# Properties

 Label 117.4.b.c.64.3 Level $117$ Weight $4$ Character 117.64 Analytic conductor $6.903$ Analytic rank $0$ 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] = [117,4,Mod(64,117)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(117, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

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

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

chi := DirichletCharacter("117.64");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$117 = 3^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 117.b (of order $$2$$, degree $$1$$, minimal)

## 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: no Analytic conductor: $$6.90322347067$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: 4.0.8112.1 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: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## Embedding invariants

 Embedding label 64.3 Root $$2.07431i$$ of defining polynomial Character $$\chi$$ $$=$$ 117.64 Dual form 117.4.b.c.64.2

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+3.52058i q^{2} -4.39445 q^{4} +9.17304i q^{5} +12.6936i q^{8} +O(q^{10})$$ $$q+3.52058i q^{2} -4.39445 q^{4} +9.17304i q^{5} +12.6936i q^{8} -32.2944 q^{10} +20.4780i q^{11} -46.8722 q^{13} -79.8444 q^{16} -40.3104i q^{20} -72.0942 q^{22} +40.8554 q^{25} -165.017i q^{26} -179.549i q^{32} -116.439 q^{40} +196.898i q^{41} +452.000 q^{43} -89.9894i q^{44} +640.490i q^{47} +343.000 q^{49} +143.834i q^{50} +205.977 q^{52} -187.845 q^{55} +579.506i q^{59} +944.654 q^{61} -6.63840 q^{64} -429.960i q^{65} -1190.09i q^{71} -418.244 q^{79} -732.416i q^{80} -693.193 q^{82} -94.6440i q^{83} +1591.30i q^{86} -259.939 q^{88} -1672.06i q^{89} -2254.89 q^{94} +1207.56i q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 32 q^{4}+O(q^{10})$$ 4 * q - 32 * q^4 $$4 q - 32 q^{4} + 116 q^{10} - 204 q^{16} + 476 q^{22} - 500 q^{25} - 884 q^{40} + 1808 q^{43} + 1372 q^{49} - 676 q^{52} - 3232 q^{55} + 1632 q^{64} + 2924 q^{82} - 2756 q^{88} - 5140 q^{94}+O(q^{100})$$ 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 - 676 * q^52 - 3232 * q^55 + 1632 * q^64 + 2924 * q^82 - 2756 * q^88 - 5140 * q^94

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/117\mathbb{Z}\right)^\times$$.

 $$n$$ $$28$$ $$92$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 3.52058i 1.24471i 0.782735 + 0.622356i $$0.213824\pi$$
−0.782735 + 0.622356i $$0.786176\pi$$
$$3$$ 0 0
$$4$$ −4.39445 −0.549306
$$5$$ 9.17304i 0.820462i 0.911982 + 0.410231i $$0.134552\pi$$
−0.911982 + 0.410231i $$0.865448\pi$$
$$6$$ 0 0
$$7$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$8$$ 12.6936i 0.560984i
$$9$$ 0 0
$$10$$ −32.2944 −1.02124
$$11$$ 20.4780i 0.561304i 0.959810 + 0.280652i $$0.0905506\pi$$
−0.959810 + 0.280652i $$0.909449\pi$$
$$12$$ 0 0
$$13$$ −46.8722 −1.00000
$$14$$ 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 1.00000 $$0$$
−1.00000 $$\pi$$
$$20$$ − 40.3104i − 0.450685i
$$21$$ 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
$$26$$ − 165.017i − 1.24471i
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$30$$ 0 0
$$31$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$32$$ − 179.549i − 0.991879i
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ −116.439 −0.460266
$$41$$ 196.898i 0.750006i 0.927024 + 0.375003i $$0.122358\pi$$
−0.927024 + 0.375003i $$0.877642\pi$$
$$42$$ 0 0
$$43$$ 452.000 1.60301 0.801504 0.597989i $$-0.204033\pi$$
0.801504 + 0.597989i $$0.204033\pi$$
$$44$$ − 89.9894i − 0.308327i
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 640.490i 1.98777i 0.110431 + 0.993884i $$0.464777\pi$$
−0.110431 + 0.993884i $$0.535223\pi$$
$$48$$ 0 0
$$49$$ 343.000 1.00000
$$50$$ 143.834i 0.406825i
$$51$$ 0 0
$$52$$ 205.977 0.549306
$$53$$ 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
$$59$$ 579.506i 1.27873i 0.768902 + 0.639367i $$0.220803\pi$$
−0.768902 + 0.639367i $$0.779197\pi$$
$$60$$ 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
$$65$$ − 429.960i − 0.820462i
$$66$$ 0 0
$$67$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ − 1190.09i − 1.98926i −0.103474 0.994632i $$-0.532996\pi$$
0.103474 0.994632i $$-0.467004\pi$$
$$72$$ 0 0
$$73$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$74$$ 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 $$0.596261\pi$$
$$80$$ − 732.416i − 1.02358i
$$81$$ 0 0
$$82$$ −693.193 −0.933541
$$83$$ − 94.6440i − 0.125163i −0.998040 0.0625815i $$-0.980067\pi$$
0.998040 0.0625815i $$-0.0199333\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 1591.30i 1.99528i
$$87$$ 0 0
$$88$$ −259.939 −0.314882
$$89$$ − 1672.06i − 1.99143i −0.0924493 0.995717i $$-0.529470\pi$$
0.0924493 0.995717i $$-0.470530\pi$$
$$90$$ 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 1.00000 $$0$$
−1.00000 $$\pi$$
$$98$$ 1207.56i 1.24471i
$$99$$ 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 $$-0.367059\pi$$
0.405611 + 0.914046i $$0.367059\pi$$
$$104$$ − 594.977i − 0.560984i
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$108$$ 0 0
$$109$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$110$$ − 661.323i − 0.573224i
$$111$$ 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
$$122$$ 3325.73i 2.46801i
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 1521.40i 1.08862i
$$126$$ 0 0
$$127$$ −2495.04 −1.74330 −0.871650 0.490129i $$-0.836950\pi$$
−0.871650 + 0.490129i $$0.836950\pi$$
$$128$$ − 1459.77i − 1.00802i
$$129$$ 0 0
$$130$$ 1513.71 1.02124
$$131$$ 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
$$137$$ 3018.79i 1.88258i 0.337604 + 0.941288i $$0.390384\pi$$
−0.337604 + 0.941288i $$0.609616\pi$$
$$138$$ 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
$$143$$ − 959.847i − 0.561304i
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ − 2477.09i − 1.36196i −0.732304 0.680978i $$-0.761555\pi$$
0.732304 0.680978i $$-0.238445\pi$$
$$150$$ 0 0
$$151$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$152$$ 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 $$0.576293\pi$$
$$158$$ − 1472.46i − 0.741409i
$$159$$ 0 0
$$160$$ 1647.01 0.813799
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$164$$ − 865.256i − 0.411983i
$$165$$ 0 0
$$166$$ 333.201 0.155792
$$167$$ − 2446.51i − 1.13363i −0.823845 0.566815i $$-0.808175\pi$$
0.823845 0.566815i $$-0.191825\pi$$
$$168$$ 0 0
$$169$$ 2197.00 1.00000
$$170$$ 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
$$176$$ − 1635.05i − 0.700265i
$$177$$ 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 $$-0.136379\pi$$
0.909611 + 0.415460i $$0.136379\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 0 0
$$188$$ − 2814.60i − 1.09189i
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$192$$ 0 0
$$193$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ −1507.30 −0.549306
$$197$$ 3977.33i 1.43844i 0.694782 + 0.719220i $$0.255501\pi$$
−0.694782 + 0.719220i $$0.744499\pi$$
$$198$$ 0 0
$$199$$ −5610.24 −1.99849 −0.999244 0.0388706i $$-0.987624\pi$$
−0.999244 + 0.0388706i $$0.987624\pi$$
$$200$$ 518.602i 0.183354i
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ −1806.15 −0.615351
$$206$$ 2985.45i 1.00974i
$$207$$ 0 0
$$208$$ 3742.48 1.24757
$$209$$ 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
$$215$$ 4146.21i 1.31521i
$$216$$ 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 1.00000 $$0$$
−1.00000 $$\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 6668.65i 1.94984i 0.222554 + 0.974920i $$0.428561\pi$$
−0.222554 + 0.974920i $$0.571439\pi$$
$$228$$ 0 0
$$229$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$230$$ 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
$$236$$ − 2546.61i − 0.702416i
$$237$$ 0 0
$$238$$ 0 0
$$239$$ − 6085.88i − 1.64712i −0.567226 0.823562i $$-0.691984\pi$$
0.567226 0.823562i $$-0.308016\pi$$
$$240$$ 0 0
$$241$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$242$$ 3209.54i 0.852550i
$$243$$ 0 0
$$244$$ −4151.24 −1.08916
$$245$$ 3146.35i 0.820462i
$$246$$ 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
$$254$$ − 8783.98i − 2.16991i
$$255$$ 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
$$260$$ 1889.44i 0.450685i
$$261$$ 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 1.00000 $$0$$
−1.00000 $$\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ −10627.9 −2.34326
$$275$$ 836.635i 0.183458i
$$276$$ 0 0
$$277$$ 7634.00 1.65589 0.827947 0.560806i $$-0.189509\pi$$
0.827947 + 0.560806i $$0.189509\pi$$
$$278$$ − 1197.00i − 0.258241i
$$279$$ 0 0
$$280$$ 0 0
$$281$$ − 8255.10i − 1.75252i −0.481839 0.876260i $$-0.660031\pi$$
0.481839 0.876260i $$-0.339969\pi$$
$$282$$ 0 0
$$283$$ 490.355 0.102999 0.0514993 0.998673i $$-0.483600\pi$$
0.0514993 + 0.998673i $$0.483600\pi$$
$$284$$ 5229.79i 1.09272i
$$285$$ 0 0
$$286$$ 3379.21 0.698661
$$287$$ 0 0
$$288$$ 0 0
$$289$$ −4913.00 −1.00000
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ − 9933.00i − 1.98052i −0.139232 0.990260i $$-0.544463\pi$$
0.139232 0.990260i $$-0.455537\pi$$
$$294$$ 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
$$305$$ 8665.35i 1.62681i
$$306$$ 0 0
$$307$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$308$$ 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 $$0.304014\pi$$
$$314$$ − 3288.22i − 0.590971i
$$315$$ 0 0
$$316$$ 1837.95 0.327193
$$317$$ − 7133.53i − 1.26391i −0.775006 0.631954i $$-0.782253\pi$$
0.775006 0.631954i $$-0.217747\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ − 60.8943i − 0.0106378i
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 0 0
$$324$$ 0 0
$$325$$ −1914.98 −0.326843
$$326$$ 0 0
$$327$$ 0 0
$$328$$ −2499.34 −0.420741
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$332$$ 415.908i 0.0687528i
$$333$$ 0 0
$$334$$ 8613.11 1.41104
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 10420.0 1.68432 0.842160 0.539228i $$-0.181284\pi$$
0.842160 + 0.539228i $$0.181284\pi$$
$$338$$ 7734.70i 1.24471i
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 5737.51i 0.899262i
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$348$$ 0 0
$$349$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 3676.81 0.556745
$$353$$ 7331.69i 1.10546i 0.833361 + 0.552728i $$0.186413\pi$$
−0.833361 + 0.552728i $$0.813587\pi$$
$$354$$ 0 0
$$355$$ 10916.7 1.63211
$$356$$ 7347.77i 1.09391i
$$357$$ 0 0
$$358$$ 0 0
$$359$$ − 11077.3i − 1.62852i −0.580502 0.814259i $$-0.697143\pi$$
0.580502 0.814259i $$-0.302857\pi$$
$$360$$ 0 0
$$361$$ 6859.00 1.00000
$$362$$ 15596.1i 2.26441i
$$363$$ 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 1.00000 $$0$$
−1.00000 $$\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ 12702.2i 1.69465i 0.531071 + 0.847327i $$0.321790\pi$$
−0.531071 + 0.847327i $$0.678210\pi$$
$$384$$ 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
$$392$$ 4353.91i 0.560984i
$$393$$ 0 0
$$394$$ −14002.5 −1.79044
$$395$$ − 3836.57i − 0.488706i
$$396$$ 0 0
$$397$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$398$$ − 19751.3i − 2.48754i
$$399$$ 0 0
$$400$$ −3262.07 −0.407759
$$401$$ − 15342.6i − 1.91066i −0.295549 0.955328i $$-0.595503\pi$$
0.295549 0.955328i $$-0.404497\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 0 0
$$409$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$410$$ − 6358.68i − 0.765934i
$$411$$ 0 0
$$412$$ −3726.49 −0.445609
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 868.173 0.102691
$$416$$ 8415.87i 0.991879i
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$420$$ 0 0
$$421$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$422$$ − 20208.2i − 2.33109i
$$423$$ 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
$$431$$ − 320.305i − 0.0357971i −0.999840 0.0178985i $$-0.994302\pi$$
0.999840 0.0178985i $$-0.00569759\pi$$
$$432$$ 0 0
$$433$$ 36.0555 0.00400166 0.00200083 0.999998i $$-0.499363\pi$$
0.00200083 + 0.999998i $$0.499363\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 0 0
$$438$$ 0 0
$$439$$ −11320.0 −1.23069 −0.615346 0.788257i $$-0.710984\pi$$
−0.615346 + 0.788257i $$0.710984\pi$$
$$440$$ − 2384.43i − 0.258349i
$$441$$ 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
$$449$$ − 333.683i − 0.0350723i −0.999846 0.0175361i $$-0.994418\pi$$
0.999846 0.0175361i $$-0.00558221\pi$$
$$450$$ 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 1.00000 $$0$$
−1.00000 $$\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 17433.8i 1.76133i 0.473738 + 0.880666i $$0.342905\pi$$
−0.473738 + 0.880666i $$0.657095\pi$$
$$462$$ 0 0
$$463$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$464$$ 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
$$470$$ − 20684.2i − 2.02998i
$$471$$ 0 0
$$472$$ −7356.03 −0.717349
$$473$$ 9256.04i 0.899774i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 21425.8 2.05019
$$479$$ 5200.84i 0.496101i 0.968747 + 0.248050i $$0.0797898\pi$$
−0.968747 + 0.248050i $$0.920210\pi$$
$$480$$ 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 1.00000 $$0$$
−1.00000 $$\pi$$
$$488$$ 11991.1i 1.11232i
$$489$$ 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 1.00000 $$0$$
−1.00000 $$\pi$$
$$500$$ − 6685.70i − 0.597988i
$$501$$ 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
$$509$$ 22966.8i 1.99997i 0.00508931 + 0.999987i $$0.498380\pi$$
−0.00508931 + 0.999987i $$0.501620\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 6227.90i 0.537572i
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 7778.74i 0.665577i
$$516$$ 0 0
$$517$$ −13115.9 −1.11574
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 5457.75 0.460266
$$521$$ 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
$$533$$ − 9229.02i − 0.750006i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 7023.94i 0.561304i
$$540$$ 0 0
$$541$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$542$$ 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 $$0.112824\pi$$
$$548$$ − 13265.9i − 1.03411i
$$549$$ 0 0
$$550$$ −2945.44 −0.228352
$$551$$ 0 0
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 26876.1i 2.06111i
$$555$$ 0 0
$$556$$ 1494.11 0.113965
$$557$$ − 22258.1i − 1.69319i −0.532237 0.846595i $$-0.678648\pi$$
0.532237 0.846595i $$-0.321352\pi$$
$$558$$ 0 0
$$559$$ −21186.2 −1.60301
$$560$$ 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
$$566$$ 1726.33i 0.128203i
$$567$$ 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 $$-0.118743\pi$$
0.931222 + 0.364451i $$0.118743\pi$$
$$572$$ 4218.00i 0.308327i
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$578$$ − 17296.6i − 1.24471i
$$579$$ 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
$$587$$ − 19338.0i − 1.35974i −0.733335 0.679868i $$-0.762037\pi$$
0.733335 0.679868i $$-0.237963\pi$$
$$588$$ 0 0
$$589$$ 0 0
$$590$$ − 18714.8i − 1.30589i
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 25958.2i 1.79760i 0.438363 + 0.898798i $$0.355558\pi$$
−0.438363 + 0.898798i $$0.644442\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 10885.5i 0.748131i
$$597$$ 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
$$605$$ 8362.63i 0.561966i
$$606$$ 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
$$611$$ − 30021.2i − 1.98777i
$$612$$ 0 0
$$613$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ − 29859.9i − 1.94832i −0.225860 0.974160i $$-0.572519\pi$$
0.225860 0.974160i $$-0.427481\pi$$
$$618$$ 0 0
$$619$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ −8848.92 −0.566331
$$626$$ 22518.5i 1.43773i
$$627$$ 0 0
$$628$$ 4104.42 0.260803
$$629$$ 0 0
$$630$$ 0 0
$$631$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$632$$ − 5309.03i − 0.334148i
$$633$$ 0 0
$$634$$ 25114.1 1.57320
$$635$$ − 22887.1i − 1.43031i
$$636$$ 0 0
$$637$$ −16077.2 −1.00000
$$638$$ 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 1.00000 $$0$$
−1.00000 $$\pi$$
$$644$$ 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
$$650$$ − 6741.83i − 0.406825i
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ − 15721.2i − 0.935684i
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$660$$ 0 0
$$661$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 1201.37 0.0702144
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 0 0
$$668$$ 10751.0i 0.622710i
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 19344.6i 1.11295i
$$672$$ 0 0
$$673$$ −25522.0 −1.46181 −0.730907 0.682477i $$-0.760903\pi$$
−0.730907 + 0.682477i $$0.760903\pi$$
$$674$$ 36684.5i 2.09649i
$$675$$ 0 0
$$676$$ −9654.60 −0.549306
$$677$$ 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
$$683$$ 32750.6i 1.83480i 0.397968 + 0.917399i $$0.369715\pi$$
−0.397968 + 0.917399i $$0.630285\pi$$
$$684$$ 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 1.00000 $$0$$
−1.00000 $$\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ − 3118.83i − 0.170222i
$$696$$ 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
$$704$$ − 135.941i − 0.00727765i
$$705$$ 0 0
$$706$$ −25811.8 −1.37597
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$710$$ 38433.2i 2.03151i
$$711$$ 0 0
$$712$$ 21224.4 1.11716
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 8804.71 0.460528
$$716$$ 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
$$722$$ 24147.6i 1.24471i
$$723$$ 0 0
$$724$$ −19467.4 −0.999310
$$725$$ 0 0
$$726$$ 0 0
$$727$$ −28455.0 −1.45163 −0.725817 0.687888i $$-0.758538\pi$$
−0.725817 + 0.687888i $$0.758538\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 0 0
$$732$$ 0 0
$$733$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$734$$ − 29206.7i − 1.46872i
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 0 0
$$739$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 19936.4i 0.984382i 0.870487 + 0.492191i $$0.163804\pi$$
−0.870487 + 0.492191i $$0.836196\pi$$
$$744$$ 0 0
$$745$$ 22722.5 1.11743
$$746$$ 22975.4i 1.12760i
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 26120.0 1.26915 0.634575 0.772861i $$-0.281175\pi$$
0.634575 + 0.772861i $$0.281175\pi$$
$$752$$ − 51139.6i − 2.47988i
$$753$$ 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
$$761$$ − 28936.4i − 1.37838i −0.724582 0.689189i $$-0.757967\pi$$
0.724582 0.689189i $$-0.242033\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ 0 0
$$766$$ −44719.1 −2.10936
$$767$$ − 27162.7i − 1.27873i
$$768$$ 0 0
$$769$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ 3484.00i 0.162110i 0.996710 + 0.0810548i $$0.0258289\pi$$
−0.996710 + 0.0810548i $$0.974171\pi$$
$$774$$ 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
$$785$$ − 8567.62i − 0.389543i
$$786$$ 0 0
$$787$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$788$$ − 17478.2i − 0.790144i
$$789$$ 0 0
$$790$$ 13506.9 0.608297
$$791$$ 0 0
$$792$$ 0 0
$$793$$ −44278.0 −1.98280
$$794$$ 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
$$800$$ − 7335.55i − 0.324189i
$$801$$ 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 1.00000 $$0$$
−1.00000 $$\pi$$
$$812$$ 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
$$821$$ − 33048.4i − 1.40487i −0.711749 0.702433i $$-0.752097\pi$$
0.711749 0.702433i $$-0.247903\pi$$
$$822$$ 0 0
$$823$$ 37352.0 1.58203 0.791014 0.611798i $$-0.209554\pi$$
0.791014 + 0.611798i $$0.209554\pi$$
$$824$$ 10764.2i 0.455083i
$$825$$ 0 0
$$826$$ 0 0
$$827$$ − 31167.2i − 1.31051i −0.755409 0.655253i $$-0.772562\pi$$
0.755409 0.655253i $$-0.227438\pi$$
$$828$$ 0 0
$$829$$ −3079.14 −0.129002 −0.0645012 0.997918i $$-0.520546\pi$$
−0.0645012 + 0.997918i $$0.520546\pi$$
$$830$$ 3056.47i 0.127821i
$$831$$ 0 0
$$832$$ 311.156 0.0129656
$$833$$ 0 0
$$834$$ 0 0
$$835$$ 22441.9 0.930101
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ 46872.5i 1.92875i 0.264543 + 0.964374i $$0.414779\pi$$
−0.264543 + 0.964374i $$0.585221\pi$$
$$840$$ 0 0
$$841$$ −24389.0 −1.00000
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 25224.3 1.02874
$$845$$ 20153.2i 0.820462i
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 0 0
$$852$$ 0 0
$$853$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$854$$ 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 $$0.170200\pi$$
$$860$$ − 18220.3i − 0.722451i
$$861$$ 0 0
$$862$$ 1127.66 0.0445571
$$863$$ 23594.8i 0.930678i 0.885133 + 0.465339i $$0.154068\pi$$
−0.885133 + 0.465339i $$0.845932\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 126.936i 0.00498091i
$$867$$ 0 0
$$868$$ 0 0
$$869$$ − 8564.79i − 0.334339i
$$870$$ 0 0
$$871$$ 0 0
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$878$$ − 39852.9i − 1.53186i
$$879$$ 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 $$-0.0160811\pi$$
0.998724 + 0.0504988i $$0.0160811\pi$$
$$884$$ 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
$$890$$ 53998.0i 2.03373i
$$891$$ 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
$$902$$ − 14195.2i − 0.524000i
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 40636.6i 1.49260i
$$906$$ 0 0
$$907$$ 35017.1 1.28195 0.640973 0.767564i $$-0.278531\pi$$
0.640973 + 0.767564i $$0.278531\pi$$
$$908$$ − 29305.0i − 1.07106i
$$909$$ 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
$$923$$ 55782.1i 1.98926i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ 7352.53i 0.259665i 0.991536 + 0.129832i $$0.0414439\pi$$
−0.991536 + 0.129832i $$0.958556\pi$$
$$930$$ 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
$$941$$ 15462.1i 0.535652i 0.963467 + 0.267826i $$0.0863052\pi$$
−0.963467 + 0.267826i $$0.913695\pi$$
$$942$$ 0 0
$$943$$ 0 0
$$944$$ − 46270.3i − 1.59531i
$$945$$ 0 0
$$946$$ −32586.6 −1.11996
$$947$$ 56498.4i 1.93870i 0.245678 + 0.969352i $$0.420989\pi$$
−0.245678 + 0.969352i $$0.579011\pi$$
$$948$$ 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
$$956$$ 26744.1i 0.904775i
$$957$$ 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 1.00000 $$0$$
−1.00000 $$\pi$$
$$968$$ 11572.2i 0.384239i
$$969$$ 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
$$977$$ 52089.2i 1.70571i 0.522146 + 0.852856i $$0.325132\pi$$
−0.522146 + 0.852856i $$0.674868\pi$$
$$978$$ 0 0
$$979$$ 34240.3 1.11780
$$980$$ − 13826.5i − 0.450685i
$$981$$ 0 0
$$982$$ 0 0
$$983$$ − 60639.0i − 1.96753i −0.179454 0.983766i $$-0.557433\pi$$
0.179454 0.983766i $$-0.442567\pi$$
$$984$$ 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
$$995$$ − 51462.9i − 1.63968i
$$996$$ 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 117.4.b.c.64.3 yes 4
3.2 odd 2 inner 117.4.b.c.64.2 4
13.5 odd 4 1521.4.a.y.1.3 4
13.8 odd 4 1521.4.a.y.1.2 4
13.12 even 2 inner 117.4.b.c.64.2 4
39.5 even 4 1521.4.a.y.1.2 4
39.8 even 4 1521.4.a.y.1.3 4
39.38 odd 2 CM 117.4.b.c.64.3 yes 4

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