# Properties

 Label 1323.4.a.bo.1.8 Level $1323$ Weight $4$ Character 1323.1 Self dual yes Analytic conductor $78.060$ Analytic rank $0$ Dimension $8$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1323,4,Mod(1,1323)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1323, 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("1323.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1323 = 3^{3} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1323.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: $$78.0595269376$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} - 54x^{6} - 6x^{5} + 555x^{4} + 642x^{3} - 218x^{2} - 54x + 9$$ x^8 - 54*x^6 - 6*x^5 + 555*x^4 + 642*x^3 - 218*x^2 - 54*x + 9 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{5}\cdot 3^{4}\cdot 7^{2}$$ Twist minimal: no (minimal twist has level 189) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.8 Root $$-2.12000$$ of defining polynomial Character $$\chi$$ $$=$$ 1323.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+5.46178 q^{2} +21.8311 q^{4} +0.199136 q^{5} +75.5424 q^{8} +O(q^{10})$$ $$q+5.46178 q^{2} +21.8311 q^{4} +0.199136 q^{5} +75.5424 q^{8} +1.08764 q^{10} +28.4233 q^{11} +32.5809 q^{13} +237.948 q^{16} -115.488 q^{17} +21.1500 q^{19} +4.34736 q^{20} +155.242 q^{22} +93.7656 q^{23} -124.960 q^{25} +177.950 q^{26} +231.571 q^{29} +281.742 q^{31} +695.280 q^{32} -630.770 q^{34} -146.554 q^{37} +115.517 q^{38} +15.0432 q^{40} -111.001 q^{41} +392.361 q^{43} +620.513 q^{44} +512.128 q^{46} +273.168 q^{47} -682.506 q^{50} +711.277 q^{52} -340.403 q^{53} +5.66011 q^{55} +1264.79 q^{58} -696.817 q^{59} -370.002 q^{61} +1538.81 q^{62} +1893.89 q^{64} +6.48804 q^{65} +87.1362 q^{67} -2521.23 q^{68} -88.3772 q^{71} -803.036 q^{73} -800.446 q^{74} +461.727 q^{76} +364.616 q^{79} +47.3840 q^{80} -606.264 q^{82} +921.684 q^{83} -22.9978 q^{85} +2142.99 q^{86} +2147.17 q^{88} -211.396 q^{89} +2047.01 q^{92} +1491.99 q^{94} +4.21172 q^{95} +845.718 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 48 q^{4}+O(q^{10})$$ 8 * q + 48 * q^4 $$8 q + 48 q^{4} + 44 q^{10} - 84 q^{13} + 156 q^{16} + 12 q^{19} + 224 q^{22} + 408 q^{25} + 800 q^{31} - 948 q^{34} + 692 q^{37} + 96 q^{40} + 1456 q^{43} + 1524 q^{46} + 1972 q^{52} - 1280 q^{55} + 2372 q^{58} + 216 q^{61} + 4964 q^{64} + 684 q^{67} - 4564 q^{73} - 380 q^{76} + 556 q^{79} + 3340 q^{82} + 1296 q^{85} + 6696 q^{88} - 492 q^{94} + 584 q^{97}+O(q^{100})$$ 8 * q + 48 * q^4 + 44 * q^10 - 84 * q^13 + 156 * q^16 + 12 * q^19 + 224 * q^22 + 408 * q^25 + 800 * q^31 - 948 * q^34 + 692 * q^37 + 96 * q^40 + 1456 * q^43 + 1524 * q^46 + 1972 * q^52 - 1280 * q^55 + 2372 * q^58 + 216 * q^61 + 4964 * q^64 + 684 * q^67 - 4564 * q^73 - 380 * q^76 + 556 * q^79 + 3340 * q^82 + 1296 * q^85 + 6696 * q^88 - 492 * q^94 + 584 * q^97

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 5.46178 1.93103 0.965516 0.260343i $$-0.0838356\pi$$
0.965516 + 0.260343i $$0.0838356\pi$$
$$3$$ 0 0
$$4$$ 21.8311 2.72889
$$5$$ 0.199136 0.0178113 0.00890564 0.999960i $$-0.497165\pi$$
0.00890564 + 0.999960i $$0.497165\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 75.5424 3.33854
$$9$$ 0 0
$$10$$ 1.08764 0.0343941
$$11$$ 28.4233 0.779087 0.389544 0.921008i $$-0.372633\pi$$
0.389544 + 0.921008i $$0.372633\pi$$
$$12$$ 0 0
$$13$$ 32.5809 0.695101 0.347551 0.937661i $$-0.387013\pi$$
0.347551 + 0.937661i $$0.387013\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 237.948 3.71793
$$17$$ −115.488 −1.64764 −0.823821 0.566849i $$-0.808162\pi$$
−0.823821 + 0.566849i $$0.808162\pi$$
$$18$$ 0 0
$$19$$ 21.1500 0.255376 0.127688 0.991814i $$-0.459244\pi$$
0.127688 + 0.991814i $$0.459244\pi$$
$$20$$ 4.34736 0.0486049
$$21$$ 0 0
$$22$$ 155.242 1.50444
$$23$$ 93.7656 0.850065 0.425032 0.905178i $$-0.360263\pi$$
0.425032 + 0.905178i $$0.360263\pi$$
$$24$$ 0 0
$$25$$ −124.960 −0.999683
$$26$$ 177.950 1.34226
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 231.571 1.48282 0.741408 0.671055i $$-0.234159\pi$$
0.741408 + 0.671055i $$0.234159\pi$$
$$30$$ 0 0
$$31$$ 281.742 1.63233 0.816167 0.577817i $$-0.196095\pi$$
0.816167 + 0.577817i $$0.196095\pi$$
$$32$$ 695.280 3.84092
$$33$$ 0 0
$$34$$ −630.770 −3.18165
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −146.554 −0.651171 −0.325585 0.945513i $$-0.605561\pi$$
−0.325585 + 0.945513i $$0.605561\pi$$
$$38$$ 115.517 0.493138
$$39$$ 0 0
$$40$$ 15.0432 0.0594636
$$41$$ −111.001 −0.422816 −0.211408 0.977398i $$-0.567805\pi$$
−0.211408 + 0.977398i $$0.567805\pi$$
$$42$$ 0 0
$$43$$ 392.361 1.39150 0.695750 0.718284i $$-0.255072\pi$$
0.695750 + 0.718284i $$0.255072\pi$$
$$44$$ 620.513 2.12604
$$45$$ 0 0
$$46$$ 512.128 1.64150
$$47$$ 273.168 0.847780 0.423890 0.905714i $$-0.360664\pi$$
0.423890 + 0.905714i $$0.360664\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ −682.506 −1.93042
$$51$$ 0 0
$$52$$ 711.277 1.89685
$$53$$ −340.403 −0.882225 −0.441113 0.897452i $$-0.645416\pi$$
−0.441113 + 0.897452i $$0.645416\pi$$
$$54$$ 0 0
$$55$$ 5.66011 0.0138765
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 1264.79 2.86336
$$59$$ −696.817 −1.53759 −0.768795 0.639495i $$-0.779143\pi$$
−0.768795 + 0.639495i $$0.779143\pi$$
$$60$$ 0 0
$$61$$ −370.002 −0.776622 −0.388311 0.921528i $$-0.626941\pi$$
−0.388311 + 0.921528i $$0.626941\pi$$
$$62$$ 1538.81 3.15209
$$63$$ 0 0
$$64$$ 1893.89 3.69900
$$65$$ 6.48804 0.0123806
$$66$$ 0 0
$$67$$ 87.1362 0.158886 0.0794431 0.996839i $$-0.474686\pi$$
0.0794431 + 0.996839i $$0.474686\pi$$
$$68$$ −2521.23 −4.49623
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −88.3772 −0.147725 −0.0738623 0.997268i $$-0.523533\pi$$
−0.0738623 + 0.997268i $$0.523533\pi$$
$$72$$ 0 0
$$73$$ −803.036 −1.28751 −0.643755 0.765231i $$-0.722625\pi$$
−0.643755 + 0.765231i $$0.722625\pi$$
$$74$$ −800.446 −1.25743
$$75$$ 0 0
$$76$$ 461.727 0.696891
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 364.616 0.519272 0.259636 0.965707i $$-0.416397\pi$$
0.259636 + 0.965707i $$0.416397\pi$$
$$80$$ 47.3840 0.0662211
$$81$$ 0 0
$$82$$ −606.264 −0.816472
$$83$$ 921.684 1.21889 0.609446 0.792828i $$-0.291392\pi$$
0.609446 + 0.792828i $$0.291392\pi$$
$$84$$ 0 0
$$85$$ −22.9978 −0.0293466
$$86$$ 2142.99 2.68703
$$87$$ 0 0
$$88$$ 2147.17 2.60101
$$89$$ −211.396 −0.251774 −0.125887 0.992045i $$-0.540178\pi$$
−0.125887 + 0.992045i $$0.540178\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 2047.01 2.31973
$$93$$ 0 0
$$94$$ 1491.99 1.63709
$$95$$ 4.21172 0.00454856
$$96$$ 0 0
$$97$$ 845.718 0.885254 0.442627 0.896706i $$-0.354047\pi$$
0.442627 + 0.896706i $$0.354047\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ −2728.02 −2.72802
$$101$$ −156.952 −0.154626 −0.0773132 0.997007i $$-0.524634\pi$$
−0.0773132 + 0.997007i $$0.524634\pi$$
$$102$$ 0 0
$$103$$ −139.325 −0.133283 −0.0666414 0.997777i $$-0.521228\pi$$
−0.0666414 + 0.997777i $$0.521228\pi$$
$$104$$ 2461.24 2.32062
$$105$$ 0 0
$$106$$ −1859.21 −1.70361
$$107$$ −415.476 −0.375379 −0.187690 0.982228i $$-0.560100\pi$$
−0.187690 + 0.982228i $$0.560100\pi$$
$$108$$ 0 0
$$109$$ −844.080 −0.741727 −0.370863 0.928687i $$-0.620938\pi$$
−0.370863 + 0.928687i $$0.620938\pi$$
$$110$$ 30.9143 0.0267961
$$111$$ 0 0
$$112$$ 0 0
$$113$$ −1342.21 −1.11738 −0.558692 0.829375i $$-0.688697\pi$$
−0.558692 + 0.829375i $$0.688697\pi$$
$$114$$ 0 0
$$115$$ 18.6721 0.0151407
$$116$$ 5055.45 4.04643
$$117$$ 0 0
$$118$$ −3805.86 −2.96914
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −523.113 −0.393023
$$122$$ −2020.87 −1.49968
$$123$$ 0 0
$$124$$ 6150.73 4.45445
$$125$$ −49.7761 −0.0356169
$$126$$ 0 0
$$127$$ 978.750 0.683858 0.341929 0.939726i $$-0.388920\pi$$
0.341929 + 0.939726i $$0.388920\pi$$
$$128$$ 4781.76 3.30197
$$129$$ 0 0
$$130$$ 35.4362 0.0239074
$$131$$ 2372.81 1.58254 0.791271 0.611465i $$-0.209420\pi$$
0.791271 + 0.611465i $$0.209420\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 475.919 0.306815
$$135$$ 0 0
$$136$$ −8724.24 −5.50071
$$137$$ 1602.67 0.999458 0.499729 0.866182i $$-0.333433\pi$$
0.499729 + 0.866182i $$0.333433\pi$$
$$138$$ 0 0
$$139$$ 2101.78 1.28252 0.641262 0.767322i $$-0.278411\pi$$
0.641262 + 0.767322i $$0.278411\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ −482.697 −0.285261
$$143$$ 926.059 0.541545
$$144$$ 0 0
$$145$$ 46.1141 0.0264108
$$146$$ −4386.01 −2.48623
$$147$$ 0 0
$$148$$ −3199.43 −1.77697
$$149$$ 524.400 0.288326 0.144163 0.989554i $$-0.453951\pi$$
0.144163 + 0.989554i $$0.453951\pi$$
$$150$$ 0 0
$$151$$ −867.666 −0.467613 −0.233807 0.972283i $$-0.575118\pi$$
−0.233807 + 0.972283i $$0.575118\pi$$
$$152$$ 1597.72 0.852580
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 56.1050 0.0290739
$$156$$ 0 0
$$157$$ −5.91999 −0.00300934 −0.00150467 0.999999i $$-0.500479\pi$$
−0.00150467 + 0.999999i $$0.500479\pi$$
$$158$$ 1991.45 1.00273
$$159$$ 0 0
$$160$$ 138.455 0.0684116
$$161$$ 0 0
$$162$$ 0 0
$$163$$ −832.959 −0.400260 −0.200130 0.979769i $$-0.564136\pi$$
−0.200130 + 0.979769i $$0.564136\pi$$
$$164$$ −2423.28 −1.15382
$$165$$ 0 0
$$166$$ 5034.04 2.35372
$$167$$ −566.798 −0.262636 −0.131318 0.991340i $$-0.541921\pi$$
−0.131318 + 0.991340i $$0.541921\pi$$
$$168$$ 0 0
$$169$$ −1135.48 −0.516834
$$170$$ −125.609 −0.0566693
$$171$$ 0 0
$$172$$ 8565.66 3.79724
$$173$$ −4125.77 −1.81316 −0.906579 0.422036i $$-0.861316\pi$$
−0.906579 + 0.422036i $$0.861316\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 6763.27 2.89660
$$177$$ 0 0
$$178$$ −1154.60 −0.486184
$$179$$ 927.646 0.387349 0.193675 0.981066i $$-0.437959\pi$$
0.193675 + 0.981066i $$0.437959\pi$$
$$180$$ 0 0
$$181$$ 1211.67 0.497585 0.248792 0.968557i $$-0.419966\pi$$
0.248792 + 0.968557i $$0.419966\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 7083.28 2.83797
$$185$$ −29.1842 −0.0115982
$$186$$ 0 0
$$187$$ −3282.55 −1.28366
$$188$$ 5963.56 2.31350
$$189$$ 0 0
$$190$$ 23.0035 0.00878342
$$191$$ −1827.49 −0.692318 −0.346159 0.938176i $$-0.612514\pi$$
−0.346159 + 0.938176i $$0.612514\pi$$
$$192$$ 0 0
$$193$$ −822.230 −0.306660 −0.153330 0.988175i $$-0.549000\pi$$
−0.153330 + 0.988175i $$0.549000\pi$$
$$194$$ 4619.13 1.70945
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −4873.12 −1.76241 −0.881207 0.472731i $$-0.843268\pi$$
−0.881207 + 0.472731i $$0.843268\pi$$
$$198$$ 0 0
$$199$$ −1482.23 −0.528001 −0.264001 0.964523i $$-0.585042\pi$$
−0.264001 + 0.964523i $$0.585042\pi$$
$$200$$ −9439.81 −3.33748
$$201$$ 0 0
$$202$$ −857.236 −0.298589
$$203$$ 0 0
$$204$$ 0 0
$$205$$ −22.1043 −0.00753090
$$206$$ −760.965 −0.257373
$$207$$ 0 0
$$208$$ 7752.55 2.58434
$$209$$ 601.153 0.198960
$$210$$ 0 0
$$211$$ −4359.21 −1.42228 −0.711138 0.703053i $$-0.751820\pi$$
−0.711138 + 0.703053i $$0.751820\pi$$
$$212$$ −7431.37 −2.40749
$$213$$ 0 0
$$214$$ −2269.24 −0.724869
$$215$$ 78.1332 0.0247844
$$216$$ 0 0
$$217$$ 0 0
$$218$$ −4610.18 −1.43230
$$219$$ 0 0
$$220$$ 123.566 0.0378675
$$221$$ −3762.70 −1.14528
$$222$$ 0 0
$$223$$ −3312.73 −0.994784 −0.497392 0.867526i $$-0.665709\pi$$
−0.497392 + 0.867526i $$0.665709\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ −7330.85 −2.15770
$$227$$ −798.321 −0.233420 −0.116710 0.993166i $$-0.537235\pi$$
−0.116710 + 0.993166i $$0.537235\pi$$
$$228$$ 0 0
$$229$$ −354.914 −0.102417 −0.0512083 0.998688i $$-0.516307\pi$$
−0.0512083 + 0.998688i $$0.516307\pi$$
$$230$$ 101.983 0.0292372
$$231$$ 0 0
$$232$$ 17493.4 4.95043
$$233$$ −4473.91 −1.25792 −0.628960 0.777438i $$-0.716519\pi$$
−0.628960 + 0.777438i $$0.716519\pi$$
$$234$$ 0 0
$$235$$ 54.3976 0.0151000
$$236$$ −15212.3 −4.19591
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −1494.65 −0.404523 −0.202262 0.979332i $$-0.564829\pi$$
−0.202262 + 0.979332i $$0.564829\pi$$
$$240$$ 0 0
$$241$$ −156.885 −0.0419329 −0.0209665 0.999780i $$-0.506674\pi$$
−0.0209665 + 0.999780i $$0.506674\pi$$
$$242$$ −2857.13 −0.758940
$$243$$ 0 0
$$244$$ −8077.55 −2.11931
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 689.085 0.177512
$$248$$ 21283.5 5.44960
$$249$$ 0 0
$$250$$ −271.866 −0.0687774
$$251$$ 3498.68 0.879819 0.439909 0.898042i $$-0.355011\pi$$
0.439909 + 0.898042i $$0.355011\pi$$
$$252$$ 0 0
$$253$$ 2665.13 0.662275
$$254$$ 5345.72 1.32055
$$255$$ 0 0
$$256$$ 10965.9 2.67721
$$257$$ 4739.10 1.15026 0.575130 0.818062i $$-0.304951\pi$$
0.575130 + 0.818062i $$0.304951\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 141.641 0.0337854
$$261$$ 0 0
$$262$$ 12959.8 3.05594
$$263$$ −3364.44 −0.788823 −0.394411 0.918934i $$-0.629052\pi$$
−0.394411 + 0.918934i $$0.629052\pi$$
$$264$$ 0 0
$$265$$ −67.7865 −0.0157136
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 1902.28 0.433583
$$269$$ −5207.27 −1.18027 −0.590136 0.807304i $$-0.700926\pi$$
−0.590136 + 0.807304i $$0.700926\pi$$
$$270$$ 0 0
$$271$$ −2312.00 −0.518244 −0.259122 0.965845i $$-0.583433\pi$$
−0.259122 + 0.965845i $$0.583433\pi$$
$$272$$ −27480.1 −6.12583
$$273$$ 0 0
$$274$$ 8753.46 1.92999
$$275$$ −3551.79 −0.778840
$$276$$ 0 0
$$277$$ 2606.72 0.565425 0.282713 0.959205i $$-0.408766\pi$$
0.282713 + 0.959205i $$0.408766\pi$$
$$278$$ 11479.5 2.47659
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −5271.62 −1.11914 −0.559570 0.828783i $$-0.689034\pi$$
−0.559570 + 0.828783i $$0.689034\pi$$
$$282$$ 0 0
$$283$$ 5808.21 1.22001 0.610004 0.792398i $$-0.291168\pi$$
0.610004 + 0.792398i $$0.291168\pi$$
$$284$$ −1929.37 −0.403124
$$285$$ 0 0
$$286$$ 5057.93 1.04574
$$287$$ 0 0
$$288$$ 0 0
$$289$$ 8424.45 1.71473
$$290$$ 251.865 0.0510002
$$291$$ 0 0
$$292$$ −17531.2 −3.51347
$$293$$ −1979.70 −0.394729 −0.197364 0.980330i $$-0.563238\pi$$
−0.197364 + 0.980330i $$0.563238\pi$$
$$294$$ 0 0
$$295$$ −138.761 −0.0273864
$$296$$ −11071.0 −2.17396
$$297$$ 0 0
$$298$$ 2864.16 0.556766
$$299$$ 3054.97 0.590881
$$300$$ 0 0
$$301$$ 0 0
$$302$$ −4739.00 −0.902977
$$303$$ 0 0
$$304$$ 5032.59 0.949469
$$305$$ −73.6808 −0.0138326
$$306$$ 0 0
$$307$$ 924.005 0.171778 0.0858888 0.996305i $$-0.472627\pi$$
0.0858888 + 0.996305i $$0.472627\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 306.433 0.0561427
$$311$$ −10108.6 −1.84311 −0.921554 0.388251i $$-0.873080\pi$$
−0.921554 + 0.388251i $$0.873080\pi$$
$$312$$ 0 0
$$313$$ 6830.52 1.23349 0.616747 0.787161i $$-0.288450\pi$$
0.616747 + 0.787161i $$0.288450\pi$$
$$314$$ −32.3337 −0.00581114
$$315$$ 0 0
$$316$$ 7959.96 1.41703
$$317$$ −7623.62 −1.35074 −0.675371 0.737478i $$-0.736016\pi$$
−0.675371 + 0.737478i $$0.736016\pi$$
$$318$$ 0 0
$$319$$ 6582.02 1.15524
$$320$$ 377.141 0.0658839
$$321$$ 0 0
$$322$$ 0 0
$$323$$ −2442.56 −0.420768
$$324$$ 0 0
$$325$$ −4071.32 −0.694881
$$326$$ −4549.45 −0.772916
$$327$$ 0 0
$$328$$ −8385.30 −1.41159
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 2483.21 0.412355 0.206177 0.978515i $$-0.433898\pi$$
0.206177 + 0.978515i $$0.433898\pi$$
$$332$$ 20121.4 3.32622
$$333$$ 0 0
$$334$$ −3095.73 −0.507158
$$335$$ 17.3520 0.00282997
$$336$$ 0 0
$$337$$ 7895.47 1.27624 0.638121 0.769936i $$-0.279712\pi$$
0.638121 + 0.769936i $$0.279712\pi$$
$$338$$ −6201.77 −0.998023
$$339$$ 0 0
$$340$$ −502.067 −0.0800836
$$341$$ 8008.05 1.27173
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 29639.9 4.64557
$$345$$ 0 0
$$346$$ −22534.1 −3.50127
$$347$$ 889.423 0.137599 0.0687994 0.997631i $$-0.478083\pi$$
0.0687994 + 0.997631i $$0.478083\pi$$
$$348$$ 0 0
$$349$$ 6962.20 1.06785 0.533923 0.845533i $$-0.320717\pi$$
0.533923 + 0.845533i $$0.320717\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 19762.2 2.99241
$$353$$ −11560.6 −1.74308 −0.871542 0.490320i $$-0.836880\pi$$
−0.871542 + 0.490320i $$0.836880\pi$$
$$354$$ 0 0
$$355$$ −17.5991 −0.00263116
$$356$$ −4615.00 −0.687064
$$357$$ 0 0
$$358$$ 5066.60 0.747984
$$359$$ 8457.02 1.24330 0.621649 0.783296i $$-0.286463\pi$$
0.621649 + 0.783296i $$0.286463\pi$$
$$360$$ 0 0
$$361$$ −6411.68 −0.934783
$$362$$ 6617.89 0.960853
$$363$$ 0 0
$$364$$ 0 0
$$365$$ −159.914 −0.0229322
$$366$$ 0 0
$$367$$ 6912.92 0.983247 0.491623 0.870808i $$-0.336404\pi$$
0.491623 + 0.870808i $$0.336404\pi$$
$$368$$ 22311.3 3.16048
$$369$$ 0 0
$$370$$ −159.398 −0.0223965
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 423.911 0.0588453 0.0294226 0.999567i $$-0.490633\pi$$
0.0294226 + 0.999567i $$0.490633\pi$$
$$374$$ −17928.6 −2.47879
$$375$$ 0 0
$$376$$ 20635.8 2.83034
$$377$$ 7544.79 1.03071
$$378$$ 0 0
$$379$$ −9714.33 −1.31660 −0.658300 0.752756i $$-0.728724\pi$$
−0.658300 + 0.752756i $$0.728724\pi$$
$$380$$ 91.9465 0.0124125
$$381$$ 0 0
$$382$$ −9981.37 −1.33689
$$383$$ −2214.15 −0.295399 −0.147699 0.989032i $$-0.547187\pi$$
−0.147699 + 0.989032i $$0.547187\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −4490.84 −0.592171
$$387$$ 0 0
$$388$$ 18462.9 2.41576
$$389$$ −7665.46 −0.999111 −0.499556 0.866282i $$-0.666503\pi$$
−0.499556 + 0.866282i $$0.666503\pi$$
$$390$$ 0 0
$$391$$ −10828.8 −1.40060
$$392$$ 0 0
$$393$$ 0 0
$$394$$ −26615.9 −3.40328
$$395$$ 72.6081 0.00924889
$$396$$ 0 0
$$397$$ 4978.20 0.629341 0.314671 0.949201i $$-0.398106\pi$$
0.314671 + 0.949201i $$0.398106\pi$$
$$398$$ −8095.60 −1.01959
$$399$$ 0 0
$$400$$ −29734.0 −3.71675
$$401$$ −4770.21 −0.594047 −0.297023 0.954870i $$-0.595994\pi$$
−0.297023 + 0.954870i $$0.595994\pi$$
$$402$$ 0 0
$$403$$ 9179.41 1.13464
$$404$$ −3426.43 −0.421958
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −4165.56 −0.507319
$$408$$ 0 0
$$409$$ −597.257 −0.0722065 −0.0361033 0.999348i $$-0.511495\pi$$
−0.0361033 + 0.999348i $$0.511495\pi$$
$$410$$ −120.729 −0.0145424
$$411$$ 0 0
$$412$$ −3041.62 −0.363714
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 183.541 0.0217100
$$416$$ 22652.9 2.66983
$$417$$ 0 0
$$418$$ 3283.37 0.384198
$$419$$ 9571.90 1.11603 0.558016 0.829830i $$-0.311563\pi$$
0.558016 + 0.829830i $$0.311563\pi$$
$$420$$ 0 0
$$421$$ −5954.33 −0.689303 −0.344651 0.938731i $$-0.612003\pi$$
−0.344651 + 0.938731i $$0.612003\pi$$
$$422$$ −23809.0 −2.74646
$$423$$ 0 0
$$424$$ −25714.9 −2.94534
$$425$$ 14431.4 1.64712
$$426$$ 0 0
$$427$$ 0 0
$$428$$ −9070.29 −1.02437
$$429$$ 0 0
$$430$$ 426.747 0.0478594
$$431$$ −2543.44 −0.284254 −0.142127 0.989848i $$-0.545394\pi$$
−0.142127 + 0.989848i $$0.545394\pi$$
$$432$$ 0 0
$$433$$ 9781.07 1.08556 0.542781 0.839874i $$-0.317371\pi$$
0.542781 + 0.839874i $$0.317371\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ −18427.2 −2.02409
$$437$$ 1983.14 0.217086
$$438$$ 0 0
$$439$$ −16820.3 −1.82867 −0.914336 0.404957i $$-0.867287\pi$$
−0.914336 + 0.404957i $$0.867287\pi$$
$$440$$ 427.579 0.0463273
$$441$$ 0 0
$$442$$ −20551.1 −2.21157
$$443$$ 3726.75 0.399691 0.199846 0.979827i $$-0.435956\pi$$
0.199846 + 0.979827i $$0.435956\pi$$
$$444$$ 0 0
$$445$$ −42.0965 −0.00448442
$$446$$ −18093.4 −1.92096
$$447$$ 0 0
$$448$$ 0 0
$$449$$ −9287.05 −0.976131 −0.488065 0.872807i $$-0.662297\pi$$
−0.488065 + 0.872807i $$0.662297\pi$$
$$450$$ 0 0
$$451$$ −3155.02 −0.329411
$$452$$ −29301.9 −3.04921
$$453$$ 0 0
$$454$$ −4360.26 −0.450742
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 12019.4 1.23029 0.615147 0.788413i $$-0.289097\pi$$
0.615147 + 0.788413i $$0.289097\pi$$
$$458$$ −1938.47 −0.197770
$$459$$ 0 0
$$460$$ 407.633 0.0413173
$$461$$ −9571.71 −0.967026 −0.483513 0.875337i $$-0.660639\pi$$
−0.483513 + 0.875337i $$0.660639\pi$$
$$462$$ 0 0
$$463$$ 2265.15 0.227366 0.113683 0.993517i $$-0.463735\pi$$
0.113683 + 0.993517i $$0.463735\pi$$
$$464$$ 55101.8 5.51301
$$465$$ 0 0
$$466$$ −24435.5 −2.42908
$$467$$ −5320.94 −0.527246 −0.263623 0.964626i $$-0.584918\pi$$
−0.263623 + 0.964626i $$0.584918\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 297.108 0.0291587
$$471$$ 0 0
$$472$$ −52639.2 −5.13330
$$473$$ 11152.2 1.08410
$$474$$ 0 0
$$475$$ −2642.91 −0.255294
$$476$$ 0 0
$$477$$ 0 0
$$478$$ −8163.48 −0.781148
$$479$$ 3766.98 0.359327 0.179663 0.983728i $$-0.442499\pi$$
0.179663 + 0.983728i $$0.442499\pi$$
$$480$$ 0 0
$$481$$ −4774.86 −0.452630
$$482$$ −856.871 −0.0809739
$$483$$ 0 0
$$484$$ −11420.1 −1.07251
$$485$$ 168.413 0.0157675
$$486$$ 0 0
$$487$$ 8857.92 0.824211 0.412105 0.911136i $$-0.364794\pi$$
0.412105 + 0.911136i $$0.364794\pi$$
$$488$$ −27950.9 −2.59278
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −17311.1 −1.59112 −0.795560 0.605874i $$-0.792823\pi$$
−0.795560 + 0.605874i $$0.792823\pi$$
$$492$$ 0 0
$$493$$ −26743.6 −2.44315
$$494$$ 3763.63 0.342781
$$495$$ 0 0
$$496$$ 67039.8 6.06891
$$497$$ 0 0
$$498$$ 0 0
$$499$$ −7030.45 −0.630714 −0.315357 0.948973i $$-0.602124\pi$$
−0.315357 + 0.948973i $$0.602124\pi$$
$$500$$ −1086.67 −0.0971945
$$501$$ 0 0
$$502$$ 19109.0 1.69896
$$503$$ 1519.74 0.134716 0.0673578 0.997729i $$-0.478543\pi$$
0.0673578 + 0.997729i $$0.478543\pi$$
$$504$$ 0 0
$$505$$ −31.2547 −0.00275409
$$506$$ 14556.4 1.27887
$$507$$ 0 0
$$508$$ 21367.2 1.86617
$$509$$ −19455.4 −1.69419 −0.847096 0.531440i $$-0.821651\pi$$
−0.847096 + 0.531440i $$0.821651\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 21639.1 1.86782
$$513$$ 0 0
$$514$$ 25883.9 2.22119
$$515$$ −27.7447 −0.00237394
$$516$$ 0 0
$$517$$ 7764.35 0.660495
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 490.122 0.0413332
$$521$$ 10224.0 0.859731 0.429866 0.902893i $$-0.358561\pi$$
0.429866 + 0.902893i $$0.358561\pi$$
$$522$$ 0 0
$$523$$ 16607.0 1.38847 0.694237 0.719746i $$-0.255742\pi$$
0.694237 + 0.719746i $$0.255742\pi$$
$$524$$ 51801.0 4.31858
$$525$$ 0 0
$$526$$ −18375.9 −1.52324
$$527$$ −32537.8 −2.68950
$$528$$ 0 0
$$529$$ −3375.01 −0.277390
$$530$$ −370.235 −0.0303434
$$531$$ 0 0
$$532$$ 0 0
$$533$$ −3616.52 −0.293900
$$534$$ 0 0
$$535$$ −82.7363 −0.00668598
$$536$$ 6582.48 0.530447
$$537$$ 0 0
$$538$$ −28441.0 −2.27914
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 6582.54 0.523116 0.261558 0.965188i $$-0.415764\pi$$
0.261558 + 0.965188i $$0.415764\pi$$
$$542$$ −12627.7 −1.00075
$$543$$ 0 0
$$544$$ −80296.4 −6.32846
$$545$$ −168.087 −0.0132111
$$546$$ 0 0
$$547$$ 3407.73 0.266369 0.133185 0.991091i $$-0.457480\pi$$
0.133185 + 0.991091i $$0.457480\pi$$
$$548$$ 34988.1 2.72741
$$549$$ 0 0
$$550$$ −19399.1 −1.50397
$$551$$ 4897.72 0.378675
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 14237.4 1.09185
$$555$$ 0 0
$$556$$ 45884.2 3.49986
$$557$$ −9480.11 −0.721158 −0.360579 0.932729i $$-0.617421\pi$$
−0.360579 + 0.932729i $$0.617421\pi$$
$$558$$ 0 0
$$559$$ 12783.5 0.967233
$$560$$ 0 0
$$561$$ 0 0
$$562$$ −28792.5 −2.16110
$$563$$ −21864.4 −1.63673 −0.818363 0.574702i $$-0.805118\pi$$
−0.818363 + 0.574702i $$0.805118\pi$$
$$564$$ 0 0
$$565$$ −267.282 −0.0199020
$$566$$ 31723.2 2.35588
$$567$$ 0 0
$$568$$ −6676.23 −0.493184
$$569$$ −1143.39 −0.0842414 −0.0421207 0.999113i $$-0.513411\pi$$
−0.0421207 + 0.999113i $$0.513411\pi$$
$$570$$ 0 0
$$571$$ 10471.1 0.767429 0.383715 0.923452i $$-0.374645\pi$$
0.383715 + 0.923452i $$0.374645\pi$$
$$572$$ 20216.9 1.47781
$$573$$ 0 0
$$574$$ 0 0
$$575$$ −11717.0 −0.849795
$$576$$ 0 0
$$577$$ 4759.60 0.343405 0.171703 0.985149i $$-0.445073\pi$$
0.171703 + 0.985149i $$0.445073\pi$$
$$578$$ 46012.6 3.31119
$$579$$ 0 0
$$580$$ 1006.72 0.0720721
$$581$$ 0 0
$$582$$ 0 0
$$583$$ −9675.39 −0.687331
$$584$$ −60663.3 −4.29840
$$585$$ 0 0
$$586$$ −10812.7 −0.762234
$$587$$ −6307.45 −0.443503 −0.221751 0.975103i $$-0.571177\pi$$
−0.221751 + 0.975103i $$0.571177\pi$$
$$588$$ 0 0
$$589$$ 5958.83 0.416858
$$590$$ −757.885 −0.0528841
$$591$$ 0 0
$$592$$ −34872.2 −2.42101
$$593$$ 17731.9 1.22793 0.613963 0.789335i $$-0.289574\pi$$
0.613963 + 0.789335i $$0.289574\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 11448.2 0.786808
$$597$$ 0 0
$$598$$ 16685.6 1.14101
$$599$$ 19110.1 1.30353 0.651766 0.758420i $$-0.274028\pi$$
0.651766 + 0.758420i $$0.274028\pi$$
$$600$$ 0 0
$$601$$ −13118.3 −0.890357 −0.445179 0.895442i $$-0.646860\pi$$
−0.445179 + 0.895442i $$0.646860\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ −18942.1 −1.27606
$$605$$ −104.171 −0.00700024
$$606$$ 0 0
$$607$$ 17473.4 1.16841 0.584204 0.811607i $$-0.301407\pi$$
0.584204 + 0.811607i $$0.301407\pi$$
$$608$$ 14705.1 0.980876
$$609$$ 0 0
$$610$$ −402.429 −0.0267112
$$611$$ 8900.06 0.589293
$$612$$ 0 0
$$613$$ 3095.39 0.203951 0.101975 0.994787i $$-0.467484\pi$$
0.101975 + 0.994787i $$0.467484\pi$$
$$614$$ 5046.71 0.331708
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 26334.8 1.71831 0.859157 0.511712i $$-0.170989\pi$$
0.859157 + 0.511712i $$0.170989\pi$$
$$618$$ 0 0
$$619$$ −2686.04 −0.174412 −0.0872061 0.996190i $$-0.527794\pi$$
−0.0872061 + 0.996190i $$0.527794\pi$$
$$620$$ 1224.83 0.0793395
$$621$$ 0 0
$$622$$ −55211.0 −3.55910
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 15610.1 0.999048
$$626$$ 37306.8 2.38192
$$627$$ 0 0
$$628$$ −129.240 −0.00821216
$$629$$ 16925.2 1.07290
$$630$$ 0 0
$$631$$ 22414.2 1.41409 0.707047 0.707167i $$-0.250027\pi$$
0.707047 + 0.707167i $$0.250027\pi$$
$$632$$ 27543.9 1.73361
$$633$$ 0 0
$$634$$ −41638.6 −2.60833
$$635$$ 194.904 0.0121804
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 35949.6 2.23081
$$639$$ 0 0
$$640$$ 952.222 0.0588123
$$641$$ 22139.1 1.36418 0.682092 0.731266i $$-0.261070\pi$$
0.682092 + 0.731266i $$0.261070\pi$$
$$642$$ 0 0
$$643$$ 22523.7 1.38141 0.690707 0.723134i $$-0.257299\pi$$
0.690707 + 0.723134i $$0.257299\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ −13340.8 −0.812516
$$647$$ 11523.5 0.700209 0.350105 0.936711i $$-0.386146\pi$$
0.350105 + 0.936711i $$0.386146\pi$$
$$648$$ 0 0
$$649$$ −19805.9 −1.19792
$$650$$ −22236.7 −1.34184
$$651$$ 0 0
$$652$$ −18184.4 −1.09226
$$653$$ 461.937 0.0276830 0.0138415 0.999904i $$-0.495594\pi$$
0.0138415 + 0.999904i $$0.495594\pi$$
$$654$$ 0 0
$$655$$ 472.511 0.0281871
$$656$$ −26412.5 −1.57200
$$657$$ 0 0
$$658$$ 0 0
$$659$$ −14373.3 −0.849627 −0.424813 0.905281i $$-0.639660\pi$$
−0.424813 + 0.905281i $$0.639660\pi$$
$$660$$ 0 0
$$661$$ 67.8490 0.00399246 0.00199623 0.999998i $$-0.499365\pi$$
0.00199623 + 0.999998i $$0.499365\pi$$
$$662$$ 13562.8 0.796271
$$663$$ 0 0
$$664$$ 69626.3 4.06931
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 21713.4 1.26049
$$668$$ −12373.8 −0.716702
$$669$$ 0 0
$$670$$ 94.7727 0.00546476
$$671$$ −10516.7 −0.605056
$$672$$ 0 0
$$673$$ 21970.2 1.25838 0.629190 0.777252i $$-0.283387\pi$$
0.629190 + 0.777252i $$0.283387\pi$$
$$674$$ 43123.4 2.46447
$$675$$ 0 0
$$676$$ −24788.9 −1.41038
$$677$$ −2993.29 −0.169928 −0.0849642 0.996384i $$-0.527078\pi$$
−0.0849642 + 0.996384i $$0.527078\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ −1737.31 −0.0979747
$$681$$ 0 0
$$682$$ 43738.2 2.45575
$$683$$ −5461.90 −0.305994 −0.152997 0.988227i $$-0.548892\pi$$
−0.152997 + 0.988227i $$0.548892\pi$$
$$684$$ 0 0
$$685$$ 319.150 0.0178016
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 93361.4 5.17350
$$689$$ −11090.6 −0.613236
$$690$$ 0 0
$$691$$ −15779.0 −0.868685 −0.434342 0.900748i $$-0.643019\pi$$
−0.434342 + 0.900748i $$0.643019\pi$$
$$692$$ −90070.0 −4.94790
$$693$$ 0 0
$$694$$ 4857.84 0.265708
$$695$$ 418.540 0.0228434
$$696$$ 0 0
$$697$$ 12819.3 0.696650
$$698$$ 38026.0 2.06204
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −8058.91 −0.434209 −0.217105 0.976148i $$-0.569661\pi$$
−0.217105 + 0.976148i $$0.569661\pi$$
$$702$$ 0 0
$$703$$ −3099.61 −0.166293
$$704$$ 53830.6 2.88184
$$705$$ 0 0
$$706$$ −63141.5 −3.36595
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 26018.9 1.37823 0.689113 0.724654i $$-0.258001\pi$$
0.689113 + 0.724654i $$0.258001\pi$$
$$710$$ −96.1225 −0.00508086
$$711$$ 0 0
$$712$$ −15969.4 −0.840558
$$713$$ 26417.7 1.38759
$$714$$ 0 0
$$715$$ 184.412 0.00964560
$$716$$ 20251.5 1.05703
$$717$$ 0 0
$$718$$ 46190.4 2.40085
$$719$$ −4663.46 −0.241889 −0.120944 0.992659i $$-0.538592\pi$$
−0.120944 + 0.992659i $$0.538592\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ −35019.2 −1.80510
$$723$$ 0 0
$$724$$ 26452.1 1.35785
$$725$$ −28937.2 −1.48234
$$726$$ 0 0
$$727$$ 35484.5 1.81024 0.905121 0.425154i $$-0.139780\pi$$
0.905121 + 0.425154i $$0.139780\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ −873.413 −0.0442828
$$731$$ −45312.9 −2.29269
$$732$$ 0 0
$$733$$ −15257.1 −0.768805 −0.384402 0.923166i $$-0.625592\pi$$
−0.384402 + 0.923166i $$0.625592\pi$$
$$734$$ 37756.9 1.89868
$$735$$ 0 0
$$736$$ 65193.4 3.26503
$$737$$ 2476.70 0.123786
$$738$$ 0 0
$$739$$ 25753.7 1.28196 0.640978 0.767559i $$-0.278529\pi$$
0.640978 + 0.767559i $$0.278529\pi$$
$$740$$ −637.123 −0.0316501
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 5703.55 0.281619 0.140809 0.990037i $$-0.455029\pi$$
0.140809 + 0.990037i $$0.455029\pi$$
$$744$$ 0 0
$$745$$ 104.427 0.00513545
$$746$$ 2315.31 0.113632
$$747$$ 0 0
$$748$$ −71661.7 −3.50296
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 15450.9 0.750747 0.375373 0.926874i $$-0.377515\pi$$
0.375373 + 0.926874i $$0.377515\pi$$
$$752$$ 64999.7 3.15199
$$753$$ 0 0
$$754$$ 41208.0 1.99033
$$755$$ −172.784 −0.00832879
$$756$$ 0 0
$$757$$ 12434.7 0.597025 0.298513 0.954406i $$-0.403509\pi$$
0.298513 + 0.954406i $$0.403509\pi$$
$$758$$ −53057.6 −2.54240
$$759$$ 0 0
$$760$$ 318.164 0.0151855
$$761$$ 2318.62 0.110447 0.0552234 0.998474i $$-0.482413\pi$$
0.0552234 + 0.998474i $$0.482413\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ −39896.1 −1.88926
$$765$$ 0 0
$$766$$ −12093.2 −0.570425
$$767$$ −22702.9 −1.06878
$$768$$ 0 0
$$769$$ 23104.3 1.08344 0.541718 0.840560i $$-0.317774\pi$$
0.541718 + 0.840560i $$0.317774\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ −17950.2 −0.836841
$$773$$ 12433.6 0.578532 0.289266 0.957249i $$-0.406589\pi$$
0.289266 + 0.957249i $$0.406589\pi$$
$$774$$ 0 0
$$775$$ −35206.6 −1.63182
$$776$$ 63887.6 2.95545
$$777$$ 0 0
$$778$$ −41867.1 −1.92932
$$779$$ −2347.67 −0.107977
$$780$$ 0 0
$$781$$ −2511.98 −0.115090
$$782$$ −59144.5 −2.70461
$$783$$ 0 0
$$784$$ 0 0
$$785$$ −1.17888 −5.36002e−5 0
$$786$$ 0 0
$$787$$ −24085.4 −1.09092 −0.545458 0.838138i $$-0.683644\pi$$
−0.545458 + 0.838138i $$0.683644\pi$$
$$788$$ −106386. −4.80943
$$789$$ 0 0
$$790$$ 396.570 0.0178599
$$791$$ 0 0
$$792$$ 0 0
$$793$$ −12055.0 −0.539831
$$794$$ 27189.8 1.21528
$$795$$ 0 0
$$796$$ −32358.6 −1.44085
$$797$$ −36596.8 −1.62651 −0.813253 0.581910i $$-0.802306\pi$$
−0.813253 + 0.581910i $$0.802306\pi$$
$$798$$ 0 0
$$799$$ −31547.6 −1.39684
$$800$$ −86882.4 −3.83970
$$801$$ 0 0
$$802$$ −26053.8 −1.14712
$$803$$ −22825.0 −1.00308
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 50135.9 2.19102
$$807$$ 0 0
$$808$$ −11856.5 −0.516226
$$809$$ −20746.3 −0.901610 −0.450805 0.892622i $$-0.648863\pi$$
−0.450805 + 0.892622i $$0.648863\pi$$
$$810$$ 0 0
$$811$$ −8538.03 −0.369680 −0.184840 0.982769i $$-0.559177\pi$$
−0.184840 + 0.982769i $$0.559177\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ −22751.4 −0.979650
$$815$$ −165.872 −0.00712915
$$816$$ 0 0
$$817$$ 8298.42 0.355355
$$818$$ −3262.09 −0.139433
$$819$$ 0 0
$$820$$ −482.562 −0.0205510
$$821$$ 20949.8 0.890564 0.445282 0.895390i $$-0.353103\pi$$
0.445282 + 0.895390i $$0.353103\pi$$
$$822$$ 0 0
$$823$$ −10130.5 −0.429073 −0.214536 0.976716i $$-0.568824\pi$$
−0.214536 + 0.976716i $$0.568824\pi$$
$$824$$ −10525.0 −0.444969
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 25327.3 1.06495 0.532476 0.846445i $$-0.321262\pi$$
0.532476 + 0.846445i $$0.321262\pi$$
$$828$$ 0 0
$$829$$ −19975.9 −0.836901 −0.418451 0.908240i $$-0.637427\pi$$
−0.418451 + 0.908240i $$0.637427\pi$$
$$830$$ 1002.46 0.0419227
$$831$$ 0 0
$$832$$ 61704.6 2.57118
$$833$$ 0 0
$$834$$ 0 0
$$835$$ −112.870 −0.00467787
$$836$$ 13123.8 0.542939
$$837$$ 0 0
$$838$$ 52279.6 2.15510
$$839$$ 3888.98 0.160027 0.0800134 0.996794i $$-0.474504\pi$$
0.0800134 + 0.996794i $$0.474504\pi$$
$$840$$ 0 0
$$841$$ 29236.1 1.19874
$$842$$ −32521.3 −1.33107
$$843$$ 0 0
$$844$$ −95166.2 −3.88123
$$845$$ −226.116 −0.00920547
$$846$$ 0 0
$$847$$ 0 0
$$848$$ −80998.1 −3.28006
$$849$$ 0 0
$$850$$ 78821.2 3.18064
$$851$$ −13741.7 −0.553537
$$852$$ 0 0
$$853$$ −32093.5 −1.28823 −0.644116 0.764928i $$-0.722775\pi$$
−0.644116 + 0.764928i $$0.722775\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ −31386.1 −1.25322
$$857$$ 29879.8 1.19098 0.595492 0.803361i $$-0.296957\pi$$
0.595492 + 0.803361i $$0.296957\pi$$
$$858$$ 0 0
$$859$$ 32988.3 1.31030 0.655150 0.755499i $$-0.272606\pi$$
0.655150 + 0.755499i $$0.272606\pi$$
$$860$$ 1705.73 0.0676337
$$861$$ 0 0
$$862$$ −13891.7 −0.548903
$$863$$ 26716.8 1.05382 0.526912 0.849920i $$-0.323350\pi$$
0.526912 + 0.849920i $$0.323350\pi$$
$$864$$ 0 0
$$865$$ −821.590 −0.0322947
$$866$$ 53422.1 2.09626
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 10363.6 0.404558
$$870$$ 0 0
$$871$$ 2838.98 0.110442
$$872$$ −63763.9 −2.47628
$$873$$ 0 0
$$874$$ 10831.5 0.419199
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −17965.7 −0.691742 −0.345871 0.938282i $$-0.612417\pi$$
−0.345871 + 0.938282i $$0.612417\pi$$
$$878$$ −91868.6 −3.53122
$$879$$ 0 0
$$880$$ 1346.81 0.0515921
$$881$$ 15852.9 0.606239 0.303119 0.952953i $$-0.401972\pi$$
0.303119 + 0.952953i $$0.401972\pi$$
$$882$$ 0 0
$$883$$ −22050.7 −0.840392 −0.420196 0.907433i $$-0.638039\pi$$
−0.420196 + 0.907433i $$0.638039\pi$$
$$884$$ −82143.9 −3.12534
$$885$$ 0 0
$$886$$ 20354.7 0.771817
$$887$$ 27772.8 1.05132 0.525658 0.850696i $$-0.323819\pi$$
0.525658 + 0.850696i $$0.323819\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ −229.922 −0.00865956
$$891$$ 0 0
$$892$$ −72320.5 −2.71465
$$893$$ 5777.50 0.216502
$$894$$ 0 0
$$895$$ 184.728 0.00689918
$$896$$ 0 0
$$897$$ 0 0
$$898$$ −50723.8 −1.88494
$$899$$ 65243.2 2.42045
$$900$$ 0 0
$$901$$ 39312.4 1.45359
$$902$$ −17232.1 −0.636103
$$903$$ 0 0
$$904$$ −101394. −3.73043
$$905$$ 241.288 0.00886262
$$906$$ 0 0
$$907$$ −5896.03 −0.215848 −0.107924 0.994159i $$-0.534420\pi$$
−0.107924 + 0.994159i $$0.534420\pi$$
$$908$$ −17428.2 −0.636978
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 42197.8 1.53466 0.767331 0.641251i $$-0.221584\pi$$
0.767331 + 0.641251i $$0.221584\pi$$
$$912$$ 0 0
$$913$$ 26197.4 0.949623
$$914$$ 65647.4 2.37574
$$915$$ 0 0
$$916$$ −7748.17 −0.279483
$$917$$ 0 0
$$918$$ 0 0
$$919$$ −40928.7 −1.46911 −0.734555 0.678549i $$-0.762609\pi$$
−0.734555 + 0.678549i $$0.762609\pi$$
$$920$$ 1410.54 0.0505479
$$921$$ 0 0
$$922$$ −52278.6 −1.86736
$$923$$ −2879.41 −0.102684
$$924$$ 0 0
$$925$$ 18313.4 0.650964
$$926$$ 12371.7 0.439050
$$927$$ 0 0
$$928$$ 161007. 5.69537
$$929$$ −21478.9 −0.758558 −0.379279 0.925282i $$-0.623828\pi$$
−0.379279 + 0.925282i $$0.623828\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ −97670.3 −3.43272
$$933$$ 0 0
$$934$$ −29061.9 −1.01813
$$935$$ −653.675 −0.0228636
$$936$$ 0 0
$$937$$ 19560.1 0.681964 0.340982 0.940070i $$-0.389241\pi$$
0.340982 + 0.940070i $$0.389241\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 1187.56 0.0412063
$$941$$ −48602.4 −1.68373 −0.841867 0.539685i $$-0.818543\pi$$
−0.841867 + 0.539685i $$0.818543\pi$$
$$942$$ 0 0
$$943$$ −10408.1 −0.359421
$$944$$ −165806. −5.71666
$$945$$ 0 0
$$946$$ 60911.0 2.09343
$$947$$ −28778.0 −0.987496 −0.493748 0.869605i $$-0.664373\pi$$
−0.493748 + 0.869605i $$0.664373\pi$$
$$948$$ 0 0
$$949$$ −26163.7 −0.894951
$$950$$ −14435.0 −0.492982
$$951$$ 0 0
$$952$$ 0 0
$$953$$ −26009.2 −0.884072 −0.442036 0.896997i $$-0.645744\pi$$
−0.442036 + 0.896997i $$0.645744\pi$$
$$954$$ 0 0
$$955$$ −363.920 −0.0123311
$$956$$ −32629.9 −1.10390
$$957$$ 0 0
$$958$$ 20574.4 0.693872
$$959$$ 0 0
$$960$$ 0 0
$$961$$ 49587.5 1.66451
$$962$$ −26079.3 −0.874043
$$963$$ 0 0
$$964$$ −3424.97 −0.114430
$$965$$ −163.736 −0.00546201
$$966$$ 0 0
$$967$$ −13284.5 −0.441779 −0.220889 0.975299i $$-0.570896\pi$$
−0.220889 + 0.975299i $$0.570896\pi$$
$$968$$ −39517.2 −1.31212
$$969$$ 0 0
$$970$$ 919.836 0.0304476
$$971$$ 44248.6 1.46242 0.731208 0.682155i $$-0.238957\pi$$
0.731208 + 0.682155i $$0.238957\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 48380.0 1.59158
$$975$$ 0 0
$$976$$ −88041.2 −2.88743
$$977$$ 48967.4 1.60349 0.801744 0.597668i $$-0.203906\pi$$
0.801744 + 0.597668i $$0.203906\pi$$
$$978$$ 0 0
$$979$$ −6008.58 −0.196154
$$980$$ 0 0
$$981$$ 0 0
$$982$$ −94549.7 −3.07251
$$983$$ 49856.3 1.61767 0.808835 0.588035i $$-0.200098\pi$$
0.808835 + 0.588035i $$0.200098\pi$$
$$984$$ 0 0
$$985$$ −970.414 −0.0313908
$$986$$ −146068. −4.71780
$$987$$ 0 0
$$988$$ 15043.5 0.484410
$$989$$ 36790.0 1.18286
$$990$$ 0 0
$$991$$ 48648.1 1.55939 0.779696 0.626158i $$-0.215373\pi$$
0.779696 + 0.626158i $$0.215373\pi$$
$$992$$ 195889. 6.26965
$$993$$ 0 0
$$994$$ 0 0
$$995$$ −295.165 −0.00940437
$$996$$ 0 0
$$997$$ 9709.15 0.308417 0.154209 0.988038i $$-0.450717\pi$$
0.154209 + 0.988038i $$0.450717\pi$$
$$998$$ −38398.8 −1.21793
$$999$$ 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 1323.4.a.bo.1.8 8
3.2 odd 2 inner 1323.4.a.bo.1.1 8
7.2 even 3 189.4.e.h.109.1 16
7.4 even 3 189.4.e.h.163.1 yes 16
7.6 odd 2 1323.4.a.bn.1.8 8
21.2 odd 6 189.4.e.h.109.8 yes 16
21.11 odd 6 189.4.e.h.163.8 yes 16
21.20 even 2 1323.4.a.bn.1.1 8

By twisted newform
Twist Min Dim Char Parity Ord Type
189.4.e.h.109.1 16 7.2 even 3
189.4.e.h.109.8 yes 16 21.2 odd 6
189.4.e.h.163.1 yes 16 7.4 even 3
189.4.e.h.163.8 yes 16 21.11 odd 6
1323.4.a.bn.1.1 8 21.20 even 2
1323.4.a.bn.1.8 8 7.6 odd 2
1323.4.a.bo.1.1 8 3.2 odd 2 inner
1323.4.a.bo.1.8 8 1.1 even 1 trivial