# Properties

 Label 1323.4.a.bb.1.4 Level $1323$ Weight $4$ Character 1323.1 Self dual yes Analytic conductor $78.060$ Analytic rank $1$ Dimension $6$ 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: $$1$$ Dimension: $$6$$ Coefficient field: 6.6.346909504.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - x^{5} - 12x^{4} + 2x^{3} + 39x^{2} + 25x + 2$$ x^6 - x^5 - 12*x^4 + 2*x^3 + 39*x^2 + 25*x + 2 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{4}\cdot 3^{4}$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.4 Root $$-2.24419$$ 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)

 $$f(q)$$ $$=$$ $$q+1.57109 q^{2} -5.53167 q^{4} +6.97204 q^{5} -21.2595 q^{8} +O(q^{10})$$ $$q+1.57109 q^{2} -5.53167 q^{4} +6.97204 q^{5} -21.2595 q^{8} +10.9537 q^{10} -32.5591 q^{11} +19.3425 q^{13} +10.8528 q^{16} +22.2459 q^{17} +155.215 q^{19} -38.5671 q^{20} -51.1534 q^{22} -76.9512 q^{23} -76.3906 q^{25} +30.3888 q^{26} +122.079 q^{29} -164.774 q^{31} +187.127 q^{32} +34.9504 q^{34} +66.2329 q^{37} +243.858 q^{38} -148.222 q^{40} -231.844 q^{41} -210.361 q^{43} +180.106 q^{44} -120.897 q^{46} +206.307 q^{47} -120.017 q^{50} -106.996 q^{52} -419.641 q^{53} -227.004 q^{55} +191.797 q^{58} +300.800 q^{59} -24.2225 q^{61} -258.875 q^{62} +207.171 q^{64} +134.857 q^{65} -274.603 q^{67} -123.057 q^{68} -336.736 q^{71} -693.601 q^{73} +104.058 q^{74} -858.601 q^{76} -584.526 q^{79} +75.6661 q^{80} -364.248 q^{82} -1209.75 q^{83} +155.100 q^{85} -330.497 q^{86} +692.191 q^{88} -1258.68 q^{89} +425.669 q^{92} +324.127 q^{94} +1082.17 q^{95} -1085.60 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + 18 q^{4} - 24 q^{5}+O(q^{10})$$ 6 * q + 18 * q^4 - 24 * q^5 $$6 q + 18 q^{4} - 24 q^{5} - 30 q^{16} - 42 q^{17} - 12 q^{20} + 132 q^{22} + 222 q^{25} - 366 q^{26} - 312 q^{37} - 336 q^{38} - 360 q^{41} + 654 q^{43} + 774 q^{46} - 1812 q^{47} - 378 q^{58} + 6 q^{59} - 2058 q^{62} + 66 q^{64} + 42 q^{67} - 2910 q^{68} + 1956 q^{79} + 2868 q^{80} - 2892 q^{83} - 1944 q^{85} - 2532 q^{88} - 1518 q^{89}+O(q^{100})$$ 6 * q + 18 * q^4 - 24 * q^5 - 30 * q^16 - 42 * q^17 - 12 * q^20 + 132 * q^22 + 222 * q^25 - 366 * q^26 - 312 * q^37 - 336 * q^38 - 360 * q^41 + 654 * q^43 + 774 * q^46 - 1812 * q^47 - 378 * q^58 + 6 * q^59 - 2058 * q^62 + 66 * q^64 + 42 * q^67 - 2910 * q^68 + 1956 * q^79 + 2868 * q^80 - 2892 * q^83 - 1944 * q^85 - 2532 * q^88 - 1518 * q^89

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.57109 0.555465 0.277732 0.960659i $$-0.410417\pi$$
0.277732 + 0.960659i $$0.410417\pi$$
$$3$$ 0 0
$$4$$ −5.53167 −0.691459
$$5$$ 6.97204 0.623599 0.311799 0.950148i $$-0.399068\pi$$
0.311799 + 0.950148i $$0.399068\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ −21.2595 −0.939546
$$9$$ 0 0
$$10$$ 10.9537 0.346387
$$11$$ −32.5591 −0.892450 −0.446225 0.894921i $$-0.647232\pi$$
−0.446225 + 0.894921i $$0.647232\pi$$
$$12$$ 0 0
$$13$$ 19.3425 0.412664 0.206332 0.978482i $$-0.433847\pi$$
0.206332 + 0.978482i $$0.433847\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 10.8528 0.169575
$$17$$ 22.2459 0.317379 0.158689 0.987329i $$-0.449273\pi$$
0.158689 + 0.987329i $$0.449273\pi$$
$$18$$ 0 0
$$19$$ 155.215 1.87415 0.937075 0.349127i $$-0.113522\pi$$
0.937075 + 0.349127i $$0.113522\pi$$
$$20$$ −38.5671 −0.431193
$$21$$ 0 0
$$22$$ −51.1534 −0.495724
$$23$$ −76.9512 −0.697628 −0.348814 0.937192i $$-0.613415\pi$$
−0.348814 + 0.937192i $$0.613415\pi$$
$$24$$ 0 0
$$25$$ −76.3906 −0.611125
$$26$$ 30.3888 0.229220
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 122.079 0.781707 0.390854 0.920453i $$-0.372180\pi$$
0.390854 + 0.920453i $$0.372180\pi$$
$$30$$ 0 0
$$31$$ −164.774 −0.954653 −0.477326 0.878726i $$-0.658394\pi$$
−0.477326 + 0.878726i $$0.658394\pi$$
$$32$$ 187.127 1.03374
$$33$$ 0 0
$$34$$ 34.9504 0.176293
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 66.2329 0.294287 0.147143 0.989115i $$-0.452992\pi$$
0.147143 + 0.989115i $$0.452992\pi$$
$$38$$ 243.858 1.04102
$$39$$ 0 0
$$40$$ −148.222 −0.585899
$$41$$ −231.844 −0.883121 −0.441560 0.897232i $$-0.645575\pi$$
−0.441560 + 0.897232i $$0.645575\pi$$
$$42$$ 0 0
$$43$$ −210.361 −0.746042 −0.373021 0.927823i $$-0.621678\pi$$
−0.373021 + 0.927823i $$0.621678\pi$$
$$44$$ 180.106 0.617093
$$45$$ 0 0
$$46$$ −120.897 −0.387508
$$47$$ 206.307 0.640276 0.320138 0.947371i $$-0.396271\pi$$
0.320138 + 0.947371i $$0.396271\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ −120.017 −0.339458
$$51$$ 0 0
$$52$$ −106.996 −0.285340
$$53$$ −419.641 −1.08759 −0.543794 0.839219i $$-0.683013\pi$$
−0.543794 + 0.839219i $$0.683013\pi$$
$$54$$ 0 0
$$55$$ −227.004 −0.556530
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 191.797 0.434211
$$59$$ 300.800 0.663742 0.331871 0.943325i $$-0.392320\pi$$
0.331871 + 0.943325i $$0.392320\pi$$
$$60$$ 0 0
$$61$$ −24.2225 −0.0508423 −0.0254211 0.999677i $$-0.508093\pi$$
−0.0254211 + 0.999677i $$0.508093\pi$$
$$62$$ −258.875 −0.530276
$$63$$ 0 0
$$64$$ 207.171 0.404630
$$65$$ 134.857 0.257337
$$66$$ 0 0
$$67$$ −274.603 −0.500718 −0.250359 0.968153i $$-0.580549\pi$$
−0.250359 + 0.968153i $$0.580549\pi$$
$$68$$ −123.057 −0.219454
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −336.736 −0.562863 −0.281431 0.959581i $$-0.590809\pi$$
−0.281431 + 0.959581i $$0.590809\pi$$
$$72$$ 0 0
$$73$$ −693.601 −1.11205 −0.556027 0.831164i $$-0.687675\pi$$
−0.556027 + 0.831164i $$0.687675\pi$$
$$74$$ 104.058 0.163466
$$75$$ 0 0
$$76$$ −858.601 −1.29590
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −584.526 −0.832459 −0.416230 0.909260i $$-0.636649\pi$$
−0.416230 + 0.909260i $$0.636649\pi$$
$$80$$ 75.6661 0.105747
$$81$$ 0 0
$$82$$ −364.248 −0.490542
$$83$$ −1209.75 −1.59984 −0.799922 0.600104i $$-0.795126\pi$$
−0.799922 + 0.600104i $$0.795126\pi$$
$$84$$ 0 0
$$85$$ 155.100 0.197917
$$86$$ −330.497 −0.414400
$$87$$ 0 0
$$88$$ 692.191 0.838497
$$89$$ −1258.68 −1.49910 −0.749550 0.661948i $$-0.769730\pi$$
−0.749550 + 0.661948i $$0.769730\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 425.669 0.482381
$$93$$ 0 0
$$94$$ 324.127 0.355651
$$95$$ 1082.17 1.16872
$$96$$ 0 0
$$97$$ −1085.60 −1.13636 −0.568178 0.822906i $$-0.692351\pi$$
−0.568178 + 0.822906i $$0.692351\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 422.568 0.422568
$$101$$ 844.120 0.831615 0.415807 0.909453i $$-0.363499\pi$$
0.415807 + 0.909453i $$0.363499\pi$$
$$102$$ 0 0
$$103$$ 1290.47 1.23450 0.617250 0.786767i $$-0.288247\pi$$
0.617250 + 0.786767i $$0.288247\pi$$
$$104$$ −411.211 −0.387717
$$105$$ 0 0
$$106$$ −659.295 −0.604117
$$107$$ −212.075 −0.191608 −0.0958039 0.995400i $$-0.530542\pi$$
−0.0958039 + 0.995400i $$0.530542\pi$$
$$108$$ 0 0
$$109$$ 524.864 0.461219 0.230609 0.973046i $$-0.425928\pi$$
0.230609 + 0.973046i $$0.425928\pi$$
$$110$$ −356.644 −0.309133
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 590.941 0.491956 0.245978 0.969275i $$-0.420891\pi$$
0.245978 + 0.969275i $$0.420891\pi$$
$$114$$ 0 0
$$115$$ −536.507 −0.435040
$$116$$ −675.302 −0.540519
$$117$$ 0 0
$$118$$ 472.584 0.368685
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −270.903 −0.203533
$$122$$ −38.0558 −0.0282411
$$123$$ 0 0
$$124$$ 911.474 0.660103
$$125$$ −1404.10 −1.00470
$$126$$ 0 0
$$127$$ 2374.15 1.65883 0.829415 0.558633i $$-0.188674\pi$$
0.829415 + 0.558633i $$0.188674\pi$$
$$128$$ −1171.53 −0.808981
$$129$$ 0 0
$$130$$ 211.872 0.142941
$$131$$ −2661.28 −1.77494 −0.887469 0.460868i $$-0.847538\pi$$
−0.887469 + 0.460868i $$0.847538\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ −431.426 −0.278131
$$135$$ 0 0
$$136$$ −472.938 −0.298192
$$137$$ −1405.23 −0.876327 −0.438164 0.898895i $$-0.644371\pi$$
−0.438164 + 0.898895i $$0.644371\pi$$
$$138$$ 0 0
$$139$$ 858.529 0.523881 0.261941 0.965084i $$-0.415638\pi$$
0.261941 + 0.965084i $$0.415638\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ −529.044 −0.312650
$$143$$ −629.774 −0.368282
$$144$$ 0 0
$$145$$ 851.141 0.487472
$$146$$ −1089.71 −0.617706
$$147$$ 0 0
$$148$$ −366.378 −0.203487
$$149$$ 3114.11 1.71220 0.856101 0.516808i $$-0.172880\pi$$
0.856101 + 0.516808i $$0.172880\pi$$
$$150$$ 0 0
$$151$$ −26.7635 −0.0144237 −0.00721186 0.999974i $$-0.502296\pi$$
−0.00721186 + 0.999974i $$0.502296\pi$$
$$152$$ −3299.80 −1.76085
$$153$$ 0 0
$$154$$ 0 0
$$155$$ −1148.81 −0.595320
$$156$$ 0 0
$$157$$ −2362.94 −1.20117 −0.600583 0.799562i $$-0.705065\pi$$
−0.600583 + 0.799562i $$0.705065\pi$$
$$158$$ −918.343 −0.462402
$$159$$ 0 0
$$160$$ 1304.66 0.644638
$$161$$ 0 0
$$162$$ 0 0
$$163$$ −284.500 −0.136710 −0.0683550 0.997661i $$-0.521775\pi$$
−0.0683550 + 0.997661i $$0.521775\pi$$
$$164$$ 1282.49 0.610642
$$165$$ 0 0
$$166$$ −1900.62 −0.888657
$$167$$ −3642.66 −1.68789 −0.843946 0.536429i $$-0.819773\pi$$
−0.843946 + 0.536429i $$0.819773\pi$$
$$168$$ 0 0
$$169$$ −1822.87 −0.829708
$$170$$ 243.676 0.109936
$$171$$ 0 0
$$172$$ 1163.65 0.515857
$$173$$ −2613.40 −1.14852 −0.574258 0.818674i $$-0.694709\pi$$
−0.574258 + 0.818674i $$0.694709\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ −353.357 −0.151337
$$177$$ 0 0
$$178$$ −1977.50 −0.832697
$$179$$ 2757.44 1.15140 0.575700 0.817661i $$-0.304730\pi$$
0.575700 + 0.817661i $$0.304730\pi$$
$$180$$ 0 0
$$181$$ −2204.50 −0.905298 −0.452649 0.891689i $$-0.649521\pi$$
−0.452649 + 0.891689i $$0.649521\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 1635.94 0.655453
$$185$$ 461.778 0.183517
$$186$$ 0 0
$$187$$ −724.309 −0.283244
$$188$$ −1141.22 −0.442725
$$189$$ 0 0
$$190$$ 1700.19 0.649181
$$191$$ −3590.90 −1.36036 −0.680179 0.733046i $$-0.738098\pi$$
−0.680179 + 0.733046i $$0.738098\pi$$
$$192$$ 0 0
$$193$$ −3333.70 −1.24334 −0.621671 0.783278i $$-0.713546\pi$$
−0.621671 + 0.783278i $$0.713546\pi$$
$$194$$ −1705.58 −0.631205
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 3125.50 1.13037 0.565185 0.824964i $$-0.308805\pi$$
0.565185 + 0.824964i $$0.308805\pi$$
$$198$$ 0 0
$$199$$ 4384.56 1.56188 0.780938 0.624608i $$-0.214741\pi$$
0.780938 + 0.624608i $$0.214741\pi$$
$$200$$ 1624.03 0.574180
$$201$$ 0 0
$$202$$ 1326.19 0.461933
$$203$$ 0 0
$$204$$ 0 0
$$205$$ −1616.43 −0.550713
$$206$$ 2027.44 0.685721
$$207$$ 0 0
$$208$$ 209.920 0.0699774
$$209$$ −5053.68 −1.67259
$$210$$ 0 0
$$211$$ −2333.73 −0.761423 −0.380712 0.924694i $$-0.624321\pi$$
−0.380712 + 0.924694i $$0.624321\pi$$
$$212$$ 2321.32 0.752023
$$213$$ 0 0
$$214$$ −333.189 −0.106431
$$215$$ −1466.65 −0.465231
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 824.609 0.256191
$$219$$ 0 0
$$220$$ 1255.71 0.384818
$$221$$ 430.292 0.130971
$$222$$ 0 0
$$223$$ −226.956 −0.0681528 −0.0340764 0.999419i $$-0.510849\pi$$
−0.0340764 + 0.999419i $$0.510849\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 928.422 0.273264
$$227$$ 1298.84 0.379767 0.189884 0.981807i $$-0.439189\pi$$
0.189884 + 0.981807i $$0.439189\pi$$
$$228$$ 0 0
$$229$$ −445.975 −0.128694 −0.0643469 0.997928i $$-0.520496\pi$$
−0.0643469 + 0.997928i $$0.520496\pi$$
$$230$$ −842.902 −0.241649
$$231$$ 0 0
$$232$$ −2595.34 −0.734450
$$233$$ −3599.79 −1.01215 −0.506073 0.862490i $$-0.668903\pi$$
−0.506073 + 0.862490i $$0.668903\pi$$
$$234$$ 0 0
$$235$$ 1438.38 0.399275
$$236$$ −1663.93 −0.458951
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −5201.62 −1.40780 −0.703902 0.710298i $$-0.748560\pi$$
−0.703902 + 0.710298i $$0.748560\pi$$
$$240$$ 0 0
$$241$$ 5929.56 1.58488 0.792440 0.609949i $$-0.208810\pi$$
0.792440 + 0.609949i $$0.208810\pi$$
$$242$$ −425.613 −0.113055
$$243$$ 0 0
$$244$$ 133.991 0.0351553
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 3002.25 0.773395
$$248$$ 3503.01 0.896940
$$249$$ 0 0
$$250$$ −2205.98 −0.558073
$$251$$ −5101.02 −1.28276 −0.641381 0.767222i $$-0.721638\pi$$
−0.641381 + 0.767222i $$0.721638\pi$$
$$252$$ 0 0
$$253$$ 2505.47 0.622598
$$254$$ 3730.00 0.921421
$$255$$ 0 0
$$256$$ −3497.94 −0.853990
$$257$$ −1441.05 −0.349768 −0.174884 0.984589i $$-0.555955\pi$$
−0.174884 + 0.984589i $$0.555955\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ −745.982 −0.177938
$$261$$ 0 0
$$262$$ −4181.11 −0.985915
$$263$$ −7287.63 −1.70865 −0.854325 0.519740i $$-0.826029\pi$$
−0.854325 + 0.519740i $$0.826029\pi$$
$$264$$ 0 0
$$265$$ −2925.76 −0.678218
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 1519.01 0.346226
$$269$$ 1964.67 0.445310 0.222655 0.974897i $$-0.428528\pi$$
0.222655 + 0.974897i $$0.428528\pi$$
$$270$$ 0 0
$$271$$ −2304.66 −0.516599 −0.258299 0.966065i $$-0.583162\pi$$
−0.258299 + 0.966065i $$0.583162\pi$$
$$272$$ 241.430 0.0538194
$$273$$ 0 0
$$274$$ −2207.74 −0.486769
$$275$$ 2487.21 0.545398
$$276$$ 0 0
$$277$$ −1685.97 −0.365704 −0.182852 0.983140i $$-0.558533\pi$$
−0.182852 + 0.983140i $$0.558533\pi$$
$$278$$ 1348.83 0.290997
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 2841.43 0.603223 0.301612 0.953431i $$-0.402475\pi$$
0.301612 + 0.953431i $$0.402475\pi$$
$$282$$ 0 0
$$283$$ 7307.50 1.53493 0.767466 0.641090i $$-0.221517\pi$$
0.767466 + 0.641090i $$0.221517\pi$$
$$284$$ 1862.72 0.389197
$$285$$ 0 0
$$286$$ −989.432 −0.204568
$$287$$ 0 0
$$288$$ 0 0
$$289$$ −4418.12 −0.899271
$$290$$ 1337.22 0.270773
$$291$$ 0 0
$$292$$ 3836.78 0.768940
$$293$$ −6778.11 −1.35147 −0.675736 0.737143i $$-0.736174\pi$$
−0.675736 + 0.737143i $$0.736174\pi$$
$$294$$ 0 0
$$295$$ 2097.19 0.413909
$$296$$ −1408.08 −0.276496
$$297$$ 0 0
$$298$$ 4892.56 0.951068
$$299$$ −1488.43 −0.287886
$$300$$ 0 0
$$301$$ 0 0
$$302$$ −42.0479 −0.00801187
$$303$$ 0 0
$$304$$ 1684.52 0.317809
$$305$$ −168.881 −0.0317052
$$306$$ 0 0
$$307$$ 8157.51 1.51653 0.758263 0.651949i $$-0.226048\pi$$
0.758263 + 0.651949i $$0.226048\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ −1804.88 −0.330679
$$311$$ 2545.26 0.464078 0.232039 0.972706i $$-0.425460\pi$$
0.232039 + 0.972706i $$0.425460\pi$$
$$312$$ 0 0
$$313$$ 2777.18 0.501519 0.250760 0.968049i $$-0.419320\pi$$
0.250760 + 0.968049i $$0.419320\pi$$
$$314$$ −3712.39 −0.667205
$$315$$ 0 0
$$316$$ 3233.40 0.575611
$$317$$ −4832.35 −0.856188 −0.428094 0.903734i $$-0.640815\pi$$
−0.428094 + 0.903734i $$0.640815\pi$$
$$318$$ 0 0
$$319$$ −3974.79 −0.697635
$$320$$ 1444.40 0.252327
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 3452.91 0.594815
$$324$$ 0 0
$$325$$ −1477.58 −0.252189
$$326$$ −446.975 −0.0759376
$$327$$ 0 0
$$328$$ 4928.89 0.829732
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 2609.24 0.433284 0.216642 0.976251i $$-0.430490\pi$$
0.216642 + 0.976251i $$0.430490\pi$$
$$332$$ 6691.93 1.10623
$$333$$ 0 0
$$334$$ −5722.96 −0.937564
$$335$$ −1914.54 −0.312247
$$336$$ 0 0
$$337$$ −8134.50 −1.31488 −0.657440 0.753507i $$-0.728360\pi$$
−0.657440 + 0.753507i $$0.728360\pi$$
$$338$$ −2863.89 −0.460874
$$339$$ 0 0
$$340$$ −857.961 −0.136851
$$341$$ 5364.89 0.851980
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 4472.17 0.700940
$$345$$ 0 0
$$346$$ −4105.89 −0.637960
$$347$$ 10500.5 1.62449 0.812246 0.583314i $$-0.198244\pi$$
0.812246 + 0.583314i $$0.198244\pi$$
$$348$$ 0 0
$$349$$ 10622.4 1.62924 0.814618 0.579998i $$-0.196947\pi$$
0.814618 + 0.579998i $$0.196947\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ −6092.68 −0.922560
$$353$$ −7575.12 −1.14216 −0.571081 0.820894i $$-0.693476\pi$$
−0.571081 + 0.820894i $$0.693476\pi$$
$$354$$ 0 0
$$355$$ −2347.74 −0.351000
$$356$$ 6962.61 1.03657
$$357$$ 0 0
$$358$$ 4332.18 0.639562
$$359$$ 9131.59 1.34247 0.671235 0.741244i $$-0.265764\pi$$
0.671235 + 0.741244i $$0.265764\pi$$
$$360$$ 0 0
$$361$$ 17232.8 2.51244
$$362$$ −3463.47 −0.502861
$$363$$ 0 0
$$364$$ 0 0
$$365$$ −4835.82 −0.693475
$$366$$ 0 0
$$367$$ −3582.52 −0.509554 −0.254777 0.967000i $$-0.582002\pi$$
−0.254777 + 0.967000i $$0.582002\pi$$
$$368$$ −835.135 −0.118300
$$369$$ 0 0
$$370$$ 725.496 0.101937
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 5192.26 0.720765 0.360382 0.932805i $$-0.382646\pi$$
0.360382 + 0.932805i $$0.382646\pi$$
$$374$$ −1137.96 −0.157332
$$375$$ 0 0
$$376$$ −4385.99 −0.601569
$$377$$ 2361.31 0.322583
$$378$$ 0 0
$$379$$ 10012.2 1.35696 0.678482 0.734617i $$-0.262638\pi$$
0.678482 + 0.734617i $$0.262638\pi$$
$$380$$ −5986.20 −0.808120
$$381$$ 0 0
$$382$$ −5641.63 −0.755631
$$383$$ 3347.99 0.446670 0.223335 0.974742i $$-0.428306\pi$$
0.223335 + 0.974742i $$0.428306\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −5237.55 −0.690633
$$387$$ 0 0
$$388$$ 6005.21 0.785743
$$389$$ 510.839 0.0665824 0.0332912 0.999446i $$-0.489401\pi$$
0.0332912 + 0.999446i $$0.489401\pi$$
$$390$$ 0 0
$$391$$ −1711.85 −0.221412
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 4910.45 0.627880
$$395$$ −4075.34 −0.519120
$$396$$ 0 0
$$397$$ 11828.7 1.49538 0.747691 0.664046i $$-0.231162\pi$$
0.747691 + 0.664046i $$0.231162\pi$$
$$398$$ 6888.55 0.867567
$$399$$ 0 0
$$400$$ −829.051 −0.103631
$$401$$ 7881.42 0.981494 0.490747 0.871302i $$-0.336724\pi$$
0.490747 + 0.871302i $$0.336724\pi$$
$$402$$ 0 0
$$403$$ −3187.13 −0.393951
$$404$$ −4669.40 −0.575028
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −2156.48 −0.262636
$$408$$ 0 0
$$409$$ 1579.66 0.190976 0.0954881 0.995431i $$-0.469559\pi$$
0.0954881 + 0.995431i $$0.469559\pi$$
$$410$$ −2539.55 −0.305901
$$411$$ 0 0
$$412$$ −7138.44 −0.853606
$$413$$ 0 0
$$414$$ 0 0
$$415$$ −8434.42 −0.997661
$$416$$ 3619.49 0.426587
$$417$$ 0 0
$$418$$ −7939.79 −0.929062
$$419$$ −433.887 −0.0505889 −0.0252944 0.999680i $$-0.508052\pi$$
−0.0252944 + 0.999680i $$0.508052\pi$$
$$420$$ 0 0
$$421$$ 813.228 0.0941432 0.0470716 0.998892i $$-0.485011\pi$$
0.0470716 + 0.998892i $$0.485011\pi$$
$$422$$ −3666.50 −0.422944
$$423$$ 0 0
$$424$$ 8921.36 1.02184
$$425$$ −1699.38 −0.193958
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 1173.13 0.132489
$$429$$ 0 0
$$430$$ −2304.24 −0.258419
$$431$$ −12624.4 −1.41090 −0.705450 0.708759i $$-0.749255\pi$$
−0.705450 + 0.708759i $$0.749255\pi$$
$$432$$ 0 0
$$433$$ −8037.33 −0.892031 −0.446016 0.895025i $$-0.647157\pi$$
−0.446016 + 0.895025i $$0.647157\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ −2903.38 −0.318914
$$437$$ −11944.0 −1.30746
$$438$$ 0 0
$$439$$ 6820.72 0.741538 0.370769 0.928725i $$-0.379094\pi$$
0.370769 + 0.928725i $$0.379094\pi$$
$$440$$ 4825.98 0.522886
$$441$$ 0 0
$$442$$ 676.027 0.0727496
$$443$$ 678.956 0.0728175 0.0364088 0.999337i $$-0.488408\pi$$
0.0364088 + 0.999337i $$0.488408\pi$$
$$444$$ 0 0
$$445$$ −8775.58 −0.934837
$$446$$ −356.568 −0.0378565
$$447$$ 0 0
$$448$$ 0 0
$$449$$ −9068.89 −0.953201 −0.476601 0.879120i $$-0.658131\pi$$
−0.476601 + 0.879120i $$0.658131\pi$$
$$450$$ 0 0
$$451$$ 7548.64 0.788141
$$452$$ −3268.89 −0.340168
$$453$$ 0 0
$$454$$ 2040.60 0.210947
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 4566.21 0.467392 0.233696 0.972310i $$-0.424918\pi$$
0.233696 + 0.972310i $$0.424918\pi$$
$$458$$ −700.668 −0.0714848
$$459$$ 0 0
$$460$$ 2967.78 0.300812
$$461$$ 7604.79 0.768308 0.384154 0.923269i $$-0.374493\pi$$
0.384154 + 0.923269i $$0.374493\pi$$
$$462$$ 0 0
$$463$$ −10608.6 −1.06484 −0.532422 0.846479i $$-0.678718\pi$$
−0.532422 + 0.846479i $$0.678718\pi$$
$$464$$ 1324.90 0.132558
$$465$$ 0 0
$$466$$ −5655.60 −0.562212
$$467$$ 13597.3 1.34734 0.673668 0.739034i $$-0.264718\pi$$
0.673668 + 0.739034i $$0.264718\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 2259.83 0.221783
$$471$$ 0 0
$$472$$ −6394.85 −0.623616
$$473$$ 6849.18 0.665805
$$474$$ 0 0
$$475$$ −11857.0 −1.14534
$$476$$ 0 0
$$477$$ 0 0
$$478$$ −8172.22 −0.781985
$$479$$ −11950.4 −1.13993 −0.569965 0.821669i $$-0.693043\pi$$
−0.569965 + 0.821669i $$0.693043\pi$$
$$480$$ 0 0
$$481$$ 1281.11 0.121442
$$482$$ 9315.87 0.880345
$$483$$ 0 0
$$484$$ 1498.54 0.140735
$$485$$ −7568.88 −0.708629
$$486$$ 0 0
$$487$$ 16336.6 1.52009 0.760044 0.649872i $$-0.225178\pi$$
0.760044 + 0.649872i $$0.225178\pi$$
$$488$$ 514.959 0.0477686
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 21307.1 1.95841 0.979204 0.202880i $$-0.0650302\pi$$
0.979204 + 0.202880i $$0.0650302\pi$$
$$492$$ 0 0
$$493$$ 2715.77 0.248097
$$494$$ 4716.81 0.429594
$$495$$ 0 0
$$496$$ −1788.25 −0.161885
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 16006.8 1.43600 0.718000 0.696043i $$-0.245058\pi$$
0.718000 + 0.696043i $$0.245058\pi$$
$$500$$ 7767.04 0.694706
$$501$$ 0 0
$$502$$ −8014.16 −0.712529
$$503$$ 14633.8 1.29719 0.648597 0.761132i $$-0.275356\pi$$
0.648597 + 0.761132i $$0.275356\pi$$
$$504$$ 0 0
$$505$$ 5885.24 0.518594
$$506$$ 3936.31 0.345831
$$507$$ 0 0
$$508$$ −13133.0 −1.14701
$$509$$ 3992.22 0.347646 0.173823 0.984777i $$-0.444388\pi$$
0.173823 + 0.984777i $$0.444388\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 3876.64 0.334619
$$513$$ 0 0
$$514$$ −2264.03 −0.194284
$$515$$ 8997.19 0.769832
$$516$$ 0 0
$$517$$ −6717.18 −0.571415
$$518$$ 0 0
$$519$$ 0 0
$$520$$ −2866.98 −0.241780
$$521$$ −21464.7 −1.80496 −0.902480 0.430731i $$-0.858256\pi$$
−0.902480 + 0.430731i $$0.858256\pi$$
$$522$$ 0 0
$$523$$ −5217.23 −0.436201 −0.218101 0.975926i $$-0.569986\pi$$
−0.218101 + 0.975926i $$0.569986\pi$$
$$524$$ 14721.3 1.22730
$$525$$ 0 0
$$526$$ −11449.5 −0.949094
$$527$$ −3665.55 −0.302986
$$528$$ 0 0
$$529$$ −6245.51 −0.513316
$$530$$ −4596.63 −0.376726
$$531$$ 0 0
$$532$$ 0 0
$$533$$ −4484.44 −0.364432
$$534$$ 0 0
$$535$$ −1478.59 −0.119486
$$536$$ 5837.92 0.470447
$$537$$ 0 0
$$538$$ 3086.68 0.247354
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −9563.03 −0.759976 −0.379988 0.924991i $$-0.624072\pi$$
−0.379988 + 0.924991i $$0.624072\pi$$
$$542$$ −3620.83 −0.286952
$$543$$ 0 0
$$544$$ 4162.81 0.328086
$$545$$ 3659.37 0.287615
$$546$$ 0 0
$$547$$ 2206.54 0.172477 0.0862383 0.996275i $$-0.472515\pi$$
0.0862383 + 0.996275i $$0.472515\pi$$
$$548$$ 7773.27 0.605944
$$549$$ 0 0
$$550$$ 3907.64 0.302949
$$551$$ 18948.6 1.46504
$$552$$ 0 0
$$553$$ 0 0
$$554$$ −2648.81 −0.203136
$$555$$ 0 0
$$556$$ −4749.10 −0.362242
$$557$$ −4723.55 −0.359324 −0.179662 0.983728i $$-0.557500\pi$$
−0.179662 + 0.983728i $$0.557500\pi$$
$$558$$ 0 0
$$559$$ −4068.91 −0.307865
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 4464.15 0.335069
$$563$$ −14140.9 −1.05856 −0.529280 0.848447i $$-0.677538\pi$$
−0.529280 + 0.848447i $$0.677538\pi$$
$$564$$ 0 0
$$565$$ 4120.07 0.306783
$$566$$ 11480.7 0.852600
$$567$$ 0 0
$$568$$ 7158.85 0.528835
$$569$$ −14817.9 −1.09174 −0.545868 0.837871i $$-0.683800\pi$$
−0.545868 + 0.837871i $$0.683800\pi$$
$$570$$ 0 0
$$571$$ −20877.9 −1.53015 −0.765073 0.643944i $$-0.777297\pi$$
−0.765073 + 0.643944i $$0.777297\pi$$
$$572$$ 3483.70 0.254652
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 5878.35 0.426338
$$576$$ 0 0
$$577$$ −25816.5 −1.86266 −0.931330 0.364176i $$-0.881351\pi$$
−0.931330 + 0.364176i $$0.881351\pi$$
$$578$$ −6941.27 −0.499513
$$579$$ 0 0
$$580$$ −4708.23 −0.337067
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 13663.2 0.970618
$$584$$ 14745.6 1.04483
$$585$$ 0 0
$$586$$ −10649.0 −0.750695
$$587$$ −1727.78 −0.121487 −0.0607435 0.998153i $$-0.519347\pi$$
−0.0607435 + 0.998153i $$0.519347\pi$$
$$588$$ 0 0
$$589$$ −25575.4 −1.78916
$$590$$ 3294.88 0.229912
$$591$$ 0 0
$$592$$ 718.811 0.0499036
$$593$$ 14580.1 1.00966 0.504832 0.863217i $$-0.331554\pi$$
0.504832 + 0.863217i $$0.331554\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −17226.3 −1.18392
$$597$$ 0 0
$$598$$ −2338.45 −0.159911
$$599$$ −27691.5 −1.88889 −0.944444 0.328673i $$-0.893398\pi$$
−0.944444 + 0.328673i $$0.893398\pi$$
$$600$$ 0 0
$$601$$ 783.554 0.0531811 0.0265906 0.999646i $$-0.491535\pi$$
0.0265906 + 0.999646i $$0.491535\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 148.047 0.00997342
$$605$$ −1888.74 −0.126923
$$606$$ 0 0
$$607$$ 23576.1 1.57648 0.788241 0.615366i $$-0.210992\pi$$
0.788241 + 0.615366i $$0.210992\pi$$
$$608$$ 29044.9 1.93738
$$609$$ 0 0
$$610$$ −265.327 −0.0176111
$$611$$ 3990.49 0.264219
$$612$$ 0 0
$$613$$ 25158.3 1.65764 0.828820 0.559515i $$-0.189012\pi$$
0.828820 + 0.559515i $$0.189012\pi$$
$$614$$ 12816.2 0.842377
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −12873.3 −0.839968 −0.419984 0.907532i $$-0.637964\pi$$
−0.419984 + 0.907532i $$0.637964\pi$$
$$618$$ 0 0
$$619$$ −14591.6 −0.947476 −0.473738 0.880666i $$-0.657096\pi$$
−0.473738 + 0.880666i $$0.657096\pi$$
$$620$$ 6354.84 0.411640
$$621$$ 0 0
$$622$$ 3998.83 0.257779
$$623$$ 0 0
$$624$$ 0 0
$$625$$ −240.648 −0.0154015
$$626$$ 4363.20 0.278576
$$627$$ 0 0
$$628$$ 13071.0 0.830557
$$629$$ 1473.41 0.0934003
$$630$$ 0 0
$$631$$ 20163.1 1.27208 0.636039 0.771657i $$-0.280572\pi$$
0.636039 + 0.771657i $$0.280572\pi$$
$$632$$ 12426.7 0.782133
$$633$$ 0 0
$$634$$ −7592.05 −0.475582
$$635$$ 16552.6 1.03444
$$636$$ 0 0
$$637$$ 0 0
$$638$$ −6244.76 −0.387511
$$639$$ 0 0
$$640$$ −8167.95 −0.504479
$$641$$ 10687.8 0.658572 0.329286 0.944230i $$-0.393192\pi$$
0.329286 + 0.944230i $$0.393192\pi$$
$$642$$ 0 0
$$643$$ 9248.53 0.567226 0.283613 0.958939i $$-0.408467\pi$$
0.283613 + 0.958939i $$0.408467\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 5424.84 0.330399
$$647$$ −14521.5 −0.882380 −0.441190 0.897414i $$-0.645444\pi$$
−0.441190 + 0.897414i $$0.645444\pi$$
$$648$$ 0 0
$$649$$ −9793.78 −0.592357
$$650$$ −2321.42 −0.140082
$$651$$ 0 0
$$652$$ 1573.76 0.0945294
$$653$$ 5894.47 0.353244 0.176622 0.984279i $$-0.443483\pi$$
0.176622 + 0.984279i $$0.443483\pi$$
$$654$$ 0 0
$$655$$ −18554.5 −1.10685
$$656$$ −2516.15 −0.149755
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 8398.63 0.496455 0.248228 0.968702i $$-0.420152\pi$$
0.248228 + 0.968702i $$0.420152\pi$$
$$660$$ 0 0
$$661$$ −3513.82 −0.206765 −0.103383 0.994642i $$-0.532967\pi$$
−0.103383 + 0.994642i $$0.532967\pi$$
$$662$$ 4099.36 0.240674
$$663$$ 0 0
$$664$$ 25718.6 1.50313
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −9394.13 −0.545341
$$668$$ 20150.0 1.16711
$$669$$ 0 0
$$670$$ −3007.92 −0.173442
$$671$$ 788.665 0.0453742
$$672$$ 0 0
$$673$$ −14499.1 −0.830457 −0.415229 0.909717i $$-0.636298\pi$$
−0.415229 + 0.909717i $$0.636298\pi$$
$$674$$ −12780.0 −0.730369
$$675$$ 0 0
$$676$$ 10083.5 0.573709
$$677$$ 26741.4 1.51810 0.759051 0.651031i $$-0.225663\pi$$
0.759051 + 0.651031i $$0.225663\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ −3297.34 −0.185952
$$681$$ 0 0
$$682$$ 8428.73 0.473245
$$683$$ 25949.3 1.45377 0.726883 0.686761i $$-0.240968\pi$$
0.726883 + 0.686761i $$0.240968\pi$$
$$684$$ 0 0
$$685$$ −9797.32 −0.546476
$$686$$ 0 0
$$687$$ 0 0
$$688$$ −2283.01 −0.126510
$$689$$ −8116.90 −0.448809
$$690$$ 0 0
$$691$$ −3583.56 −0.197286 −0.0986432 0.995123i $$-0.531450\pi$$
−0.0986432 + 0.995123i $$0.531450\pi$$
$$692$$ 14456.5 0.794152
$$693$$ 0 0
$$694$$ 16497.3 0.902348
$$695$$ 5985.70 0.326691
$$696$$ 0 0
$$697$$ −5157.59 −0.280284
$$698$$ 16688.7 0.904983
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 15096.1 0.813367 0.406683 0.913569i $$-0.366685\pi$$
0.406683 + 0.913569i $$0.366685\pi$$
$$702$$ 0 0
$$703$$ 10280.4 0.551538
$$704$$ −6745.30 −0.361112
$$705$$ 0 0
$$706$$ −11901.2 −0.634430
$$707$$ 0 0
$$708$$ 0 0
$$709$$ −29170.5 −1.54517 −0.772583 0.634914i $$-0.781036\pi$$
−0.772583 + 0.634914i $$0.781036\pi$$
$$710$$ −3688.52 −0.194968
$$711$$ 0 0
$$712$$ 26758.9 1.40847
$$713$$ 12679.5 0.665992
$$714$$ 0 0
$$715$$ −4390.81 −0.229660
$$716$$ −15253.2 −0.796146
$$717$$ 0 0
$$718$$ 14346.6 0.745695
$$719$$ 9145.94 0.474390 0.237195 0.971462i $$-0.423772\pi$$
0.237195 + 0.971462i $$0.423772\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 27074.4 1.39557
$$723$$ 0 0
$$724$$ 12194.6 0.625977
$$725$$ −9325.70 −0.477721
$$726$$ 0 0
$$727$$ −35509.8 −1.81154 −0.905768 0.423773i $$-0.860705\pi$$
−0.905768 + 0.423773i $$0.860705\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ −7597.51 −0.385201
$$731$$ −4679.69 −0.236778
$$732$$ 0 0
$$733$$ −6476.19 −0.326335 −0.163168 0.986598i $$-0.552171\pi$$
−0.163168 + 0.986598i $$0.552171\pi$$
$$734$$ −5628.47 −0.283039
$$735$$ 0 0
$$736$$ −14399.6 −0.721165
$$737$$ 8940.83 0.446866
$$738$$ 0 0
$$739$$ 31421.5 1.56408 0.782042 0.623226i $$-0.214178\pi$$
0.782042 + 0.623226i $$0.214178\pi$$
$$740$$ −2554.41 −0.126894
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −12845.7 −0.634268 −0.317134 0.948381i $$-0.602721\pi$$
−0.317134 + 0.948381i $$0.602721\pi$$
$$744$$ 0 0
$$745$$ 21711.7 1.06773
$$746$$ 8157.52 0.400359
$$747$$ 0 0
$$748$$ 4006.64 0.195852
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 24892.6 1.20951 0.604756 0.796411i $$-0.293271\pi$$
0.604756 + 0.796411i $$0.293271\pi$$
$$752$$ 2239.01 0.108575
$$753$$ 0 0
$$754$$ 3709.83 0.179183
$$755$$ −186.596 −0.00899461
$$756$$ 0 0
$$757$$ 34261.1 1.64497 0.822483 0.568789i $$-0.192588\pi$$
0.822483 + 0.568789i $$0.192588\pi$$
$$758$$ 15730.0 0.753746
$$759$$ 0 0
$$760$$ −23006.4 −1.09806
$$761$$ 8054.98 0.383696 0.191848 0.981425i $$-0.438552\pi$$
0.191848 + 0.981425i $$0.438552\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 19863.7 0.940632
$$765$$ 0 0
$$766$$ 5260.00 0.248109
$$767$$ 5818.21 0.273903
$$768$$ 0 0
$$769$$ 1915.85 0.0898403 0.0449202 0.998991i $$-0.485697\pi$$
0.0449202 + 0.998991i $$0.485697\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 18441.0 0.859721
$$773$$ −15431.6 −0.718027 −0.359014 0.933332i $$-0.616887\pi$$
−0.359014 + 0.933332i $$0.616887\pi$$
$$774$$ 0 0
$$775$$ 12587.2 0.583412
$$776$$ 23079.4 1.06766
$$777$$ 0 0
$$778$$ 802.574 0.0369842
$$779$$ −35985.8 −1.65510
$$780$$ 0 0
$$781$$ 10963.8 0.502327
$$782$$ −2689.48 −0.122987
$$783$$ 0 0
$$784$$ 0 0
$$785$$ −16474.5 −0.749046
$$786$$ 0 0
$$787$$ −18487.3 −0.837358 −0.418679 0.908134i $$-0.637507\pi$$
−0.418679 + 0.908134i $$0.637507\pi$$
$$788$$ −17289.3 −0.781604
$$789$$ 0 0
$$790$$ −6402.73 −0.288353
$$791$$ 0 0
$$792$$ 0 0
$$793$$ −468.524 −0.0209808
$$794$$ 18584.0 0.830632
$$795$$ 0 0
$$796$$ −24254.0 −1.07997
$$797$$ −7017.35 −0.311879 −0.155939 0.987767i $$-0.549840\pi$$
−0.155939 + 0.987767i $$0.549840\pi$$
$$798$$ 0 0
$$799$$ 4589.50 0.203210
$$800$$ −14294.7 −0.631743
$$801$$ 0 0
$$802$$ 12382.4 0.545185
$$803$$ 22583.1 0.992452
$$804$$ 0 0
$$805$$ 0 0
$$806$$ −5007.27 −0.218826
$$807$$ 0 0
$$808$$ −17945.6 −0.781340
$$809$$ −13231.2 −0.575012 −0.287506 0.957779i $$-0.592826\pi$$
−0.287506 + 0.957779i $$0.592826\pi$$
$$810$$ 0 0
$$811$$ 8930.40 0.386669 0.193335 0.981133i $$-0.438070\pi$$
0.193335 + 0.981133i $$0.438070\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ −3388.03 −0.145885
$$815$$ −1983.54 −0.0852521
$$816$$ 0 0
$$817$$ −32651.3 −1.39819
$$818$$ 2481.79 0.106081
$$819$$ 0 0
$$820$$ 8941.54 0.380795
$$821$$ 35165.3 1.49486 0.747429 0.664341i $$-0.231288\pi$$
0.747429 + 0.664341i $$0.231288\pi$$
$$822$$ 0 0
$$823$$ −9236.94 −0.391227 −0.195613 0.980681i $$-0.562670\pi$$
−0.195613 + 0.980681i $$0.562670\pi$$
$$824$$ −27434.7 −1.15987
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −22176.8 −0.932483 −0.466241 0.884658i $$-0.654392\pi$$
−0.466241 + 0.884658i $$0.654392\pi$$
$$828$$ 0 0
$$829$$ 36563.5 1.53185 0.765925 0.642930i $$-0.222281\pi$$
0.765925 + 0.642930i $$0.222281\pi$$
$$830$$ −13251.2 −0.554165
$$831$$ 0 0
$$832$$ 4007.19 0.166976
$$833$$ 0 0
$$834$$ 0 0
$$835$$ −25396.8 −1.05257
$$836$$ 27955.3 1.15652
$$837$$ 0 0
$$838$$ −681.675 −0.0281003
$$839$$ −44231.4 −1.82007 −0.910035 0.414531i $$-0.863945\pi$$
−0.910035 + 0.414531i $$0.863945\pi$$
$$840$$ 0 0
$$841$$ −9485.70 −0.388933
$$842$$ 1277.65 0.0522932
$$843$$ 0 0
$$844$$ 12909.4 0.526493
$$845$$ −12709.1 −0.517405
$$846$$ 0 0
$$847$$ 0 0
$$848$$ −4554.28 −0.184427
$$849$$ 0 0
$$850$$ −2669.88 −0.107737
$$851$$ −5096.70 −0.205303
$$852$$ 0 0
$$853$$ 19451.0 0.780762 0.390381 0.920653i $$-0.372343\pi$$
0.390381 + 0.920653i $$0.372343\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 4508.60 0.180024
$$857$$ −26386.1 −1.05173 −0.525865 0.850568i $$-0.676258\pi$$
−0.525865 + 0.850568i $$0.676258\pi$$
$$858$$ 0 0
$$859$$ −27227.7 −1.08149 −0.540744 0.841187i $$-0.681857\pi$$
−0.540744 + 0.841187i $$0.681857\pi$$
$$860$$ 8113.02 0.321688
$$861$$ 0 0
$$862$$ −19834.2 −0.783705
$$863$$ 20901.0 0.824425 0.412212 0.911088i $$-0.364756\pi$$
0.412212 + 0.911088i $$0.364756\pi$$
$$864$$ 0 0
$$865$$ −18220.8 −0.716213
$$866$$ −12627.4 −0.495492
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 19031.6 0.742928
$$870$$ 0 0
$$871$$ −5311.50 −0.206628
$$872$$ −11158.3 −0.433336
$$873$$ 0 0
$$874$$ −18765.1 −0.726248
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 12516.3 0.481921 0.240961 0.970535i $$-0.422538\pi$$
0.240961 + 0.970535i $$0.422538\pi$$
$$878$$ 10716.0 0.411898
$$879$$ 0 0
$$880$$ −2463.62 −0.0943735
$$881$$ 1308.26 0.0500301 0.0250150 0.999687i $$-0.492037\pi$$
0.0250150 + 0.999687i $$0.492037\pi$$
$$882$$ 0 0
$$883$$ 19537.5 0.744609 0.372305 0.928111i $$-0.378568\pi$$
0.372305 + 0.928111i $$0.378568\pi$$
$$884$$ −2380.23 −0.0905609
$$885$$ 0 0
$$886$$ 1066.70 0.0404475
$$887$$ 286.574 0.0108480 0.00542402 0.999985i $$-0.498273\pi$$
0.00542402 + 0.999985i $$0.498273\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ −13787.2 −0.519269
$$891$$ 0 0
$$892$$ 1255.44 0.0471249
$$893$$ 32022.1 1.19997
$$894$$ 0 0
$$895$$ 19225.0 0.718011
$$896$$ 0 0
$$897$$ 0 0
$$898$$ −14248.1 −0.529470
$$899$$ −20115.4 −0.746259
$$900$$ 0 0
$$901$$ −9335.32 −0.345177
$$902$$ 11859.6 0.437784
$$903$$ 0 0
$$904$$ −12563.1 −0.462215
$$905$$ −15369.9 −0.564543
$$906$$ 0 0
$$907$$ −7960.25 −0.291418 −0.145709 0.989328i $$-0.546546\pi$$
−0.145709 + 0.989328i $$0.546546\pi$$
$$908$$ −7184.77 −0.262594
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −38900.9 −1.41476 −0.707378 0.706835i $$-0.750122\pi$$
−0.707378 + 0.706835i $$0.750122\pi$$
$$912$$ 0 0
$$913$$ 39388.4 1.42778
$$914$$ 7173.93 0.259620
$$915$$ 0 0
$$916$$ 2466.99 0.0889865
$$917$$ 0 0
$$918$$ 0 0
$$919$$ 1315.54 0.0472205 0.0236103 0.999721i $$-0.492484\pi$$
0.0236103 + 0.999721i $$0.492484\pi$$
$$920$$ 11405.9 0.408740
$$921$$ 0 0
$$922$$ 11947.8 0.426768
$$923$$ −6513.31 −0.232273
$$924$$ 0 0
$$925$$ −5059.57 −0.179846
$$926$$ −16667.1 −0.591483
$$927$$ 0 0
$$928$$ 22844.2 0.808081
$$929$$ −46155.4 −1.63004 −0.815021 0.579432i $$-0.803274\pi$$
−0.815021 + 0.579432i $$0.803274\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 19912.9 0.699858
$$933$$ 0 0
$$934$$ 21362.5 0.748397
$$935$$ −5049.91 −0.176631
$$936$$ 0 0
$$937$$ 41274.8 1.43905 0.719525 0.694466i $$-0.244359\pi$$
0.719525 + 0.694466i $$0.244359\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ −7956.66 −0.276083
$$941$$ −29642.6 −1.02691 −0.513454 0.858117i $$-0.671634\pi$$
−0.513454 + 0.858117i $$0.671634\pi$$
$$942$$ 0 0
$$943$$ 17840.7 0.616090
$$944$$ 3264.52 0.112554
$$945$$ 0 0
$$946$$ 10760.7 0.369831
$$947$$ −24856.8 −0.852944 −0.426472 0.904501i $$-0.640244\pi$$
−0.426472 + 0.904501i $$0.640244\pi$$
$$948$$ 0 0
$$949$$ −13416.0 −0.458905
$$950$$ −18628.4 −0.636196
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 36288.1 1.23346 0.616729 0.787175i $$-0.288457\pi$$
0.616729 + 0.787175i $$0.288457\pi$$
$$954$$ 0 0
$$955$$ −25035.9 −0.848317
$$956$$ 28773.7 0.973438
$$957$$ 0 0
$$958$$ −18775.1 −0.633190
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −2640.61 −0.0886378
$$962$$ 2012.74 0.0674566
$$963$$ 0 0
$$964$$ −32800.4 −1.09588
$$965$$ −23242.7 −0.775347
$$966$$ 0 0
$$967$$ −16773.7 −0.557815 −0.278908 0.960318i $$-0.589972\pi$$
−0.278908 + 0.960318i $$0.589972\pi$$
$$968$$ 5759.25 0.191229
$$969$$ 0 0
$$970$$ −11891.4 −0.393619
$$971$$ −4458.85 −0.147365 −0.0736825 0.997282i $$-0.523475\pi$$
−0.0736825 + 0.997282i $$0.523475\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 25666.3 0.844355
$$975$$ 0 0
$$976$$ −262.882 −0.00862156
$$977$$ −4100.18 −0.134265 −0.0671323 0.997744i $$-0.521385\pi$$
−0.0671323 + 0.997744i $$0.521385\pi$$
$$978$$ 0 0
$$979$$ 40981.6 1.33787
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 33475.5 1.08783
$$983$$ −8704.08 −0.282418 −0.141209 0.989980i $$-0.545099\pi$$
−0.141209 + 0.989980i $$0.545099\pi$$
$$984$$ 0 0
$$985$$ 21791.1 0.704897
$$986$$ 4266.71 0.137809
$$987$$ 0 0
$$988$$ −16607.5 −0.534771
$$989$$ 16187.6 0.520460
$$990$$ 0 0
$$991$$ −20466.1 −0.656030 −0.328015 0.944673i $$-0.606380\pi$$
−0.328015 + 0.944673i $$0.606380\pi$$
$$992$$ −30833.6 −0.986861
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 30569.4 0.973984
$$996$$ 0 0
$$997$$ −9354.70 −0.297158 −0.148579 0.988901i $$-0.547470\pi$$
−0.148579 + 0.988901i $$0.547470\pi$$
$$998$$ 25148.2 0.797647
$$999$$ 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.bb.1.4 yes 6
3.2 odd 2 1323.4.a.bc.1.3 yes 6
7.6 odd 2 1323.4.a.bc.1.4 yes 6
21.20 even 2 inner 1323.4.a.bb.1.3 6

By twisted newform
Twist Min Dim Char Parity Ord Type
1323.4.a.bb.1.3 6 21.20 even 2 inner
1323.4.a.bb.1.4 yes 6 1.1 even 1 trivial
1323.4.a.bc.1.3 yes 6 3.2 odd 2
1323.4.a.bc.1.4 yes 6 7.6 odd 2