882
$$89$$ −1672.06 −1.99143 −0.995717 0.0924493i $$-0.970530\pi$$
−0.995717 + 0.0924493i $$0.970530\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 0 0
$$94$$ −2254.89 −2.47420
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$98$$ 1207.56 1.24471
$$99$$ 0 0
$$100$$ −179.537 −0.179537
$$101$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$102$$ 0 0
$$103$$ −848.000 −0.811223 −0.405611 0.914046i $$-0.632941\pi$$
−0.405611 + 0.914046i $$0.632941\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$108$$ 0 0
$$109$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$110$$ 661.323 0.573224
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 0 0
$$118$$ −2040.20 −1.59165
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −911.653 −0.684938
$$122$$ −3325.73 −2.46801
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 1521.40 1.08862
$$126$$ 0 0
$$127$$ 2495.04 1.74330 0.871650 0.490129i $$-0.163050\pi$$
0.871650 + 0.490129i $$0.163050\pi$$
$$128$$ −1459.77 −1.00802
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 3018.79 1.88258 0.941288 0.337604i $$-0.109616\pi$$
0.941288 + 0.337604i $$0.109616\pi$$
$$138$$ 0 0
$$139$$ −340.000 −0.207471 −0.103735 0.994605i $$-0.533079\pi$$
−0.103735 + 0.994605i $$0.533079\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ −4189.80 −2.47606
$$143$$ 0 0
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 2477.09 1.36196 0.680978 0.732304i $$-0.261555\pi$$
0.680978 + 0.732304i $$0.261555\pi$$
$$150$$ 0 0
$$151$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ −934.000 −0.474785 −0.237393 0.971414i $$-0.576293\pi$$
−0.237393 + 0.971414i $$0.576293\pi$$
$$158$$ 1472.46 0.741409
$$159$$ 0 0
$$160$$ −1647.01 −0.813799
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$164$$ −865.256 −0.411983
$$165$$ 0 0
$$166$$ −333.201 −0.155792
$$167$$ −2446.51 −1.13363 −0.566815 0.823845i $$-0.691825\pi$$
−0.566815 + 0.823845i $$0.691825\pi$$
$$168$$ 0 0
$$169$$ 0 0
$$170$$ 0 0
$$171$$ 0 0
$$172$$ −1986.29 −0.880542
$$173$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ −1635.05 −0.700265
$$177$$ 0 0
$$178$$ 5886.60 2.47876
$$179$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$180$$ 0 0
$$181$$ −4430.00 −1.81922 −0.909611 0.415460i $$-0.863621\pi$$
−0.909611 + 0.415460i $$0.863621\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 2814.60 1.09189
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$192$$ 0 0
$$193$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ −1507.30 −0.549306
$$197$$ −3977.33 −1.43844 −0.719220 0.694782i $$-0.755501\pi$$
−0.719220 + 0.694782i $$0.755501\pi$$
$$198$$ 0 0
$$199$$ 5610.24 1.99849 0.999244 0.0388706i $$-0.0123760\pi$$
0.999244 + 0.0388706i $$0.0123760\pi$$
$$200$$ −518.602 −0.183354
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 1806.15 0.615351
$$206$$ 2985.45 1.00974
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 0 0
$$210$$ 0 0
$$211$$ −5740.04 −1.87280 −0.936399 0.350937i $$-0.885863\pi$$
−0.936399 + 0.350937i $$0.885863\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 4146.21 1.31521
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ 0 0
$$220$$ −825.476 −0.252971
$$221$$ 0 0
$$222$$ 0 0
$$223$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ −6668.65 −1.94984 −0.974920 0.222554i $$-0.928561\pi$$
−0.974920 + 0.222554i $$0.928561\pi$$
$$228$$ 0 0
$$229$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$234$$ 0 0
$$235$$ −5875.24 −1.63089
$$236$$ 2546.61 0.702416
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 6085.88 1.64712 0.823562 0.567226i $$-0.191984\pi$$
0.823562 + 0.567226i $$0.191984\pi$$
$$240$$ 0 0
$$241$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$242$$ 3209.54 0.852550
$$243$$ 0 0
$$244$$ 4151.24 1.08916
$$245$$ 3146.35 0.820462
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 0 0
$$249$$ 0 0
$$250$$ −5356.19 −1.35502
$$251$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ −8783.98 −2.16991
$$255$$ 0 0
$$256$$ 5086.11 1.24173
$$257$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$270$$ 0 0
$$271$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ −10627.9 −2.34326
$$275$$ −836.635 −0.183458
$$276$$ 0 0
$$277$$ −7634.00 −1.65589 −0.827947 0.560806i $$-0.810491\pi$$
−0.827947 + 0.560806i $$0.810491\pi$$
$$278$$ 1197.00 0.258241
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −8255.10 −1.75252 −0.876260 0.481839i $$-0.839969\pi$$
−0.876260 + 0.481839i $$0.839969\pi$$
$$282$$ 0 0
$$283$$ −490.355 −0.102999 −0.0514993 0.998673i $$-0.516400\pi$$
−0.0514993 + 0.998673i $$0.516400\pi$$
$$284$$ 5229.79 1.09272
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ −4913.00 −1.00000
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ −9933.00 −1.98052 −0.990260 0.139232i $$-0.955537\pi$$
−0.990260 + 0.139232i $$0.955537\pi$$
$$294$$ 0 0
$$295$$ −5315.83 −1.04915
$$296$$ 0 0
$$297$$ 0 0
$$298$$ −8720.79 −1.69524
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ −8665.35 −1.62681
$$306$$ 0 0
$$307$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$312$$ 0 0
$$313$$ 6396.25 1.15507 0.577536 0.816365i $$-0.304014\pi$$
0.577536 + 0.816365i $$0.304014\pi$$
$$314$$ 3288.22 0.590971
$$315$$ 0 0
$$316$$ −1837.95 −0.327193
$$317$$ 7133.53 1.26391 0.631954 0.775006i $$-0.282253\pi$$
0.631954 + 0.775006i $$0.282253\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ −60.8943 −0.0106378
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 0 0
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 0 0
$$328$$ −2499.34 −0.420741
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$332$$ 415.908 0.0687528
$$333$$ 0 0
$$334$$ 8613.11 1.41104
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −10420.0 −1.68432 −0.842160 0.539228i $$-0.818716\pi$$
−0.842160 + 0.539228i $$0.818716\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ 0 0
$$344$$ −5737.51 −0.899262
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$348$$ 0 0
$$349$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 3676.81 0.556745
$$353$$ −7331.69 −1.10546 −0.552728 0.833361i $$-0.686413\pi$$
−0.552728 + 0.833361i $$0.686413\pi$$
$$354$$ 0 0
$$355$$ −10916.7 −1.63211
$$356$$ −7347.77 −1.09391
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −11077.3 −1.62852 −0.814259 0.580502i $$-0.802857\pi$$
−0.814259 + 0.580502i $$0.802857\pi$$
$$360$$ 0 0
$$361$$ −6859.00 −1.00000
$$362$$ 15596.1 2.26441
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ −8296.00 −1.17997 −0.589983 0.807416i $$-0.700866\pi$$
−0.589983 + 0.807416i $$0.700866\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 6526.05 0.905914 0.452957 0.891532i $$-0.350369\pi$$
0.452957 + 0.891532i $$0.350369\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 8130.13 1.11511
$$377$$ 0 0
$$378$$ 0 0
$$379$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ −12702.2 −1.69465 −0.847327 0.531071i $$-0.821790\pi$$
−0.847327 + 0.531071i $$0.821790\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ −4353.91 −0.560984
$$393$$ 0 0
$$394$$ 14002.5 1.79044
$$395$$ 3836.57 0.488706
$$396$$ 0 0
$$397$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$398$$ −19751.3 −2.48754
$$399$$ 0 0
$$400$$ 3262.07 0.407759
$$401$$ −15342.6 −1.91066 −0.955328 0.295549i $$-0.904497\pi$$
−0.955328 + 0.295549i $$0.904497\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 0 0
$$409$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$410$$ −6358.68 −0.765934
$$411$$ 0 0
$$412$$ −3726.49 −0.445609
$$413$$ 0 0
$$414$$ 0 0
$$415$$ −868.173 −0.102691
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$420$$ 0 0
$$421$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$422$$ 20208.2 2.33109
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ 0 0
$$430$$ −14597.1 −1.63705
$$431$$ 320.305 0.0357971 0.0178985 0.999840i $$-0.494302\pi$$
0.0178985 + 0.999840i $$0.494302\pi$$
$$432$$ 0 0
$$433$$ −36.0555 −0.00400166 −0.00200083 0.999998i $$-0.500637\pi$$
−0.00200083 + 0.999998i $$0.500637\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 0 0
$$438$$ 0 0
$$439$$ 11320.0 1.23069 0.615346 0.788257i $$-0.289016\pi$$
0.615346 + 0.788257i $$0.289016\pi$$
$$440$$ −2384.43 −0.258349
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$444$$ 0 0
$$445$$ 15337.8 1.63390
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ −333.683 −0.0350723 −0.0175361 0.999846i $$-0.505582\pi$$
−0.0175361 + 0.999846i $$0.505582\pi$$
$$450$$ 0 0
$$451$$ −4032.06 −0.420981
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 23477.5 2.42699
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −17433.8 −1.76133 −0.880666 0.473738i $$-0.842905\pi$$
−0.880666 + 0.473738i $$0.842905\pi$$
$$462$$ 0 0
$$463$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 20684.2 2.02998
$$471$$ 0 0
$$472$$ 7356.03 0.717349
$$473$$ −9256.04 −0.899774
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 0 0
$$478$$ −21425.8 −2.05019
$$479$$ 5200.84 0.496101 0.248050 0.968747i $$-0.420210\pi$$
0.248050 + 0.968747i $$0.420210\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 0 0
$$483$$ 0 0
$$484$$ −4006.21 −0.376241
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$488$$ 11991.1 1.11232
$$489$$ 0 0
$$490$$ −11077.0 −1.02124
$$491$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$492$$ 0 0
$$493$$ 0 0
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$500$$ 6685.70 0.597988
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 10964.3 0.957605
$$509$$ −22966.8 −1.99997 −0.999987 0.00508931i $$-0.998380\pi$$
−0.999987 + 0.00508931i $$0.998380\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ −6227.90 −0.537572
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 7778.74 0.665577
$$516$$ 0 0
$$517$$ 13115.9 1.11574
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$522$$ 0 0
$$523$$ 16148.0 1.35010 0.675050 0.737772i $$-0.264122\pi$$
0.675050 + 0.737772i $$0.264122\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 0 0
$$528$$ 0 0
$$529$$ −12167.0 −1.00000
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 0 0
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ −7023.94 −0.561304
$$540$$ 0 0
$$541$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 23996.0 1.87568 0.937838 0.347073i $$-0.112824\pi$$
0.937838 + 0.347073i $$0.112824\pi$$
$$548$$ 13265.9 1.03411
$$549$$ 0 0
$$550$$ 2945.44 0.228352
$$551$$ 0 0
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 26876.1 2.06111
$$555$$ 0 0
$$556$$ −1494.11 −0.113965
$$557$$ −22258.1 −1.69319 −0.846595 0.532237i $$-0.821352\pi$$
−0.846595 + 0.532237i $$0.821352\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 29062.7 2.18138
$$563$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 1726.33 0.128203
$$567$$ 0 0
$$568$$ 15106.6 1.11595
$$569$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$570$$ 0 0
$$571$$ −25411.9 −1.86244 −0.931222 0.364451i $$-0.881257\pi$$
−0.931222 + 0.364451i $$0.881257\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$578$$ 17296.6 1.24471
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 0 0
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 34969.9 2.46517
$$587$$ 19338.0 1.35974 0.679868 0.733335i $$-0.262037\pi$$
0.679868 + 0.733335i $$0.262037\pi$$
$$588$$ 0 0
$$589$$ 0 0
$$590$$ 18714.8 1.30589
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 25958.2 1.79760 0.898798 0.438363i $$-0.144442\pi$$
0.898798 + 0.438363i $$0.144442\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 10885.5 0.748131
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$600$$ 0 0
$$601$$ 12626.6 0.856991 0.428495 0.903544i $$-0.359044\pi$$
0.428495 + 0.903544i $$0.359044\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ 8362.63 0.561966
$$606$$ 0 0
$$607$$ −25504.0 −1.70540 −0.852698 0.522404i $$-0.825035\pi$$
−0.852698 + 0.522404i $$0.825035\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 30507.0 2.02491
$$611$$ 0 0
$$612$$ 0 0
$$613$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 29859.9 1.94832 0.974160 0.225860i $$-0.0725191\pi$$
0.974160 + 0.225860i $$0.0725191\pi$$
$$618$$ 0 0
$$619$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ −8848.92 −0.566331
$$626$$ −22518.5 −1.43773
$$627$$ 0 0
$$628$$ −4104.42 −0.260803
$$629$$ 0 0
$$630$$ 0 0
$$631$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$632$$ −5309.03 −0.334148
$$633$$ 0 0
$$634$$ −25114.1 −1.57320
$$635$$ −22887.1 −1.43031
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 13390.5 0.827040
$$641$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$642$$ 0 0
$$643$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$648$$ 0 0
$$649$$ 11867.1 0.717758
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 15721.2 0.935684
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$660$$ 0 0
$$661$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 1201.37 0.0702144
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 0 0
$$668$$ −10751.0 −0.622710
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 19344.6 1.11295
$$672$$ 0 0
$$673$$ 25522.0 1.46181 0.730907 0.682477i $$-0.239097\pi$$
0.730907 + 0.682477i $$0.239097\pi$$
$$674$$ 36684.5 2.09649
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 32750.6 1.83480 0.917399 0.397968i $$-0.130285\pi$$
0.917399 + 0.397968i $$0.130285\pi$$
$$684$$ 0 0
$$685$$ −27691.5 −1.54458
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 36089.7 1.99986
$$689$$ 0 0
$$690$$ 0 0
$$691$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 3118.83 0.170222
$$696$$ 0 0
$$697$$ 0 0
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$702$$ 0 0
$$703$$ 0 0
$$704$$ 135.941 0.00727765
$$705$$ 0 0
$$706$$ 25811.8 1.37597
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$710$$ 38433.2 2.03151
$$711$$ 0 0
$$712$$ −21224.4 −1.11716
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 38998.5 2.02704
$$719$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 24147.6 1.24471
$$723$$ 0 0
$$724$$ −19467.4 −0.999310
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 28455.0 1.45163 0.725817 0.687888i $$-0.241462\pi$$
0.725817 + 0.687888i $$0.241462\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 0 0
$$732$$ 0 0
$$733$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$734$$ 29206.7 1.46872
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 0 0
$$739$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −19936.4 −0.984382 −0.492191 0.870487i $$-0.663804\pi$$
−0.492191 + 0.870487i $$0.663804\pi$$
$$744$$ 0 0
$$745$$ −22722.5 −1.11743
$$746$$ −22975.4 −1.12760
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −26120.0 −1.26915 −0.634575 0.772861i $$-0.718825\pi$$
−0.634575 + 0.772861i $$0.718825\pi$$
$$752$$ −51139.6 −2.47988
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −35009.9 −1.68092 −0.840460 0.541873i $$-0.817715\pi$$
−0.840460 + 0.541873i $$0.817715\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −28936.4 −1.37838 −0.689189 0.724582i $$-0.742033\pi$$
−0.689189 + 0.724582i $$0.742033\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 44719.1 2.10936
$$767$$ 0 0
$$768$$ 0 0
$$769$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ −3484.00 −0.162110 −0.0810548 0.996710i $$-0.525829\pi$$
−0.0810548 + 0.996710i $$0.525829\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 0 0
$$780$$ 0 0
$$781$$ 24370.6 1.11658
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 27386.6 1.24757
$$785$$ 8567.62 0.389543
$$786$$ 0 0
$$787$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$788$$ −17478.2 −0.790144
$$789$$ 0 0
$$790$$ −13506.9 −0.608297
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 0 0
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 24653.9 1.09778
$$797$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ −7335.55 −0.324189
$$801$$ 0 0
$$802$$ 54014.8 2.37821
$$803$$ 0 0
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$810$$ 0 0
$$811$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 0 0
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 7937.03 0.338016
$$821$$ 33048.4 1.40487 0.702433 0.711749i $$-0.252097\pi$$
0.702433 + 0.711749i $$0.252097\pi$$
$$822$$ 0 0
$$823$$ −37352.0 −1.58203 −0.791014 0.611798i $$-0.790446\pi$$
−0.791014 + 0.611798i $$0.790446\pi$$
$$824$$ −10764.2 −0.455083
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −31167.2 −1.31051 −0.655253 0.755409i $$-0.727438\pi$$
−0.655253 + 0.755409i $$0.727438\pi$$
$$828$$ 0 0
$$829$$ 3079.14 0.129002 0.0645012 0.997918i $$-0.479454\pi$$
0.0645012 + 0.997918i $$0.479454\pi$$
$$830$$ 3056.47 0.127821
$$831$$ 0 0
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 0 0
$$835$$ 22441.9 0.930101
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ 46872.5 1.92875 0.964374 0.264543i $$-0.0852210\pi$$
0.964374 + 0.264543i $$0.0852210\pi$$
$$840$$ 0 0
$$841$$ −24389.0 −1.00000
$$842$$ 0 0
$$843$$ 0 0
$$844$$ −25224.3 −1.02874
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 0 0
$$852$$ 0 0
$$853$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$858$$ 0 0
$$859$$ 43324.3 1.72085 0.860423 0.509581i $$-0.170200\pi$$
0.860423 + 0.509581i $$0.170200\pi$$
$$860$$ 18220.3 0.722451
$$861$$ 0 0
$$862$$ −1127.66 −0.0445571
$$863$$ −23594.8 −0.930678 −0.465339 0.885133i $$-0.654068\pi$$
−0.465339 + 0.885133i $$0.654068\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 126.936 0.00498091
$$867$$ 0 0
$$868$$ 0 0
$$869$$ −8564.79 −0.334339
$$870$$ 0 0
$$871$$ 0 0
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$878$$ −39852.9 −1.53186
$$879$$ 0 0
$$880$$ 14998.4 0.574540
$$881$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$882$$ 0 0
$$883$$ −52410.3 −1.99745 −0.998724 0.0504988i $$-0.983919\pi$$
−0.998724 + 0.0504988i $$0.983919\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ −53998.0 −2.03373
$$891$$ 0 0
$$892$$ 0 0
$$893$$ 0 0
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 1174.75 0.0436549
$$899$$ 0 0
$$900$$ 0 0
$$901$$ 0 0
$$902$$ 14195.2 0.524000
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 40636.6 1.49260
$$906$$ 0 0
$$907$$ −35017.1 −1.28195 −0.640973 0.767564i $$-0.721469\pi$$
−0.640973 + 0.767564i $$0.721469\pi$$
$$908$$ −29305.0 −1.07106
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$912$$ 0 0
$$913$$ 1938.12 0.0702545
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ −36762.2 −1.31956 −0.659779 0.751460i $$-0.729350\pi$$
−0.659779 + 0.751460i $$0.729350\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 61377.1 2.19235
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ −7352.53 −0.259665 −0.129832 0.991536i $$-0.541444\pi$$
−0.129832 + 0.991536i $$0.541444\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 13275.6 0.462856 0.231428 0.972852i $$-0.425660\pi$$
0.231428 + 0.972852i $$0.425660\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ −25818.4 −0.895856
$$941$$ −15462.1 −0.535652 −0.267826 0.963467i $$-0.586305\pi$$
−0.267826 + 0.963467i $$0.586305\pi$$
$$942$$ 0 0
$$943$$ 0 0
$$944$$ −46270.3 −1.59531
$$945$$ 0 0
$$946$$ 32586.6 1.11996
$$947$$ 56498.4 1.93870 0.969352 0.245678i $$-0.0790105\pi$$
0.969352 + 0.245678i $$0.0790105\pi$$
$$948$$ 0 0
$$949$$ 0 0
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 26744.1 0.904775
$$957$$ 0 0
$$958$$ −18309.9 −0.617502
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −29791.0 −1.00000
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$968$$ −11572.2 −0.384239
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 0 0
$$975$$ 0 0
$$976$$ −75425.4 −2.47368
$$977$$ −52089.2 −1.70571 −0.852856 0.522146i $$-0.825132\pi$$
−0.852856 + 0.522146i $$0.825132\pi$$
$$978$$ 0 0
$$979$$ −34240.3 −1.11780
$$980$$ 13826.5 0.450685
$$981$$ 0 0
$$982$$ 0 0
$$983$$ −60639.0 −1.96753 −0.983766 0.179454i $$-0.942567\pi$$
−0.983766 + 0.179454i $$0.942567\pi$$
$$984$$ 0 0
$$985$$ 36484.2 1.18019
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 0 0
$$990$$ 0 0
$$991$$ −23272.0 −0.745973 −0.372987 0.927837i $$-0.621666\pi$$
−0.372987 + 0.927837i $$0.621666\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ −51462.9 −1.63968
$$996$$ 0 0
$$997$$ −21634.0 −0.687217 −0.343609 0.939113i $$-0.611649\pi$$
−0.343609 + 0.939113i $$0.611649\pi$$
$$998$$ 0 0
$$999$$ 0 0
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1521.4.a.y.1.2 4
3.2 odd 2 inner 1521.4.a.y.1.3 4
13.5 odd 4 117.4.b.c.64.3 yes 4
13.8 odd 4 117.4.b.c.64.2 4
13.12 even 2 inner 1521.4.a.y.1.3 4
39.5 even 4 117.4.b.c.64.2 4
39.8 even 4 117.4.b.c.64.3 yes 4
39.38 odd 2 CM 1521.4.a.y.1.2 4

By twisted newform
Twist Min Dim Char Parity Ord Type
117.4.b.c.64.2 4 13.8 odd 4
117.4.b.c.64.2 4 39.5 even 4
117.4.b.c.64.3 yes 4 13.5 odd 4
117.4.b.c.64.3 yes 4 39.8 even 4
1521.4.a.y.1.2 4 1.1 even 1 trivial
1521.4.a.y.1.2 4 39.38 odd 2 CM
1521.4.a.y.1.3 4 3.2 odd 2 inner
1521.4.a.y.1.3 4 13.12 even 2 inner