# Properties

 Label 117.4.b.c.64.1 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 64.1 Root $$-0.835000i$$ of defining polynomial Character $$\chi$$ $$=$$ 117.64 Dual form 117.4.b.c.64.4

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-4.42782i q^{2} -11.6056 q^{4} +20.3925i q^{5} +15.9647i q^{8} +O(q^{10})$$ $$q-4.42782i q^{2} -11.6056 q^{4} +20.3925i q^{5} +15.9647i q^{8} +90.2944 q^{10} +70.0332i q^{11} +46.8722 q^{13} -22.1556 q^{16} -236.667i q^{20} +310.094 q^{22} -290.855 q^{25} -207.541i q^{26} +225.819i q^{32} -325.561 q^{40} +486.739i q^{41} +452.000 q^{43} -812.774i q^{44} -71.1653i q^{47} +343.000 q^{49} +1287.85i q^{50} -543.977 q^{52} -1428.15 q^{55} -696.914i q^{59} -944.654 q^{61} +822.638 q^{64} +955.842i q^{65} +123.807i q^{71} +418.244 q^{79} -451.809i q^{80} +2155.19 q^{82} -1509.37i q^{83} -2001.37i q^{86} -1118.06 q^{88} +155.245i q^{89} -315.107 q^{94} -1518.74i q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 32 q^{4}+O(q^{10})$$ 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ − 4.42782i − 1.56547i −0.622356 0.782735i $$-0.713824\pi$$
0.622356 0.782735i $$-0.286176\pi$$
$$3$$ 0 0
$$4$$ −11.6056 −1.45069
$$5$$ 20.3925i 1.82396i 0.410231 + 0.911982i $$0.365448\pi$$
−0.410231 + 0.911982i $$0.634552\pi$$
$$6$$ 0 0
$$7$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$8$$ 15.9647i 0.705547i
$$9$$ 0 0
$$10$$ 90.2944 2.85536
$$11$$ 70.0332i 1.91962i 0.280652 + 0.959810i $$0.409449\pi$$
−0.280652 + 0.959810i $$0.590551\pi$$
$$12$$ 0 0
$$13$$ 46.8722 1.00000
$$14$$ 0 0
$$15$$ 0 0
$$16$$ −22.1556 −0.346181
$$17$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$18$$ 0 0
$$19$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$20$$ − 236.667i − 2.64601i
$$21$$ 0 0
$$22$$ 310.094 3.00510
$$23$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$24$$ 0 0
$$25$$ −290.855 −2.32684
$$26$$ − 207.541i − 1.56547i
$$27$$ 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 $$\pi$$
$$32$$ 225.819i 1.24748i
$$33$$ 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
$$40$$ −325.561 −1.28689
$$41$$ 486.739i 1.85405i 0.375003 + 0.927024i $$0.377642\pi$$
−0.375003 + 0.927024i $$0.622358\pi$$
$$42$$ 0 0
$$43$$ 452.000 1.60301 0.801504 0.597989i $$-0.204033\pi$$
0.801504 + 0.597989i $$0.204033\pi$$
$$44$$ − 812.774i − 2.78478i
$$45$$ 0 0
$$46$$ 0 0
$$47$$ − 71.1653i − 0.220862i −0.993884 0.110431i $$-0.964777\pi$$
0.993884 0.110431i $$-0.0352231\pi$$
$$48$$ 0 0
$$49$$ 343.000 1.00000
$$50$$ 1287.85i 3.64260i
$$51$$ 0 0
$$52$$ −543.977 −1.45069
$$53$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$54$$ 0 0
$$55$$ −1428.15 −3.50132
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ − 696.914i − 1.53780i −0.639367 0.768902i $$-0.720803\pi$$
0.639367 0.768902i $$-0.279197\pi$$
$$60$$ 0 0
$$61$$ −944.654 −1.98280 −0.991398 0.130879i $$-0.958220\pi$$
−0.991398 + 0.130879i $$0.958220\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 822.638 1.60672
$$65$$ 955.842i 1.82396i
$$66$$ 0 0
$$67$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 123.807i 0.206947i 0.994632 + 0.103474i $$0.0329957\pi$$
−0.994632 + 0.103474i $$0.967004\pi$$
$$72$$ 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 $$-0.403739\pi$$
0.297824 + 0.954621i $$0.403739\pi$$
$$80$$ − 451.809i − 0.631422i
$$81$$ 0 0
$$82$$ 2155.19 2.90245
$$83$$ − 1509.37i − 1.99608i −0.0625815 0.998040i $$-0.519933\pi$$
0.0625815 0.998040i $$-0.480067\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ − 2001.37i − 2.50946i
$$87$$ 0 0
$$88$$ −1118.06 −1.35438
$$89$$ 155.245i 0.184899i 0.995717 + 0.0924493i $$0.0294696\pi$$
−0.995717 + 0.0924493i $$0.970530\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 0 0
$$94$$ −315.107 −0.345753
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$98$$ − 1518.74i − 1.56547i
$$99$$ 0 0
$$100$$ 3375.54 3.37554
$$101$$ 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 $$0.367059\pi$$
$$104$$ 748.301i 0.705547i
$$105$$ 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 $$\pi$$
$$110$$ 6323.61i 5.48120i
$$111$$ 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
$$118$$ −3085.80 −2.40738
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −3573.65 −2.68494
$$122$$ 4182.76i 3.10401i
$$123$$ 0 0
$$124$$ 0 0
$$125$$ − 3382.21i − 2.42011i
$$126$$ 0 0
$$127$$ 2495.04 1.74330 0.871650 0.490129i $$-0.163050\pi$$
0.871650 + 0.490129i $$0.163050\pi$$
$$128$$ − 1835.94i − 1.26778i
$$129$$ 0 0
$$130$$ 4232.29 2.85536
$$131$$ 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
$$137$$ − 1082.73i − 0.675208i −0.941288 0.337604i $$-0.890384\pi$$
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
$$142$$ 548.197 0.323969
$$143$$ 3282.61i 1.91962i
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ − 2663.80i − 1.46461i −0.680978 0.732304i $$-0.738445\pi$$
0.680978 0.732304i $$-0.261555\pi$$
$$150$$ 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 $$0.576293\pi$$
$$158$$ − 1851.91i − 0.932467i
$$159$$ 0 0
$$160$$ −4605.01 −2.27536
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$164$$ − 5648.88i − 2.68966i
$$165$$ 0 0
$$166$$ −6683.20 −3.12480
$$167$$ 3555.90i 1.64769i 0.566815 + 0.823845i $$0.308175\pi$$
−0.566815 + 0.823845i $$0.691825\pi$$
$$168$$ 0 0
$$169$$ 2197.00 1.00000
$$170$$ 0 0
$$171$$ 0 0
$$172$$ −5245.71 −2.32547
$$173$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ − 1551.63i − 0.664536i
$$177$$ 0 0
$$178$$ 687.398 0.289453
$$179$$ 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
$$188$$ 825.912i 0.320403i
$$189$$ 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
$$196$$ −3980.70 −1.45069
$$197$$ 3842.18i 1.38956i 0.719220 + 0.694782i $$0.244499\pi$$
−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 $$0.0123760\pi$$
$$200$$ − 4643.42i − 1.64170i
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ −9925.85 −3.38171
$$206$$ − 3754.79i − 1.26994i
$$207$$ 0 0
$$208$$ −1038.48 −0.346181
$$209$$ 0 0
$$210$$ 0 0
$$211$$ 5740.04 1.87280 0.936399 0.350937i $$-0.114137\pi$$
0.936399 + 0.350937i $$0.114137\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 9217.42i 2.92383i
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 16574.5 5.07934
$$221$$ 0 0
$$222$$ 0 0
$$223$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ − 1522.31i − 0.445107i −0.974920 0.222554i $$-0.928561\pi$$
0.974920 0.222554i $$-0.0714393\pi$$
$$228$$ 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
$$235$$ 1451.24 0.402845
$$236$$ 8088.07i 2.23088i
$$237$$ 0 0
$$238$$ 0 0
$$239$$ − 4191.63i − 1.13445i −0.823562 0.567226i $$-0.808016\pi$$
0.823562 0.567226i $$-0.191984\pi$$
$$240$$ 0 0
$$241$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$242$$ 15823.5i 4.20319i
$$243$$ 0 0
$$244$$ 10963.2 2.87643
$$245$$ 6994.64i 1.82396i
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 0 0
$$249$$ 0 0
$$250$$ −14975.8 −3.78861
$$251$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ − 11047.6i − 2.72908i
$$255$$ 0 0
$$256$$ −1548.11 −0.377956
$$257$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ − 11093.1i − 2.64601i
$$261$$ 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
$$274$$ −4794.11 −1.05702
$$275$$ − 20369.5i − 4.46665i
$$276$$ 0 0
$$277$$ 7634.00 1.65589 0.827947 0.560806i $$-0.189509\pi$$
0.827947 + 0.560806i $$0.189509\pi$$
$$278$$ 1505.46i 0.324789i
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 4539.33i 0.963678i 0.876260 + 0.481839i $$0.160031\pi$$
−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 $$0.516400\pi$$
$$284$$ − 1436.85i − 0.300217i
$$285$$ 0 0
$$286$$ 14534.8 3.00510
$$287$$ 0 0
$$288$$ 0 0
$$289$$ −4913.00 −1.00000
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ − 1396.60i − 0.278465i −0.990260 0.139232i $$-0.955537\pi$$
0.990260 0.139232i $$-0.0444635\pi$$
$$294$$ 0 0
$$295$$ 14211.8 2.80490
$$296$$ 0 0
$$297$$ 0 0
$$298$$ −11794.8 −2.29280
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ − 19263.9i − 3.61655i
$$306$$ 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 $$-0.695986\pi$$
−0.577536 + 0.816365i $$0.695986\pi$$
$$314$$ 4135.58i 0.743262i
$$315$$ 0 0
$$316$$ −4853.95 −0.864102
$$317$$ 8748.31i 1.55001i 0.631954 + 0.775006i $$0.282253\pi$$
−0.631954 + 0.775006i $$0.717747\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 16775.7i 2.93059i
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 0 0
$$324$$ 0 0
$$325$$ −13633.0 −2.32684
$$326$$ 0 0
$$327$$ 0 0
$$328$$ −7770.66 −1.30812
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$332$$ 17517.0i 2.89570i
$$333$$ 0 0
$$334$$ 15744.9 2.57941
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −10420.0 −1.68432 −0.842160 0.539228i $$-0.818716\pi$$
−0.842160 + 0.539228i $$0.818716\pi$$
$$338$$ − 9727.91i − 1.56547i
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 7216.05i 1.13100i
$$345$$ 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
$$352$$ −15814.8 −2.39469
$$353$$ 11054.2i 1.66672i 0.552728 + 0.833361i $$0.313587\pi$$
−0.552728 + 0.833361i $$0.686413\pi$$
$$354$$ 0 0
$$355$$ −2524.75 −0.377464
$$356$$ − 1801.71i − 0.268231i
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 7897.23i 1.16100i 0.814259 + 0.580502i $$0.197143\pi$$
−0.814259 + 0.580502i $$0.802857\pi$$
$$360$$ 0 0
$$361$$ 6859.00 1.00000
$$362$$ − 19615.2i − 2.84794i
$$363$$ 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 $$-0.649631\pi$$
−0.452957 + 0.891532i $$0.649631\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 1136.13 0.155829
$$377$$ 0 0
$$378$$ 0 0
$$379$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ − 7961.24i − 1.06214i −0.847327 0.531071i $$-0.821790\pi$$
0.847327 0.531071i $$-0.178210\pi$$
$$384$$ 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
$$392$$ 5475.90i 0.705547i
$$393$$ 0 0
$$394$$ 17012.5 2.17532
$$395$$ 8529.05i 1.08644i
$$396$$ 0 0
$$397$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$398$$ − 24841.1i − 3.12857i
$$399$$ 0 0
$$400$$ 6444.07 0.805509
$$401$$ − 4746.53i − 0.591098i −0.955328 0.295549i $$-0.904497\pi$$
0.955328 0.295549i $$-0.0955026\pi$$
$$402$$ 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 $$\pi$$
$$410$$ 43949.8i 5.29397i
$$411$$ 0 0
$$412$$ −9841.51 −1.17684
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 30779.8 3.64078
$$416$$ 10584.6i 1.24748i
$$417$$ 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 $$\pi$$
$$422$$ − 25415.8i − 2.93181i
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 40813.1 4.57716
$$431$$ − 17892.7i − 1.99968i −0.0178985 0.999840i $$-0.505698\pi$$
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 $$-0.710984\pi$$
−0.615346 + 0.788257i $$0.710984\pi$$
$$440$$ − 22800.1i − 2.47034i
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$444$$ 0 0
$$445$$ −3165.84 −0.337248
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ − 19025.4i − 1.99969i −0.0175361 0.999846i $$-0.505582\pi$$
0.0175361 0.999846i $$-0.494418\pi$$
$$450$$ 0 0
$$451$$ −34087.9 −3.55906
$$452$$ 0 0
$$453$$ 0 0
$$454$$ −6740.52 −0.696802
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ − 9378.19i − 0.947475i −0.880666 0.473738i $$-0.842905\pi$$
0.880666 0.473738i $$-0.157095\pi$$
$$462$$ 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
$$470$$ − 6425.82i − 0.630641i
$$471$$ 0 0
$$472$$ 11126.0 1.08499
$$473$$ 31655.0i 3.07717i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 0 0
$$478$$ −18559.8 −1.77595
$$479$$ 20311.6i 1.93749i 0.248050 + 0.968747i $$0.420210\pi$$
−0.248050 + 0.968747i $$0.579790\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 41474.2 3.89502
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$488$$ − 15081.1i − 1.39896i
$$489$$ 0 0
$$490$$ 30971.0 2.85536
$$491$$ 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 $$\pi$$
$$500$$ 39252.4i 3.51084i
$$501$$ 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
$$508$$ −28956.3 −2.52900
$$509$$ − 116.887i − 0.0101786i −0.999987 0.00508931i $$-0.998380\pi$$
0.999987 0.00508931i $$-0.00161998\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ − 7832.81i − 0.676102i
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 17292.9i 1.47964i
$$516$$ 0 0
$$517$$ 4983.93 0.423971
$$518$$ 0 0
$$519$$ 0 0
$$520$$ −15259.7 −1.28689
$$521$$ 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
$$533$$ 22814.5i 1.85405i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 24021.4i 1.91962i
$$540$$ 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 $$0.112824\pi$$
$$548$$ 12565.6i 0.979520i
$$549$$ 0 0
$$550$$ −90192.6 −6.99241
$$551$$ 0 0
$$552$$ 0 0
$$553$$ 0 0
$$554$$ − 33801.9i − 2.59225i
$$555$$ 0 0
$$556$$ 3945.89 0.300976
$$557$$ − 13993.2i − 1.06447i −0.846595 0.532237i $$-0.821352\pi$$
0.846595 0.532237i $$-0.178648\pi$$
$$558$$ 0 0
$$559$$ 21186.2 1.60301
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 20099.3 1.50861
$$563$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 2171.20i 0.161241i
$$567$$ 0 0
$$568$$ −1976.55 −0.146011
$$569$$ 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 $$0.881257\pi$$
$$572$$ − 38096.5i − 2.78478i
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$578$$ 21753.9i 1.56547i
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 0 0
$$584$$ 0 0
$$585$$ 0 0
$$586$$ −6183.88 −0.435928
$$587$$ − 20858.8i − 1.46667i −0.679868 0.733335i $$-0.737963\pi$$
0.679868 0.733335i $$-0.262037\pi$$
$$588$$ 0 0
$$589$$ 0 0
$$590$$ − 62927.4i − 4.39098i
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 12660.4i 0.876726i 0.898798 + 0.438363i $$0.144442\pi$$
−0.898798 + 0.438363i $$0.855558\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 30914.8i 2.12470i
$$597$$ 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 $$-0.640956\pi$$
−0.428495 + 0.903544i $$0.640956\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ − 72875.8i − 4.89723i
$$606$$ 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
$$610$$ −85297.0 −5.66160
$$611$$ − 3335.67i − 0.220862i
$$612$$ 0 0
$$613$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ − 6923.04i − 0.451719i −0.974160 0.225860i $$-0.927481\pi$$
0.974160 0.225860i $$-0.0725191\pi$$
$$618$$ 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
$$625$$ 32614.9 2.08735
$$626$$ 28321.4i 1.80823i
$$627$$ 0 0
$$628$$ 10839.6 0.688768
$$629$$ 0 0
$$630$$ 0 0
$$631$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$632$$ 6677.15i 0.420257i
$$633$$ 0 0
$$634$$ 38735.9 2.42650
$$635$$ 50880.2i 3.17972i
$$636$$ 0 0
$$637$$ 16077.2 1.00000
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 37439.5 2.31239
$$641$$ 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
$$649$$ 48807.1 2.95200
$$650$$ 60364.5i 3.64260i
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ − 10784.0i − 0.641836i
$$657$$ 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
$$664$$ 24096.6 1.40833
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 0 0
$$668$$ − 41268.2i − 2.39029i
$$669$$ 0 0
$$670$$ 0 0
$$671$$ − 66157.2i − 3.80622i
$$672$$ 0 0
$$673$$ −25522.0 −1.46181 −0.730907 0.682477i $$-0.760903\pi$$
−0.730907 + 0.682477i $$0.760903\pi$$
$$674$$ 46138.0i 2.63675i
$$675$$ 0 0
$$676$$ −25497.4 −1.45069
$$677$$ 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
$$683$$ 14207.2i 0.795936i 0.917399 + 0.397968i $$0.130285\pi$$
−0.917399 + 0.397968i $$0.869715\pi$$
$$684$$ 0 0
$$685$$ 22079.5 1.23155
$$686$$ 0 0
$$687$$ 0 0
$$688$$ −10014.3 −0.554931
$$689$$ 0 0
$$690$$ 0 0
$$691$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ − 6933.46i − 0.378419i
$$696$$ 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
$$704$$ 57612.0i 3.08428i
$$705$$ 0 0
$$706$$ 48945.8 2.60920
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$710$$ 11179.1i 0.590908i
$$711$$ 0 0
$$712$$ −2478.45 −0.130455
$$713$$ 0 0
$$714$$ 0 0
$$715$$ −66940.7 −3.50132
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 34967.5 1.81751
$$719$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ − 30370.4i − 1.56547i
$$723$$ 0 0
$$724$$ −51412.6 −2.63914
$$725$$ 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 1.00000 $$0$$
−1.00000 $$\pi$$
$$734$$ 36733.2i 1.84720i
$$735$$ 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
$$743$$ − 35259.5i − 1.74097i −0.492191 0.870487i $$-0.663804\pi$$
0.492191 0.870487i $$-0.336196\pi$$
$$744$$ 0 0
$$745$$ 54321.5 2.67139
$$746$$ 28896.1i 1.41818i
$$747$$ 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 $$0.281175\pi$$
$$752$$ 1576.71i 0.0764583i
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 35009.9 1.68092 0.840460 0.541873i $$-0.182285\pi$$
0.840460 + 0.541873i $$0.182285\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 30422.5i 1.44916i 0.689189 + 0.724582i $$0.257967\pi$$
−0.689189 + 0.724582i $$0.742033\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ 0 0
$$766$$ −35250.9 −1.66275
$$767$$ − 32665.9i − 1.53780i
$$768$$ 0 0
$$769$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ 42841.8i 1.99342i 0.0810548 + 0.996710i $$0.474171\pi$$
−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
$$781$$ −8670.64 −0.397260
$$782$$ 0 0
$$783$$ 0 0
$$784$$ −7599.37 −0.346181
$$785$$ − 19046.6i − 0.865991i
$$786$$ 0 0
$$787$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$788$$ − 44590.6i − 2.01583i
$$789$$ 0 0
$$790$$ 37765.1 1.70079
$$791$$ 0 0
$$792$$ 0 0
$$793$$ −44278.0 −1.98280
$$794$$ 0 0
$$795$$ 0 0
$$796$$ −65109.9 −2.89920
$$797$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ − 65680.5i − 2.90270i
$$801$$ 0 0
$$802$$ −21016.8 −0.925346
$$803$$ 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
$$820$$ 115195. 4.90583
$$821$$ 33486.7i 1.42350i 0.702433 + 0.711749i $$0.252097\pi$$
−0.702433 + 0.711749i $$0.747903\pi$$
$$822$$ 0 0
$$823$$ 37352.0 1.58203 0.791014 0.611798i $$-0.209554\pi$$
0.791014 + 0.611798i $$0.209554\pi$$
$$824$$ 13538.1i 0.572356i
$$825$$ 0 0
$$826$$ 0 0
$$827$$ − 35931.1i − 1.51082i −0.655253 0.755409i $$-0.727438\pi$$
0.655253 0.755409i $$-0.272562\pi$$
$$828$$ 0 0
$$829$$ 3079.14 0.129002 0.0645012 0.997918i $$-0.479454\pi$$
0.0645012 + 0.997918i $$0.479454\pi$$
$$830$$ − 136287.i − 5.69952i
$$831$$ 0 0
$$832$$ 38558.8 1.60672
$$833$$ 0 0
$$834$$ 0 0
$$835$$ −72513.9 −3.00533
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ 12857.9i 0.529086i 0.964374 + 0.264543i $$0.0852210\pi$$
−0.964374 + 0.264543i $$0.914779\pi$$
$$840$$ 0 0
$$841$$ −24389.0 −1.00000
$$842$$ 0 0
$$843$$ 0 0
$$844$$ −66616.3 −2.71686
$$845$$ 44802.4i 1.82396i
$$846$$ 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 $$-0.829800\pi$$
−0.860423 + 0.509581i $$0.829800\pi$$
$$860$$ − 106973.i − 4.24158i
$$861$$ 0 0
$$862$$ −79225.7 −3.13044
$$863$$ 44880.2i 1.77027i 0.465339 + 0.885133i $$0.345932\pi$$
−0.465339 + 0.885133i $$0.654068\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 159.647i 0.00626447i
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 29291.0i 1.14342i
$$870$$ 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 $$\pi$$
$$878$$ 50122.9i 1.92661i
$$879$$ 0 0
$$880$$ 31641.6 1.21209
$$881$$ 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
$$890$$ 14017.8i 0.527952i
$$891$$ 0 0
$$892$$ 0 0
$$893$$ 0 0
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 0 0
$$898$$ −84240.8 −3.13046
$$899$$ 0 0
$$900$$ 0 0
$$901$$ 0 0
$$902$$ 150935.i 5.57161i
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 90338.9i 3.31820i
$$906$$ 0 0
$$907$$ −35017.1 −1.28195 −0.640973 0.767564i $$-0.721469\pi$$
−0.640973 + 0.767564i $$0.721469\pi$$
$$908$$ 17667.3i 0.645715i
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$912$$ 0 0
$$913$$ 105706. 3.83171
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ 36762.2 1.31956 0.659779 0.751460i $$-0.270650\pi$$
0.659779 + 0.751460i $$0.270650\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ −41524.9 −1.48324
$$923$$ 5803.12i 0.206947i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ − 56151.6i − 1.98307i −0.129832 0.991536i $$-0.541444\pi$$
0.129832 0.991536i $$-0.458556\pi$$
$$930$$ 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 $$-0.574340\pi$$
−0.231428 + 0.972852i $$0.574340\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ −16842.4 −0.584404
$$941$$ − 55622.6i − 1.92693i −0.267826 0.963467i $$-0.586305\pi$$
0.267826 0.963467i $$-0.413695\pi$$
$$942$$ 0 0
$$943$$ 0 0
$$944$$ 15440.5i 0.532359i
$$945$$ 0 0
$$946$$ 140163. 4.81721
$$947$$ − 14319.3i − 0.491356i −0.969352 0.245678i $$-0.920989\pi$$
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
$$956$$ 48646.2i 1.64574i
$$957$$ 0 0
$$958$$ 89935.9 3.03309
$$959$$ 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 $$\pi$$
$$968$$ − 57052.4i − 1.89435i
$$969$$ 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
$$976$$ 20929.4 0.686407
$$977$$ − 31890.7i − 1.04429i −0.852856 0.522146i $$-0.825132\pi$$
0.852856 0.522146i $$-0.174868\pi$$
$$978$$ 0 0
$$979$$ −10872.3 −0.354935
$$980$$ − 81176.6i − 2.64601i
$$981$$ 0 0
$$982$$ 0 0
$$983$$ − 11061.5i − 0.358908i −0.983766 0.179454i $$-0.942567\pi$$
0.983766 0.179454i $$-0.0574331\pi$$
$$984$$ 0 0
$$985$$ −78351.8 −2.53451
$$986$$ 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
$$995$$ 114407.i 3.64517i
$$996$$ 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.64.1 4
3.2 odd 2 inner 117.4.b.c.64.4 yes 4
13.5 odd 4 1521.4.a.y.1.1 4
13.8 odd 4 1521.4.a.y.1.4 4
13.12 even 2 inner 117.4.b.c.64.4 yes 4
39.5 even 4 1521.4.a.y.1.4 4
39.8 even 4 1521.4.a.y.1.1 4
39.38 odd 2 CM 117.4.b.c.64.1 4

By twisted newform
Twist Min Dim Char Parity Ord Type
117.4.b.c.64.1 4 1.1 even 1 trivial
117.4.b.c.64.1 4 39.38 odd 2 CM
117.4.b.c.64.4 yes 4 3.2 odd 2 inner
117.4.b.c.64.4 yes 4 13.12 even 2 inner
1521.4.a.y.1.1 4 13.5 odd 4
1521.4.a.y.1.1 4 39.8 even 4
1521.4.a.y.1.4 4 13.8 odd 4
1521.4.a.y.1.4 4 39.5 even 4