# Properties

 Label 1521.4.a.y.1.4 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 1.4 Root $$-0.835000$$ 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)

 $$f(q)$$ $$=$$ $$q+4.42782 q^{2} +11.6056 q^{4} -20.3925 q^{5} +15.9647 q^{8} +O(q^{10})$$ $$q+4.42782 q^{2} +11.6056 q^{4} -20.3925 q^{5} +15.9647 q^{8} -90.2944 q^{10} +70.0332 q^{11} -22.1556 q^{16} -236.667 q^{20} +310.094 q^{22} +290.855 q^{25} -225.819 q^{32} -325.561 q^{40} -486.739 q^{41} -452.000 q^{43} +812.774 q^{44} -71.1653 q^{47} -343.000 q^{49} +1287.85 q^{50} -1428.15 q^{55} -696.914 q^{59} -944.654 q^{61} -822.638 q^{64} -123.807 q^{71} +418.244 q^{79} +451.809 q^{80} -2155.19 q^{82} +1509.37 q^{83} -2001.37 q^{86} +1118.06 q^{88} +155.245 q^{89} -315.107 q^{94} -1518.74 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 32 q^{4}+O(q^{10})$$ 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 4.42782 1.56547 0.782735 0.622356i $$-0.213824\pi$$
0.782735 + 0.622356i $$0.213824\pi$$
$$3$$ 0 0
$$4$$ 11.6056 1.45069
$$5$$ −20.3925 −1.82396 −0.911982 0.410231i $$-0.865448\pi$$
−0.911982 + 0.410231i $$0.865448\pi$$
$$6$$ 0 0
$$7$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$8$$ 15.9647 0.705547
$$9$$ 0 0
$$10$$ −90.2944 −2.85536
$$11$$ 70.0332 1.91962 0.959810 0.280652i $$-0.0905506\pi$$
0.959810 + 0.280652i $$0.0905506\pi$$
$$12$$ 0 0
$$13$$ 0 0
$$14$$ 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 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$20$$ −236.667 −2.64601
$$21$$ 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
$$26$$ 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 $$0.5\pi$$
$$32$$ −225.819 −1.24748
$$33$$ 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
$$40$$ −325.561 −1.28689
$$41$$ −486.739 −1.85405 −0.927024 0.375003i $$-0.877642\pi$$
−0.927024 + 0.375003i $$0.877642\pi$$
$$42$$ 0 0
$$43$$ −452.000 −1.60301 −0.801504 0.597989i $$-0.795967\pi$$
−0.801504 + 0.597989i $$0.795967\pi$$
$$44$$ 812.774 2.78478
$$45$$ 0 0
$$46$$ 0 0
$$47$$ −71.1653 −0.220862 −0.110431 0.993884i $$-0.535223\pi$$
−0.110431 + 0.993884i $$0.535223\pi$$
$$48$$ 0 0
$$49$$ −343.000 −1.00000
$$50$$ 1287.85 3.64260
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 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
$$59$$ −696.914 −1.53780 −0.768902 0.639367i $$-0.779197\pi$$
−0.768902 + 0.639367i $$0.779197\pi$$
$$60$$ 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
$$65$$ 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
$$71$$ −123.807 −0.206947 −0.103474 0.994632i $$-0.532996\pi$$
−0.103474 + 0.994632i $$0.532996\pi$$
$$72$$ 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 $$-0.403739\pi$$
0.297824 + 0.954621i $$0.403739\pi$$
$$80$$ 451.809 0.631422
$$81$$ 0 0
$$82$$ −2155.19 −2.90245
$$83$$ 1509.37 1.99608 0.998040 0.0625815i $$-0.0199333\pi$$
0.998040 + 0.0625815i $$0.0199333\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ −2001.37 −2.50946
$$87$$ 0 0
$$88$$ 1118.06 1.35438
$$89$$ 155.245 0.184899 0.0924493 0.995717i $$-0.470530\pi$$
0.0924493 + 0.995717i $$0.470530\pi$$
$$90$$ 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 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$98$$ −1518.74 −1.56547
$$99$$ 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 $$-0.632941\pi$$
−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 $$0.5\pi$$
$$110$$ −6323.61 −5.48120
$$111$$ 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
$$122$$ −4182.76 −3.10401
$$123$$ 0 0
$$124$$ 0 0
$$125$$ −3382.21 −2.42011
$$126$$ 0 0
$$127$$ −2495.04 −1.74330 −0.871650 0.490129i $$-0.836950\pi$$
−0.871650 + 0.490129i $$0.836950\pi$$
$$128$$ −1835.94 −1.26778
$$129$$ 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
$$137$$ −1082.73 −0.675208 −0.337604 0.941288i $$-0.609616\pi$$
−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
$$142$$ −548.197 −0.323969
$$143$$ 0 0
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 2663.80 1.46461 0.732304 0.680978i $$-0.238445\pi$$
0.732304 + 0.680978i $$0.238445\pi$$
$$150$$ 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 $$0.576293\pi$$
$$158$$ 1851.91 0.932467
$$159$$ 0 0
$$160$$ 4605.01 2.27536
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$164$$ −5648.88 −2.68966
$$165$$ 0 0
$$166$$ 6683.20 3.12480
$$167$$ 3555.90 1.64769 0.823845 0.566815i $$-0.191825\pi$$
0.823845 + 0.566815i $$0.191825\pi$$
$$168$$ 0 0
$$169$$ 0 0
$$170$$ 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
$$176$$ −1551.63 −0.664536
$$177$$ 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 $$-0.863621\pi$$
−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
$$188$$ −825.912 −0.320403
$$189$$ 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
$$196$$ −3980.70 −1.45069
$$197$$ −3842.18 −1.38956 −0.694782 0.719220i $$-0.744499\pi$$
−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 $$0.987624\pi$$
$$200$$ 4643.42 1.64170
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 9925.85 3.38171
$$206$$ −3754.79 −1.26994
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 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
$$215$$ 9217.42 2.92383
$$216$$ 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 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 1522.31 0.445107 0.222554 0.974920i $$-0.428561\pi$$
0.222554 + 0.974920i $$0.428561\pi$$
$$228$$ 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
$$235$$ 1451.24 0.402845
$$236$$ −8088.07 −2.23088
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 4191.63 1.13445 0.567226 0.823562i $$-0.308016\pi$$
0.567226 + 0.823562i $$0.308016\pi$$
$$240$$ 0 0
$$241$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$242$$ 15823.5 4.20319
$$243$$ 0 0
$$244$$ −10963.2 −2.87643
$$245$$ 6994.64 1.82396
$$246$$ 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
$$254$$ −11047.6 −2.72908
$$255$$ 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
$$260$$ 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
$$274$$ −4794.11 −1.05702
$$275$$ 20369.5 4.46665
$$276$$ 0 0
$$277$$ −7634.00 −1.65589 −0.827947 0.560806i $$-0.810491\pi$$
−0.827947 + 0.560806i $$0.810491\pi$$
$$278$$ −1505.46 −0.324789
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 4539.33 0.963678 0.481839 0.876260i $$-0.339969\pi$$
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 $$0.483600\pi$$
$$284$$ −1436.85 −0.300217
$$285$$ 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
$$293$$ −1396.60 −0.278465 −0.139232 0.990260i $$-0.544463\pi$$
−0.139232 + 0.990260i $$0.544463\pi$$
$$294$$ 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
$$305$$ 19263.9 3.61655
$$306$$ 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 $$-0.695986\pi$$
−0.577536 + 0.816365i $$0.695986\pi$$
$$314$$ −4135.58 −0.743262
$$315$$ 0 0
$$316$$ 4853.95 0.864102
$$317$$ −8748.31 −1.55001 −0.775006 0.631954i $$-0.782253\pi$$
−0.775006 + 0.631954i $$0.782253\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 16775.7 2.93059
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 0 0
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 0 0
$$328$$ −7770.66 −1.30812
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$332$$ 17517.0 2.89570
$$333$$ 0 0
$$334$$ 15744.9 2.57941
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 10420.0 1.68432 0.842160 0.539228i $$-0.181284\pi$$
0.842160 + 0.539228i $$0.181284\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ 0 0
$$344$$ −7216.05 −1.13100
$$345$$ 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
$$352$$ −15814.8 −2.39469
$$353$$ −11054.2 −1.66672 −0.833361 0.552728i $$-0.813587\pi$$
−0.833361 + 0.552728i $$0.813587\pi$$
$$354$$ 0 0
$$355$$ 2524.75 0.377464
$$356$$ 1801.71 0.268231
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 7897.23 1.16100 0.580502 0.814259i $$-0.302857\pi$$
0.580502 + 0.814259i $$0.302857\pi$$
$$360$$ 0 0
$$361$$ −6859.00 −1.00000
$$362$$ −19615.2 −2.84794
$$363$$ 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 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ 7961.24 1.06214 0.531071 0.847327i $$-0.321790\pi$$
0.531071 + 0.847327i $$0.321790\pi$$
$$384$$ 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
$$392$$ −5475.90 −0.705547
$$393$$ 0 0
$$394$$ −17012.5 −2.17532
$$395$$ −8529.05 −1.08644
$$396$$ 0 0
$$397$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$398$$ −24841.1 −3.12857
$$399$$ 0 0
$$400$$ −6444.07 −0.805509
$$401$$ −4746.53 −0.591098 −0.295549 0.955328i $$-0.595503\pi$$
−0.295549 + 0.955328i $$0.595503\pi$$
$$402$$ 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 $$0.5\pi$$
$$410$$ 43949.8 5.29397
$$411$$ 0 0
$$412$$ −9841.51 −1.17684
$$413$$ 0 0
$$414$$ 0 0
$$415$$ −30779.8 −3.64078
$$416$$ 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 $$0.5\pi$$
$$422$$ 25415.8 2.93181
$$423$$ 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
$$431$$ 17892.7 1.99968 0.999840 0.0178985i $$-0.00569759\pi$$
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 $$-0.289016\pi$$
0.615346 + 0.788257i $$0.289016\pi$$
$$440$$ −22800.1 −2.47034
$$441$$ 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
$$449$$ −19025.4 −1.99969 −0.999846 0.0175361i $$-0.994418\pi$$
−0.999846 + 0.0175361i $$0.994418\pi$$
$$450$$ 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 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 9378.19 0.947475 0.473738 0.880666i $$-0.342905\pi$$
0.473738 + 0.880666i $$0.342905\pi$$
$$462$$ 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
$$470$$ 6425.82 0.630641
$$471$$ 0 0
$$472$$ −11126.0 −1.08499
$$473$$ −31655.0 −3.07717
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 18559.8 1.77595
$$479$$ 20311.6 1.93749 0.968747 0.248050i $$-0.0797898\pi$$
0.968747 + 0.248050i $$0.0797898\pi$$
$$480$$ 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 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$488$$ −15081.1 −1.39896
$$489$$ 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 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$500$$ −39252.4 −3.51084
$$501$$ 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
$$509$$ 116.887 0.0101786 0.00508931 0.999987i $$-0.498380\pi$$
0.00508931 + 0.999987i $$0.498380\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 7832.81 0.676102
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 17292.9 1.47964
$$516$$ 0 0
$$517$$ −4983.93 −0.423971
$$518$$ 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
$$539$$ −24021.4 −1.91962
$$540$$ 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 $$0.112824\pi$$
$$548$$ −12565.6 −0.979520
$$549$$ 0 0
$$550$$ 90192.6 6.99241
$$551$$ 0 0
$$552$$ 0 0
$$553$$ 0 0
$$554$$ −33801.9 −2.59225
$$555$$ 0 0
$$556$$ −3945.89 −0.300976
$$557$$ −13993.2 −1.06447 −0.532237 0.846595i $$-0.678648\pi$$
−0.532237 + 0.846595i $$0.678648\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ 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
$$566$$ 2171.20 0.161241
$$567$$ 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 $$-0.118743\pi$$
0.931222 + 0.364451i $$0.118743\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$578$$ −21753.9 −1.56547
$$579$$ 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
$$587$$ 20858.8 1.46667 0.733335 0.679868i $$-0.237963\pi$$
0.733335 + 0.679868i $$0.237963\pi$$
$$588$$ 0 0
$$589$$ 0 0
$$590$$ 62927.4 4.39098
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 12660.4 0.876726 0.438363 0.898798i $$-0.355558\pi$$
0.438363 + 0.898798i $$0.355558\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 30914.8 2.12470
$$597$$ 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
$$605$$ −72875.8 −4.89723
$$606$$ 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
$$611$$ 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
$$617$$ 6923.04 0.451719 0.225860 0.974160i $$-0.427481\pi$$
0.225860 + 0.974160i $$0.427481\pi$$
$$618$$ 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
$$625$$ 32614.9 2.08735
$$626$$ −28321.4 −1.80823
$$627$$ 0 0
$$628$$ −10839.6 −0.688768
$$629$$ 0 0
$$630$$ 0 0
$$631$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$632$$ 6677.15 0.420257
$$633$$ 0 0
$$634$$ −38735.9 −2.42650
$$635$$ 50880.2 3.17972
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 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 1.00000i $$-0.5\pi$$
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
$$649$$ −48807.1 −2.95200
$$650$$ 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
$$656$$ 10784.0 0.641836
$$657$$ 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
$$664$$ 24096.6 1.40833
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 0 0
$$668$$ 41268.2 2.39029
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −66157.2 −3.80622
$$672$$ 0 0
$$673$$ 25522.0 1.46181 0.730907 0.682477i $$-0.239097\pi$$
0.730907 + 0.682477i $$0.239097\pi$$
$$674$$ 46138.0 2.63675
$$675$$ 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
$$683$$ 14207.2 0.795936 0.397968 0.917399i $$-0.369715\pi$$
0.397968 + 0.917399i $$0.369715\pi$$
$$684$$ 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 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 6933.46 0.378419
$$696$$ 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
$$704$$ −57612.0 −3.08428
$$705$$ 0 0
$$706$$ −48945.8 −2.60920
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$710$$ 11179.1 0.590908
$$711$$ 0 0
$$712$$ 2478.45 0.130455
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 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
$$722$$ −30370.4 −1.56547
$$723$$ 0 0
$$724$$ −51412.6 −2.63914
$$725$$ 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 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$734$$ −36733.2 −1.84720
$$735$$ 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
$$743$$ 35259.5 1.74097 0.870487 0.492191i $$-0.163804\pi$$
0.870487 + 0.492191i $$0.163804\pi$$
$$744$$ 0 0
$$745$$ −54321.5 −2.67139
$$746$$ −28896.1 −1.41818
$$747$$ 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 $$0.718825\pi$$
$$752$$ 1576.71 0.0764583
$$753$$ 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
$$761$$ 30422.5 1.44916 0.724582 0.689189i $$-0.242033\pi$$
0.724582 + 0.689189i $$0.242033\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 35250.9 1.66275
$$767$$ 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
$$773$$ −42841.8 −1.99342 −0.996710 0.0810548i $$-0.974171\pi$$
−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
$$781$$ −8670.64 −0.397260
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 7599.37 0.346181
$$785$$ 19046.6 0.865991
$$786$$ 0 0
$$787$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$788$$ −44590.6 −2.01583
$$789$$ 0 0
$$790$$ −37765.1 −1.70079
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 0 0
$$794$$ 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
$$800$$ −65680.5 −2.90270
$$801$$ 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 1.00000i $$-0.5\pi$$
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
$$820$$ 115195. 4.90583
$$821$$ −33486.7 −1.42350 −0.711749 0.702433i $$-0.752097\pi$$
−0.711749 + 0.702433i $$0.752097\pi$$
$$822$$ 0 0
$$823$$ −37352.0 −1.58203 −0.791014 0.611798i $$-0.790446\pi$$
−0.791014 + 0.611798i $$0.790446\pi$$
$$824$$ −13538.1 −0.572356
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −35931.1 −1.51082 −0.755409 0.655253i $$-0.772562\pi$$
−0.755409 + 0.655253i $$0.772562\pi$$
$$828$$ 0 0
$$829$$ −3079.14 −0.129002 −0.0645012 0.997918i $$-0.520546\pi$$
−0.0645012 + 0.997918i $$0.520546\pi$$
$$830$$ −136287. −5.69952
$$831$$ 0 0
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 0 0
$$835$$ −72513.9 −3.00533
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ 12857.9 0.529086 0.264543 0.964374i $$-0.414779\pi$$
0.264543 + 0.964374i $$0.414779\pi$$
$$840$$ 0 0
$$841$$ −24389.0 −1.00000
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 66616.3 2.71686
$$845$$ 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 $$-0.829800\pi$$
−0.860423 + 0.509581i $$0.829800\pi$$
$$860$$ 106973. 4.24158
$$861$$ 0 0
$$862$$ 79225.7 3.13044
$$863$$ −44880.2 −1.77027 −0.885133 0.465339i $$-0.845932\pi$$
−0.885133 + 0.465339i $$0.845932\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 159.647 0.00626447
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 29291.0 1.14342
$$870$$ 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 $$0.5\pi$$
$$878$$ 50122.9 1.92661
$$879$$ 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 $$-0.0160811\pi$$
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
$$890$$ −14017.8 −0.527952
$$891$$ 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. −5.57161
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 90338.9 3.31820
$$906$$ 0 0
$$907$$ 35017.1 1.28195 0.640973 0.767564i $$-0.278531\pi$$
0.640973 + 0.767564i $$0.278531\pi$$
$$908$$ 17667.3 0.645715
$$909$$ 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
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ 56151.6 1.98307 0.991536 0.129832i $$-0.0414439\pi$$
0.991536 + 0.129832i $$0.0414439\pi$$
$$930$$ 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
$$941$$ 55622.6 1.92693 0.963467 0.267826i $$-0.0863052\pi$$
0.963467 + 0.267826i $$0.0863052\pi$$
$$942$$ 0 0
$$943$$ 0 0
$$944$$ 15440.5 0.532359
$$945$$ 0 0
$$946$$ −140163. −4.81721
$$947$$ −14319.3 −0.491356 −0.245678 0.969352i $$-0.579011\pi$$
−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
$$956$$ 48646.2 1.64574
$$957$$ 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 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$968$$ 57052.4 1.89435
$$969$$ 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
$$977$$ 31890.7 1.04429 0.522146 0.852856i $$-0.325132\pi$$
0.522146 + 0.852856i $$0.325132\pi$$
$$978$$ 0 0
$$979$$ 10872.3 0.354935
$$980$$ 81176.6 2.64601
$$981$$ 0 0
$$982$$ 0 0
$$983$$ −11061.5 −0.358908 −0.179454 0.983766i $$-0.557433\pi$$
−0.179454 + 0.983766i $$0.557433\pi$$
$$984$$ 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. 3.64517
$$996$$ 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.1.4 4
3.2 odd 2 inner 1521.4.a.y.1.1 4
13.5 odd 4 117.4.b.c.64.1 4
13.8 odd 4 117.4.b.c.64.4 yes 4
13.12 even 2 inner 1521.4.a.y.1.1 4
39.5 even 4 117.4.b.c.64.4 yes 4
39.8 even 4 117.4.b.c.64.1 4
39.38 odd 2 CM 1521.4.a.y.1.4 4

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