# Properties

 Label 1323.4.a.o.1.2 Level $1323$ Weight $4$ Character 1323.1 Self dual yes Analytic conductor $78.060$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# 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: $$2$$ Coefficient field: $$\Q(\zeta_{12})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 3$$ x^2 - 3 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 189) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$1.73205$$ 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)

 $$f(q)$$ $$=$$ $$q-1.26795 q^{2} -6.39230 q^{4} -3.92820 q^{5} +18.2487 q^{8} +O(q^{10})$$ $$q-1.26795 q^{2} -6.39230 q^{4} -3.92820 q^{5} +18.2487 q^{8} +4.98076 q^{10} -51.7128 q^{11} -67.5692 q^{13} +28.0000 q^{16} +63.4974 q^{17} +71.9230 q^{19} +25.1103 q^{20} +65.5692 q^{22} -147.779 q^{23} -109.569 q^{25} +85.6743 q^{26} -117.646 q^{29} -54.7077 q^{31} -181.492 q^{32} -80.5115 q^{34} +9.70766 q^{37} -91.1948 q^{38} -71.6846 q^{40} -236.338 q^{41} -489.785 q^{43} +330.564 q^{44} +187.377 q^{46} +613.841 q^{47} +138.928 q^{50} +431.923 q^{52} -316.543 q^{53} +203.138 q^{55} +149.169 q^{58} +2.43594 q^{59} -482.831 q^{61} +69.3665 q^{62} +6.12297 q^{64} +265.426 q^{65} +646.123 q^{67} -405.895 q^{68} +459.846 q^{71} -137.615 q^{73} -12.3088 q^{74} -459.754 q^{76} -816.307 q^{79} -109.990 q^{80} +299.665 q^{82} -1009.25 q^{83} -249.431 q^{85} +621.022 q^{86} -943.692 q^{88} -255.682 q^{89} +944.651 q^{92} -778.319 q^{94} -282.528 q^{95} +62.9076 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 6 q^{2} + 8 q^{4} + 6 q^{5} - 12 q^{8}+O(q^{10})$$ 2 * q - 6 * q^2 + 8 * q^4 + 6 * q^5 - 12 * q^8 $$2 q - 6 q^{2} + 8 q^{4} + 6 q^{5} - 12 q^{8} - 42 q^{10} - 48 q^{11} - 52 q^{13} + 56 q^{16} + 30 q^{17} - 64 q^{19} + 168 q^{20} + 48 q^{22} - 60 q^{23} - 136 q^{25} + 12 q^{26} - 360 q^{29} + 140 q^{31} - 72 q^{32} + 78 q^{34} - 230 q^{37} + 552 q^{38} - 372 q^{40} + 234 q^{41} - 938 q^{43} + 384 q^{44} - 228 q^{46} + 618 q^{47} + 264 q^{50} + 656 q^{52} + 420 q^{53} + 240 q^{55} + 1296 q^{58} + 282 q^{59} + 32 q^{61} - 852 q^{62} - 736 q^{64} + 420 q^{65} + 544 q^{67} - 888 q^{68} + 504 q^{71} + 764 q^{73} + 1122 q^{74} - 2416 q^{76} + 238 q^{79} + 168 q^{80} - 1926 q^{82} - 522 q^{83} - 582 q^{85} + 2742 q^{86} - 1056 q^{88} + 708 q^{89} + 2208 q^{92} - 798 q^{94} - 1632 q^{95} - 664 q^{97}+O(q^{100})$$ 2 * q - 6 * q^2 + 8 * q^4 + 6 * q^5 - 12 * q^8 - 42 * q^10 - 48 * q^11 - 52 * q^13 + 56 * q^16 + 30 * q^17 - 64 * q^19 + 168 * q^20 + 48 * q^22 - 60 * q^23 - 136 * q^25 + 12 * q^26 - 360 * q^29 + 140 * q^31 - 72 * q^32 + 78 * q^34 - 230 * q^37 + 552 * q^38 - 372 * q^40 + 234 * q^41 - 938 * q^43 + 384 * q^44 - 228 * q^46 + 618 * q^47 + 264 * q^50 + 656 * q^52 + 420 * q^53 + 240 * q^55 + 1296 * q^58 + 282 * q^59 + 32 * q^61 - 852 * q^62 - 736 * q^64 + 420 * q^65 + 544 * q^67 - 888 * q^68 + 504 * q^71 + 764 * q^73 + 1122 * q^74 - 2416 * q^76 + 238 * q^79 + 168 * q^80 - 1926 * q^82 - 522 * q^83 - 582 * q^85 + 2742 * q^86 - 1056 * q^88 + 708 * q^89 + 2208 * q^92 - 798 * q^94 - 1632 * q^95 - 664 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.26795 −0.448288 −0.224144 0.974556i $$-0.571959\pi$$
−0.224144 + 0.974556i $$0.571959\pi$$
$$3$$ 0 0
$$4$$ −6.39230 −0.799038
$$5$$ −3.92820 −0.351349 −0.175675 0.984448i $$-0.556211\pi$$
−0.175675 + 0.984448i $$0.556211\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 18.2487 0.806487
$$9$$ 0 0
$$10$$ 4.98076 0.157506
$$11$$ −51.7128 −1.41745 −0.708727 0.705483i $$-0.750730\pi$$
−0.708727 + 0.705483i $$0.750730\pi$$
$$12$$ 0 0
$$13$$ −67.5692 −1.44156 −0.720782 0.693162i $$-0.756217\pi$$
−0.720782 + 0.693162i $$0.756217\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 28.0000 0.437500
$$17$$ 63.4974 0.905905 0.452953 0.891535i $$-0.350371\pi$$
0.452953 + 0.891535i $$0.350371\pi$$
$$18$$ 0 0
$$19$$ 71.9230 0.868436 0.434218 0.900808i $$-0.357025\pi$$
0.434218 + 0.900808i $$0.357025\pi$$
$$20$$ 25.1103 0.280741
$$21$$ 0 0
$$22$$ 65.5692 0.635427
$$23$$ −147.779 −1.33975 −0.669873 0.742476i $$-0.733651\pi$$
−0.669873 + 0.742476i $$0.733651\pi$$
$$24$$ 0 0
$$25$$ −109.569 −0.876554
$$26$$ 85.6743 0.646235
$$27$$ 0 0
$$28$$ 0 0
$$29$$ −117.646 −0.753322 −0.376661 0.926351i $$-0.622928\pi$$
−0.376661 + 0.926351i $$0.622928\pi$$
$$30$$ 0 0
$$31$$ −54.7077 −0.316961 −0.158480 0.987362i $$-0.550659\pi$$
−0.158480 + 0.987362i $$0.550659\pi$$
$$32$$ −181.492 −1.00261
$$33$$ 0 0
$$34$$ −80.5115 −0.406106
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 9.70766 0.0431332 0.0215666 0.999767i $$-0.493135\pi$$
0.0215666 + 0.999767i $$0.493135\pi$$
$$38$$ −91.1948 −0.389309
$$39$$ 0 0
$$40$$ −71.6846 −0.283358
$$41$$ −236.338 −0.900240 −0.450120 0.892968i $$-0.648619\pi$$
−0.450120 + 0.892968i $$0.648619\pi$$
$$42$$ 0 0
$$43$$ −489.785 −1.73701 −0.868505 0.495680i $$-0.834919\pi$$
−0.868505 + 0.495680i $$0.834919\pi$$
$$44$$ 330.564 1.13260
$$45$$ 0 0
$$46$$ 187.377 0.600591
$$47$$ 613.841 1.90506 0.952531 0.304442i $$-0.0984700\pi$$
0.952531 + 0.304442i $$0.0984700\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 138.928 0.392948
$$51$$ 0 0
$$52$$ 431.923 1.15186
$$53$$ −316.543 −0.820388 −0.410194 0.911998i $$-0.634539\pi$$
−0.410194 + 0.911998i $$0.634539\pi$$
$$54$$ 0 0
$$55$$ 203.138 0.498021
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 149.169 0.337705
$$59$$ 2.43594 0.00537511 0.00268756 0.999996i $$-0.499145\pi$$
0.00268756 + 0.999996i $$0.499145\pi$$
$$60$$ 0 0
$$61$$ −482.831 −1.01344 −0.506722 0.862109i $$-0.669143\pi$$
−0.506722 + 0.862109i $$0.669143\pi$$
$$62$$ 69.3665 0.142090
$$63$$ 0 0
$$64$$ 6.12297 0.0119589
$$65$$ 265.426 0.506492
$$66$$ 0 0
$$67$$ 646.123 1.17816 0.589078 0.808076i $$-0.299491\pi$$
0.589078 + 0.808076i $$0.299491\pi$$
$$68$$ −405.895 −0.723853
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 459.846 0.768644 0.384322 0.923199i $$-0.374435\pi$$
0.384322 + 0.923199i $$0.374435\pi$$
$$72$$ 0 0
$$73$$ −137.615 −0.220639 −0.110319 0.993896i $$-0.535187\pi$$
−0.110319 + 0.993896i $$0.535187\pi$$
$$74$$ −12.3088 −0.0193361
$$75$$ 0 0
$$76$$ −459.754 −0.693913
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −816.307 −1.16255 −0.581277 0.813706i $$-0.697447\pi$$
−0.581277 + 0.813706i $$0.697447\pi$$
$$80$$ −109.990 −0.153715
$$81$$ 0 0
$$82$$ 299.665 0.403567
$$83$$ −1009.25 −1.33469 −0.667344 0.744749i $$-0.732569\pi$$
−0.667344 + 0.744749i $$0.732569\pi$$
$$84$$ 0 0
$$85$$ −249.431 −0.318289
$$86$$ 621.022 0.778681
$$87$$ 0 0
$$88$$ −943.692 −1.14316
$$89$$ −255.682 −0.304519 −0.152260 0.988341i $$-0.548655\pi$$
−0.152260 + 0.988341i $$0.548655\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 944.651 1.07051
$$93$$ 0 0
$$94$$ −778.319 −0.854016
$$95$$ −282.528 −0.305124
$$96$$ 0 0
$$97$$ 62.9076 0.0658484 0.0329242 0.999458i $$-0.489518\pi$$
0.0329242 + 0.999458i $$0.489518\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 700.400 0.700400
$$101$$ 823.128 0.810934 0.405467 0.914110i $$-0.367109\pi$$
0.405467 + 0.914110i $$0.367109\pi$$
$$102$$ 0 0
$$103$$ −983.646 −0.940986 −0.470493 0.882404i $$-0.655924\pi$$
−0.470493 + 0.882404i $$0.655924\pi$$
$$104$$ −1233.05 −1.16260
$$105$$ 0 0
$$106$$ 401.361 0.367770
$$107$$ 1176.47 1.06293 0.531464 0.847081i $$-0.321642\pi$$
0.531464 + 0.847081i $$0.321642\pi$$
$$108$$ 0 0
$$109$$ −1171.83 −1.02973 −0.514867 0.857270i $$-0.672159\pi$$
−0.514867 + 0.857270i $$0.672159\pi$$
$$110$$ −257.569 −0.223257
$$111$$ 0 0
$$112$$ 0 0
$$113$$ −426.231 −0.354836 −0.177418 0.984136i $$-0.556774\pi$$
−0.177418 + 0.984136i $$0.556774\pi$$
$$114$$ 0 0
$$115$$ 580.508 0.470718
$$116$$ 752.030 0.601933
$$117$$ 0 0
$$118$$ −3.08864 −0.00240960
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 1343.22 1.00918
$$122$$ 612.205 0.454315
$$123$$ 0 0
$$124$$ 349.708 0.253264
$$125$$ 921.436 0.659326
$$126$$ 0 0
$$127$$ −1832.03 −1.28005 −0.640025 0.768354i $$-0.721076\pi$$
−0.640025 + 0.768354i $$0.721076\pi$$
$$128$$ 1444.17 0.997252
$$129$$ 0 0
$$130$$ −336.546 −0.227054
$$131$$ 2801.24 1.86829 0.934143 0.356898i $$-0.116166\pi$$
0.934143 + 0.356898i $$0.116166\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ −819.251 −0.528153
$$135$$ 0 0
$$136$$ 1158.75 0.730600
$$137$$ −898.113 −0.560080 −0.280040 0.959988i $$-0.590348\pi$$
−0.280040 + 0.959988i $$0.590348\pi$$
$$138$$ 0 0
$$139$$ −1141.11 −0.696313 −0.348156 0.937436i $$-0.613192\pi$$
−0.348156 + 0.937436i $$0.613192\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ −583.061 −0.344573
$$143$$ 3494.19 2.04335
$$144$$ 0 0
$$145$$ 462.138 0.264679
$$146$$ 174.489 0.0989098
$$147$$ 0 0
$$148$$ −62.0543 −0.0344651
$$149$$ 3603.59 1.98133 0.990663 0.136335i $$-0.0435323\pi$$
0.990663 + 0.136335i $$0.0435323\pi$$
$$150$$ 0 0
$$151$$ 145.447 0.0783859 0.0391930 0.999232i $$-0.487521\pi$$
0.0391930 + 0.999232i $$0.487521\pi$$
$$152$$ 1312.50 0.700382
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 214.903 0.111364
$$156$$ 0 0
$$157$$ 1180.17 0.599922 0.299961 0.953951i $$-0.403026\pi$$
0.299961 + 0.953951i $$0.403026\pi$$
$$158$$ 1035.04 0.521159
$$159$$ 0 0
$$160$$ 712.939 0.352267
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 2116.31 1.01694 0.508472 0.861078i $$-0.330210\pi$$
0.508472 + 0.861078i $$0.330210\pi$$
$$164$$ 1510.75 0.719326
$$165$$ 0 0
$$166$$ 1279.67 0.598324
$$167$$ 3247.22 1.50465 0.752326 0.658790i $$-0.228932\pi$$
0.752326 + 0.658790i $$0.228932\pi$$
$$168$$ 0 0
$$169$$ 2368.60 1.07811
$$170$$ 316.266 0.142685
$$171$$ 0 0
$$172$$ 3130.85 1.38794
$$173$$ 990.195 0.435163 0.217581 0.976042i $$-0.430183\pi$$
0.217581 + 0.976042i $$0.430183\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ −1447.96 −0.620136
$$177$$ 0 0
$$178$$ 324.192 0.136512
$$179$$ −743.267 −0.310360 −0.155180 0.987886i $$-0.549596\pi$$
−0.155180 + 0.987886i $$0.549596\pi$$
$$180$$ 0 0
$$181$$ 2422.14 0.994675 0.497337 0.867557i $$-0.334311\pi$$
0.497337 + 0.867557i $$0.334311\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ −2696.78 −1.08049
$$185$$ −38.1337 −0.0151548
$$186$$ 0 0
$$187$$ −3283.63 −1.28408
$$188$$ −3923.86 −1.52222
$$189$$ 0 0
$$190$$ 358.232 0.136783
$$191$$ 361.897 0.137099 0.0685496 0.997648i $$-0.478163\pi$$
0.0685496 + 0.997648i $$0.478163\pi$$
$$192$$ 0 0
$$193$$ 1128.39 0.420844 0.210422 0.977611i $$-0.432516\pi$$
0.210422 + 0.977611i $$0.432516\pi$$
$$194$$ −79.7636 −0.0295190
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 1972.54 0.713389 0.356694 0.934221i $$-0.383904\pi$$
0.356694 + 0.934221i $$0.383904\pi$$
$$198$$ 0 0
$$199$$ −1100.51 −0.392024 −0.196012 0.980601i $$-0.562799\pi$$
−0.196012 + 0.980601i $$0.562799\pi$$
$$200$$ −1999.50 −0.706929
$$201$$ 0 0
$$202$$ −1043.68 −0.363532
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 928.385 0.316299
$$206$$ 1247.21 0.421832
$$207$$ 0 0
$$208$$ −1891.94 −0.630684
$$209$$ −3719.34 −1.23097
$$210$$ 0 0
$$211$$ −3617.08 −1.18014 −0.590071 0.807352i $$-0.700900\pi$$
−0.590071 + 0.807352i $$0.700900\pi$$
$$212$$ 2023.44 0.655522
$$213$$ 0 0
$$214$$ −1491.70 −0.476498
$$215$$ 1923.97 0.610297
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 1485.82 0.461617
$$219$$ 0 0
$$220$$ −1298.52 −0.397938
$$221$$ −4290.47 −1.30592
$$222$$ 0 0
$$223$$ −4224.15 −1.26848 −0.634238 0.773138i $$-0.718686\pi$$
−0.634238 + 0.773138i $$0.718686\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 540.439 0.159068
$$227$$ 6255.65 1.82908 0.914542 0.404491i $$-0.132551\pi$$
0.914542 + 0.404491i $$0.132551\pi$$
$$228$$ 0 0
$$229$$ 5237.89 1.51148 0.755742 0.654870i $$-0.227277\pi$$
0.755742 + 0.654870i $$0.227277\pi$$
$$230$$ −736.054 −0.211017
$$231$$ 0 0
$$232$$ −2146.89 −0.607544
$$233$$ 5433.71 1.52779 0.763893 0.645342i $$-0.223285\pi$$
0.763893 + 0.645342i $$0.223285\pi$$
$$234$$ 0 0
$$235$$ −2411.29 −0.669342
$$236$$ −15.5712 −0.00429492
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 2927.15 0.792225 0.396112 0.918202i $$-0.370359\pi$$
0.396112 + 0.918202i $$0.370359\pi$$
$$240$$ 0 0
$$241$$ −3922.03 −1.04830 −0.524150 0.851626i $$-0.675617\pi$$
−0.524150 + 0.851626i $$0.675617\pi$$
$$242$$ −1703.13 −0.452402
$$243$$ 0 0
$$244$$ 3086.40 0.809781
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −4859.78 −1.25191
$$248$$ −998.344 −0.255625
$$249$$ 0 0
$$250$$ −1168.33 −0.295568
$$251$$ −3827.00 −0.962383 −0.481191 0.876616i $$-0.659796\pi$$
−0.481191 + 0.876616i $$0.659796\pi$$
$$252$$ 0 0
$$253$$ 7642.09 1.89903
$$254$$ 2322.92 0.573831
$$255$$ 0 0
$$256$$ −1880.12 −0.459015
$$257$$ −1718.54 −0.417120 −0.208560 0.978010i $$-0.566878\pi$$
−0.208560 + 0.978010i $$0.566878\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ −1696.68 −0.404707
$$261$$ 0 0
$$262$$ −3551.83 −0.837530
$$263$$ 4181.27 0.980336 0.490168 0.871628i $$-0.336935\pi$$
0.490168 + 0.871628i $$0.336935\pi$$
$$264$$ 0 0
$$265$$ 1243.45 0.288243
$$266$$ 0 0
$$267$$ 0 0
$$268$$ −4130.22 −0.941392
$$269$$ −6158.35 −1.39584 −0.697920 0.716175i $$-0.745891\pi$$
−0.697920 + 0.716175i $$0.745891\pi$$
$$270$$ 0 0
$$271$$ 4754.98 1.06585 0.532924 0.846163i $$-0.321093\pi$$
0.532924 + 0.846163i $$0.321093\pi$$
$$272$$ 1777.93 0.396333
$$273$$ 0 0
$$274$$ 1138.76 0.251077
$$275$$ 5666.13 1.24248
$$276$$ 0 0
$$277$$ 4991.83 1.08278 0.541390 0.840772i $$-0.317898\pi$$
0.541390 + 0.840772i $$0.317898\pi$$
$$278$$ 1446.87 0.312148
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 6068.29 1.28827 0.644135 0.764912i $$-0.277217\pi$$
0.644135 + 0.764912i $$0.277217\pi$$
$$282$$ 0 0
$$283$$ 1854.58 0.389553 0.194777 0.980848i $$-0.437602\pi$$
0.194777 + 0.980848i $$0.437602\pi$$
$$284$$ −2939.48 −0.614175
$$285$$ 0 0
$$286$$ −4430.46 −0.916009
$$287$$ 0 0
$$288$$ 0 0
$$289$$ −881.077 −0.179336
$$290$$ −585.968 −0.118652
$$291$$ 0 0
$$292$$ 879.679 0.176299
$$293$$ −5938.68 −1.18410 −0.592050 0.805901i $$-0.701681\pi$$
−0.592050 + 0.805901i $$0.701681\pi$$
$$294$$ 0 0
$$295$$ −9.56885 −0.00188854
$$296$$ 177.152 0.0347864
$$297$$ 0 0
$$298$$ −4569.17 −0.888204
$$299$$ 9985.34 1.93133
$$300$$ 0 0
$$301$$ 0 0
$$302$$ −184.419 −0.0351395
$$303$$ 0 0
$$304$$ 2013.85 0.379941
$$305$$ 1896.66 0.356073
$$306$$ 0 0
$$307$$ 8870.08 1.64900 0.824498 0.565864i $$-0.191457\pi$$
0.824498 + 0.565864i $$0.191457\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ −272.486 −0.0499231
$$311$$ −2343.32 −0.427259 −0.213629 0.976915i $$-0.568528\pi$$
−0.213629 + 0.976915i $$0.568528\pi$$
$$312$$ 0 0
$$313$$ −0.262523 −4.74080e−5 0 −2.37040e−5 1.00000i $$-0.500008\pi$$
−2.37040e−5 1.00000i $$0.500008\pi$$
$$314$$ −1496.39 −0.268938
$$315$$ 0 0
$$316$$ 5218.09 0.928925
$$317$$ 3051.05 0.540581 0.270290 0.962779i $$-0.412880\pi$$
0.270290 + 0.962779i $$0.412880\pi$$
$$318$$ 0 0
$$319$$ 6083.81 1.06780
$$320$$ −24.0523 −0.00420176
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 4566.93 0.786720
$$324$$ 0 0
$$325$$ 7403.51 1.26361
$$326$$ −2683.37 −0.455884
$$327$$ 0 0
$$328$$ −4312.87 −0.726032
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 10952.6 1.81876 0.909381 0.415965i $$-0.136556\pi$$
0.909381 + 0.415965i $$0.136556\pi$$
$$332$$ 6451.41 1.06647
$$333$$ 0 0
$$334$$ −4117.30 −0.674517
$$335$$ −2538.10 −0.413944
$$336$$ 0 0
$$337$$ −3551.61 −0.574091 −0.287046 0.957917i $$-0.592673\pi$$
−0.287046 + 0.957917i $$0.592673\pi$$
$$338$$ −3003.26 −0.483302
$$339$$ 0 0
$$340$$ 1594.44 0.254325
$$341$$ 2829.09 0.449278
$$342$$ 0 0
$$343$$ 0 0
$$344$$ −8937.94 −1.40088
$$345$$ 0 0
$$346$$ −1255.52 −0.195078
$$347$$ −7089.53 −1.09679 −0.548394 0.836220i $$-0.684761\pi$$
−0.548394 + 0.836220i $$0.684761\pi$$
$$348$$ 0 0
$$349$$ 6493.29 0.995925 0.497963 0.867198i $$-0.334082\pi$$
0.497963 + 0.867198i $$0.334082\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 9385.48 1.42116
$$353$$ −5106.99 −0.770022 −0.385011 0.922912i $$-0.625802\pi$$
−0.385011 + 0.922912i $$0.625802\pi$$
$$354$$ 0 0
$$355$$ −1806.37 −0.270062
$$356$$ 1634.40 0.243323
$$357$$ 0 0
$$358$$ 942.425 0.139130
$$359$$ −9598.78 −1.41115 −0.705577 0.708633i $$-0.749312\pi$$
−0.705577 + 0.708633i $$0.749312\pi$$
$$360$$ 0 0
$$361$$ −1686.08 −0.245819
$$362$$ −3071.15 −0.445900
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 540.581 0.0775213
$$366$$ 0 0
$$367$$ −1267.23 −0.180242 −0.0901212 0.995931i $$-0.528725\pi$$
−0.0901212 + 0.995931i $$0.528725\pi$$
$$368$$ −4137.82 −0.586139
$$369$$ 0 0
$$370$$ 48.3515 0.00679372
$$371$$ 0 0
$$372$$ 0 0
$$373$$ −9312.57 −1.29273 −0.646363 0.763030i $$-0.723711\pi$$
−0.646363 + 0.763030i $$0.723711\pi$$
$$374$$ 4163.48 0.575637
$$375$$ 0 0
$$376$$ 11201.8 1.53641
$$377$$ 7949.26 1.08596
$$378$$ 0 0
$$379$$ 1366.43 0.185195 0.0925973 0.995704i $$-0.470483\pi$$
0.0925973 + 0.995704i $$0.470483\pi$$
$$380$$ 1806.01 0.243806
$$381$$ 0 0
$$382$$ −458.867 −0.0614599
$$383$$ −5096.46 −0.679940 −0.339970 0.940436i $$-0.610417\pi$$
−0.339970 + 0.940436i $$0.610417\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −1430.74 −0.188659
$$387$$ 0 0
$$388$$ −402.124 −0.0526154
$$389$$ 8662.07 1.12901 0.564504 0.825430i $$-0.309067\pi$$
0.564504 + 0.825430i $$0.309067\pi$$
$$390$$ 0 0
$$391$$ −9383.61 −1.21368
$$392$$ 0 0
$$393$$ 0 0
$$394$$ −2501.08 −0.319803
$$395$$ 3206.62 0.408462
$$396$$ 0 0
$$397$$ −15350.7 −1.94063 −0.970314 0.241850i $$-0.922246\pi$$
−0.970314 + 0.241850i $$0.922246\pi$$
$$398$$ 1395.39 0.175740
$$399$$ 0 0
$$400$$ −3067.94 −0.383492
$$401$$ −5740.02 −0.714821 −0.357410 0.933947i $$-0.616340\pi$$
−0.357410 + 0.933947i $$0.616340\pi$$
$$402$$ 0 0
$$403$$ 3696.55 0.456919
$$404$$ −5261.69 −0.647967
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −502.010 −0.0611394
$$408$$ 0 0
$$409$$ −8785.54 −1.06214 −0.531072 0.847327i $$-0.678211\pi$$
−0.531072 + 0.847327i $$0.678211\pi$$
$$410$$ −1177.15 −0.141793
$$411$$ 0 0
$$412$$ 6287.77 0.751884
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 3964.52 0.468942
$$416$$ 12263.3 1.44533
$$417$$ 0 0
$$418$$ 4715.94 0.551828
$$419$$ 4404.08 0.513492 0.256746 0.966479i $$-0.417350\pi$$
0.256746 + 0.966479i $$0.417350\pi$$
$$420$$ 0 0
$$421$$ 4027.26 0.466215 0.233108 0.972451i $$-0.425111\pi$$
0.233108 + 0.972451i $$0.425111\pi$$
$$422$$ 4586.27 0.529043
$$423$$ 0 0
$$424$$ −5776.51 −0.661632
$$425$$ −6957.36 −0.794075
$$426$$ 0 0
$$427$$ 0 0
$$428$$ −7520.33 −0.849320
$$429$$ 0 0
$$430$$ −2439.50 −0.273589
$$431$$ −4733.90 −0.529058 −0.264529 0.964378i $$-0.585216\pi$$
−0.264529 + 0.964378i $$0.585216\pi$$
$$432$$ 0 0
$$433$$ 15456.7 1.71548 0.857738 0.514087i $$-0.171869\pi$$
0.857738 + 0.514087i $$0.171869\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 7490.70 0.822797
$$437$$ −10628.7 −1.16348
$$438$$ 0 0
$$439$$ 9779.64 1.06323 0.531614 0.846987i $$-0.321586\pi$$
0.531614 + 0.846987i $$0.321586\pi$$
$$440$$ 3707.01 0.401648
$$441$$ 0 0
$$442$$ 5440.10 0.585428
$$443$$ −14671.0 −1.57346 −0.786729 0.617298i $$-0.788227\pi$$
−0.786729 + 0.617298i $$0.788227\pi$$
$$444$$ 0 0
$$445$$ 1004.37 0.106993
$$446$$ 5356.01 0.568642
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 503.293 0.0528995 0.0264497 0.999650i $$-0.491580\pi$$
0.0264497 + 0.999650i $$0.491580\pi$$
$$450$$ 0 0
$$451$$ 12221.7 1.27605
$$452$$ 2724.60 0.283527
$$453$$ 0 0
$$454$$ −7931.85 −0.819956
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −18037.2 −1.84627 −0.923136 0.384475i $$-0.874383\pi$$
−0.923136 + 0.384475i $$0.874383\pi$$
$$458$$ −6641.38 −0.677579
$$459$$ 0 0
$$460$$ −3710.78 −0.376122
$$461$$ 10432.6 1.05400 0.527001 0.849864i $$-0.323316\pi$$
0.527001 + 0.849864i $$0.323316\pi$$
$$462$$ 0 0
$$463$$ 206.922 0.0207700 0.0103850 0.999946i $$-0.496694\pi$$
0.0103850 + 0.999946i $$0.496694\pi$$
$$464$$ −3294.09 −0.329578
$$465$$ 0 0
$$466$$ −6889.67 −0.684888
$$467$$ −1808.79 −0.179231 −0.0896153 0.995976i $$-0.528564\pi$$
−0.0896153 + 0.995976i $$0.528564\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 3057.40 0.300058
$$471$$ 0 0
$$472$$ 44.4527 0.00433496
$$473$$ 25328.1 2.46213
$$474$$ 0 0
$$475$$ −7880.55 −0.761231
$$476$$ 0 0
$$477$$ 0 0
$$478$$ −3711.48 −0.355145
$$479$$ −3269.26 −0.311850 −0.155925 0.987769i $$-0.549836\pi$$
−0.155925 + 0.987769i $$0.549836\pi$$
$$480$$ 0 0
$$481$$ −655.939 −0.0621793
$$482$$ 4972.94 0.469940
$$483$$ 0 0
$$484$$ −8586.24 −0.806371
$$485$$ −247.114 −0.0231358
$$486$$ 0 0
$$487$$ −15589.4 −1.45056 −0.725282 0.688451i $$-0.758291\pi$$
−0.725282 + 0.688451i $$0.758291\pi$$
$$488$$ −8811.04 −0.817330
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −12032.1 −1.10591 −0.552954 0.833212i $$-0.686499\pi$$
−0.552954 + 0.833212i $$0.686499\pi$$
$$492$$ 0 0
$$493$$ −7470.23 −0.682438
$$494$$ 6161.96 0.561214
$$495$$ 0 0
$$496$$ −1531.81 −0.138670
$$497$$ 0 0
$$498$$ 0 0
$$499$$ −9958.64 −0.893407 −0.446704 0.894682i $$-0.647402\pi$$
−0.446704 + 0.894682i $$0.647402\pi$$
$$500$$ −5890.10 −0.526826
$$501$$ 0 0
$$502$$ 4852.44 0.431424
$$503$$ 7368.14 0.653139 0.326570 0.945173i $$-0.394107\pi$$
0.326570 + 0.945173i $$0.394107\pi$$
$$504$$ 0 0
$$505$$ −3233.41 −0.284921
$$506$$ −9689.78 −0.851311
$$507$$ 0 0
$$508$$ 11710.9 1.02281
$$509$$ 7725.18 0.672716 0.336358 0.941734i $$-0.390805\pi$$
0.336358 + 0.941734i $$0.390805\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ −9169.49 −0.791481
$$513$$ 0 0
$$514$$ 2179.03 0.186990
$$515$$ 3863.96 0.330615
$$516$$ 0 0
$$517$$ −31743.4 −2.70034
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 4843.68 0.408479
$$521$$ −3904.26 −0.328308 −0.164154 0.986435i $$-0.552489\pi$$
−0.164154 + 0.986435i $$0.552489\pi$$
$$522$$ 0 0
$$523$$ −17029.3 −1.42379 −0.711893 0.702288i $$-0.752162\pi$$
−0.711893 + 0.702288i $$0.752162\pi$$
$$524$$ −17906.4 −1.49283
$$525$$ 0 0
$$526$$ −5301.64 −0.439472
$$527$$ −3473.80 −0.287136
$$528$$ 0 0
$$529$$ 9671.77 0.794918
$$530$$ −1576.63 −0.129216
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 15969.2 1.29775
$$534$$ 0 0
$$535$$ −4621.40 −0.373459
$$536$$ 11790.9 0.950168
$$537$$ 0 0
$$538$$ 7808.47 0.625738
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 15599.2 1.23967 0.619834 0.784733i $$-0.287200\pi$$
0.619834 + 0.784733i $$0.287200\pi$$
$$542$$ −6029.08 −0.477806
$$543$$ 0 0
$$544$$ −11524.3 −0.908272
$$545$$ 4603.19 0.361796
$$546$$ 0 0
$$547$$ −22666.1 −1.77172 −0.885860 0.463953i $$-0.846431\pi$$
−0.885860 + 0.463953i $$0.846431\pi$$
$$548$$ 5741.01 0.447525
$$549$$ 0 0
$$550$$ −7184.37 −0.556986
$$551$$ −8461.47 −0.654212
$$552$$ 0 0
$$553$$ 0 0
$$554$$ −6329.39 −0.485397
$$555$$ 0 0
$$556$$ 7294.31 0.556380
$$557$$ 15151.1 1.15255 0.576276 0.817255i $$-0.304505\pi$$
0.576276 + 0.817255i $$0.304505\pi$$
$$558$$ 0 0
$$559$$ 33094.4 2.50401
$$560$$ 0 0
$$561$$ 0 0
$$562$$ −7694.29 −0.577516
$$563$$ −13957.9 −1.04486 −0.522429 0.852683i $$-0.674974\pi$$
−0.522429 + 0.852683i $$0.674974\pi$$
$$564$$ 0 0
$$565$$ 1674.32 0.124671
$$566$$ −2351.52 −0.174632
$$567$$ 0 0
$$568$$ 8391.60 0.619901
$$569$$ −3463.93 −0.255212 −0.127606 0.991825i $$-0.540729\pi$$
−0.127606 + 0.991825i $$0.540729\pi$$
$$570$$ 0 0
$$571$$ 22177.2 1.62537 0.812687 0.582701i $$-0.198004\pi$$
0.812687 + 0.582701i $$0.198004\pi$$
$$572$$ −22336.0 −1.63272
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 16192.1 1.17436
$$576$$ 0 0
$$577$$ 5083.18 0.366752 0.183376 0.983043i $$-0.441297\pi$$
0.183376 + 0.983043i $$0.441297\pi$$
$$578$$ 1117.16 0.0803941
$$579$$ 0 0
$$580$$ −2954.13 −0.211489
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 16369.4 1.16286
$$584$$ −2511.30 −0.177942
$$585$$ 0 0
$$586$$ 7529.94 0.530817
$$587$$ 26283.4 1.84809 0.924047 0.382278i $$-0.124860\pi$$
0.924047 + 0.382278i $$0.124860\pi$$
$$588$$ 0 0
$$589$$ −3934.74 −0.275260
$$590$$ 12.1328 0.000846610 0
$$591$$ 0 0
$$592$$ 271.814 0.0188708
$$593$$ −23901.8 −1.65519 −0.827596 0.561324i $$-0.810292\pi$$
−0.827596 + 0.561324i $$0.810292\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −23035.2 −1.58315
$$597$$ 0 0
$$598$$ −12660.9 −0.865791
$$599$$ −15150.3 −1.03343 −0.516715 0.856158i $$-0.672845\pi$$
−0.516715 + 0.856158i $$0.672845\pi$$
$$600$$ 0 0
$$601$$ −11860.1 −0.804962 −0.402481 0.915428i $$-0.631852\pi$$
−0.402481 + 0.915428i $$0.631852\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ −929.739 −0.0626334
$$605$$ −5276.42 −0.354574
$$606$$ 0 0
$$607$$ 7847.27 0.524730 0.262365 0.964969i $$-0.415498\pi$$
0.262365 + 0.964969i $$0.415498\pi$$
$$608$$ −13053.5 −0.870705
$$609$$ 0 0
$$610$$ −2404.86 −0.159623
$$611$$ −41476.8 −2.74627
$$612$$ 0 0
$$613$$ 2068.12 0.136265 0.0681327 0.997676i $$-0.478296\pi$$
0.0681327 + 0.997676i $$0.478296\pi$$
$$614$$ −11246.8 −0.739225
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 25241.6 1.64698 0.823492 0.567328i $$-0.192023\pi$$
0.823492 + 0.567328i $$0.192023\pi$$
$$618$$ 0 0
$$619$$ 16776.6 1.08935 0.544677 0.838646i $$-0.316652\pi$$
0.544677 + 0.838646i $$0.316652\pi$$
$$620$$ −1373.72 −0.0889840
$$621$$ 0 0
$$622$$ 2971.21 0.191535
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 10076.6 0.644900
$$626$$ 0.332866 2.12524e−5 0
$$627$$ 0 0
$$628$$ −7544.00 −0.479360
$$629$$ 616.411 0.0390746
$$630$$ 0 0
$$631$$ 22011.8 1.38871 0.694354 0.719633i $$-0.255690\pi$$
0.694354 + 0.719633i $$0.255690\pi$$
$$632$$ −14896.6 −0.937584
$$633$$ 0 0
$$634$$ −3868.58 −0.242336
$$635$$ 7196.59 0.449745
$$636$$ 0 0
$$637$$ 0 0
$$638$$ −7713.97 −0.478682
$$639$$ 0 0
$$640$$ −5673.01 −0.350384
$$641$$ −21485.3 −1.32390 −0.661950 0.749548i $$-0.730271\pi$$
−0.661950 + 0.749548i $$0.730271\pi$$
$$642$$ 0 0
$$643$$ −14305.3 −0.877365 −0.438682 0.898642i $$-0.644555\pi$$
−0.438682 + 0.898642i $$0.644555\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ −5790.63 −0.352677
$$647$$ −13848.7 −0.841498 −0.420749 0.907177i $$-0.638233\pi$$
−0.420749 + 0.907177i $$0.638233\pi$$
$$648$$ 0 0
$$649$$ −125.969 −0.00761898
$$650$$ −9387.27 −0.566460
$$651$$ 0 0
$$652$$ −13528.1 −0.812578
$$653$$ −24324.9 −1.45774 −0.728872 0.684649i $$-0.759955\pi$$
−0.728872 + 0.684649i $$0.759955\pi$$
$$654$$ 0 0
$$655$$ −11003.8 −0.656421
$$656$$ −6617.47 −0.393855
$$657$$ 0 0
$$658$$ 0 0
$$659$$ −5232.82 −0.309320 −0.154660 0.987968i $$-0.549428\pi$$
−0.154660 + 0.987968i $$0.549428\pi$$
$$660$$ 0 0
$$661$$ 33511.1 1.97191 0.985954 0.167016i $$-0.0534132\pi$$
0.985954 + 0.167016i $$0.0534132\pi$$
$$662$$ −13887.4 −0.815328
$$663$$ 0 0
$$664$$ −18417.4 −1.07641
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 17385.7 1.00926
$$668$$ −20757.2 −1.20228
$$669$$ 0 0
$$670$$ 3218.18 0.185566
$$671$$ 24968.5 1.43651
$$672$$ 0 0
$$673$$ −17861.8 −1.02306 −0.511530 0.859265i $$-0.670921\pi$$
−0.511530 + 0.859265i $$0.670921\pi$$
$$674$$ 4503.27 0.257358
$$675$$ 0 0
$$676$$ −15140.8 −0.861448
$$677$$ 28992.9 1.64592 0.822959 0.568101i $$-0.192322\pi$$
0.822959 + 0.568101i $$0.192322\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ −4551.79 −0.256696
$$681$$ 0 0
$$682$$ −3587.14 −0.201406
$$683$$ 2381.89 0.133441 0.0667206 0.997772i $$-0.478746\pi$$
0.0667206 + 0.997772i $$0.478746\pi$$
$$684$$ 0 0
$$685$$ 3527.97 0.196784
$$686$$ 0 0
$$687$$ 0 0
$$688$$ −13714.0 −0.759942
$$689$$ 21388.6 1.18264
$$690$$ 0 0
$$691$$ −8734.63 −0.480870 −0.240435 0.970665i $$-0.577290\pi$$
−0.240435 + 0.970665i $$0.577290\pi$$
$$692$$ −6329.63 −0.347712
$$693$$ 0 0
$$694$$ 8989.16 0.491677
$$695$$ 4482.50 0.244649
$$696$$ 0 0
$$697$$ −15006.9 −0.815532
$$698$$ −8233.16 −0.446461
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −24618.5 −1.32643 −0.663215 0.748429i $$-0.730809\pi$$
−0.663215 + 0.748429i $$0.730809\pi$$
$$702$$ 0 0
$$703$$ 698.204 0.0374584
$$704$$ −316.636 −0.0169512
$$705$$ 0 0
$$706$$ 6475.41 0.345192
$$707$$ 0 0
$$708$$ 0 0
$$709$$ −460.784 −0.0244078 −0.0122039 0.999926i $$-0.503885\pi$$
−0.0122039 + 0.999926i $$0.503885\pi$$
$$710$$ 2290.38 0.121066
$$711$$ 0 0
$$712$$ −4665.86 −0.245591
$$713$$ 8084.67 0.424647
$$714$$ 0 0
$$715$$ −13725.9 −0.717930
$$716$$ 4751.19 0.247989
$$717$$ 0 0
$$718$$ 12170.8 0.632603
$$719$$ 31274.8 1.62219 0.811094 0.584915i $$-0.198872\pi$$
0.811094 + 0.584915i $$0.198872\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 2137.86 0.110198
$$723$$ 0 0
$$724$$ −15483.0 −0.794783
$$725$$ 12890.4 0.660327
$$726$$ 0 0
$$727$$ −1523.12 −0.0777021 −0.0388510 0.999245i $$-0.512370\pi$$
−0.0388510 + 0.999245i $$0.512370\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ −685.429 −0.0347519
$$731$$ −31100.1 −1.57357
$$732$$ 0 0
$$733$$ 28363.5 1.42923 0.714617 0.699516i $$-0.246601\pi$$
0.714617 + 0.699516i $$0.246601\pi$$
$$734$$ 1606.78 0.0808004
$$735$$ 0 0
$$736$$ 26820.8 1.34325
$$737$$ −33412.8 −1.66998
$$738$$ 0 0
$$739$$ 4971.75 0.247481 0.123741 0.992315i $$-0.460511\pi$$
0.123741 + 0.992315i $$0.460511\pi$$
$$740$$ 243.762 0.0121093
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 8803.96 0.434705 0.217353 0.976093i $$-0.430258\pi$$
0.217353 + 0.976093i $$0.430258\pi$$
$$744$$ 0 0
$$745$$ −14155.6 −0.696137
$$746$$ 11807.9 0.579513
$$747$$ 0 0
$$748$$ 20990.0 1.02603
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −30759.0 −1.49456 −0.747278 0.664511i $$-0.768640\pi$$
−0.747278 + 0.664511i $$0.768640\pi$$
$$752$$ 17187.5 0.833465
$$753$$ 0 0
$$754$$ −10079.3 −0.486823
$$755$$ −571.344 −0.0275408
$$756$$ 0 0
$$757$$ 1104.42 0.0530261 0.0265131 0.999648i $$-0.491560\pi$$
0.0265131 + 0.999648i $$0.491560\pi$$
$$758$$ −1732.56 −0.0830205
$$759$$ 0 0
$$760$$ −5155.78 −0.246079
$$761$$ −3722.02 −0.177297 −0.0886485 0.996063i $$-0.528255\pi$$
−0.0886485 + 0.996063i $$0.528255\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ −2313.36 −0.109547
$$765$$ 0 0
$$766$$ 6462.06 0.304809
$$767$$ −164.594 −0.00774857
$$768$$ 0 0
$$769$$ −12779.6 −0.599279 −0.299640 0.954052i $$-0.596866\pi$$
−0.299640 + 0.954052i $$0.596866\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ −7212.98 −0.336271
$$773$$ −24178.0 −1.12500 −0.562498 0.826798i $$-0.690160\pi$$
−0.562498 + 0.826798i $$0.690160\pi$$
$$774$$ 0 0
$$775$$ 5994.28 0.277833
$$776$$ 1147.98 0.0531059
$$777$$ 0 0
$$778$$ −10983.1 −0.506121
$$779$$ −16998.2 −0.781801
$$780$$ 0 0
$$781$$ −23779.9 −1.08952
$$782$$ 11897.9 0.544079
$$783$$ 0 0
$$784$$ 0 0
$$785$$ −4635.94 −0.210782
$$786$$ 0 0
$$787$$ −34814.0 −1.57686 −0.788428 0.615126i $$-0.789105\pi$$
−0.788428 + 0.615126i $$0.789105\pi$$
$$788$$ −12609.1 −0.570025
$$789$$ 0 0
$$790$$ −4065.83 −0.183109
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 32624.5 1.46095
$$794$$ 19463.9 0.869959
$$795$$ 0 0
$$796$$ 7034.78 0.313243
$$797$$ −1412.03 −0.0627563 −0.0313781 0.999508i $$-0.509990\pi$$
−0.0313781 + 0.999508i $$0.509990\pi$$
$$798$$ 0 0
$$799$$ 38977.3 1.72581
$$800$$ 19886.0 0.878844
$$801$$ 0 0
$$802$$ 7278.06 0.320445
$$803$$ 7116.47 0.312746
$$804$$ 0 0
$$805$$ 0 0
$$806$$ −4687.04 −0.204831
$$807$$ 0 0
$$808$$ 15021.0 0.654007
$$809$$ −7209.04 −0.313296 −0.156648 0.987655i $$-0.550069\pi$$
−0.156648 + 0.987655i $$0.550069\pi$$
$$810$$ 0 0
$$811$$ 35027.8 1.51664 0.758318 0.651885i $$-0.226021\pi$$
0.758318 + 0.651885i $$0.226021\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 636.524 0.0274080
$$815$$ −8313.29 −0.357303
$$816$$ 0 0
$$817$$ −35226.8 −1.50848
$$818$$ 11139.6 0.476146
$$819$$ 0 0
$$820$$ −5934.52 −0.252735
$$821$$ 2387.69 0.101499 0.0507496 0.998711i $$-0.483839\pi$$
0.0507496 + 0.998711i $$0.483839\pi$$
$$822$$ 0 0
$$823$$ 11009.1 0.466287 0.233144 0.972442i $$-0.425099\pi$$
0.233144 + 0.972442i $$0.425099\pi$$
$$824$$ −17950.3 −0.758893
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −156.810 −0.00659348 −0.00329674 0.999995i $$-0.501049\pi$$
−0.00329674 + 0.999995i $$0.501049\pi$$
$$828$$ 0 0
$$829$$ −10244.8 −0.429210 −0.214605 0.976701i $$-0.568846\pi$$
−0.214605 + 0.976701i $$0.568846\pi$$
$$830$$ −5026.81 −0.210221
$$831$$ 0 0
$$832$$ −413.725 −0.0172396
$$833$$ 0 0
$$834$$ 0 0
$$835$$ −12755.7 −0.528659
$$836$$ 23775.2 0.983590
$$837$$ 0 0
$$838$$ −5584.15 −0.230192
$$839$$ 6146.56 0.252924 0.126462 0.991971i $$-0.459638\pi$$
0.126462 + 0.991971i $$0.459638\pi$$
$$840$$ 0 0
$$841$$ −10548.4 −0.432506
$$842$$ −5106.36 −0.208999
$$843$$ 0 0
$$844$$ 23121.5 0.942978
$$845$$ −9304.34 −0.378792
$$846$$ 0 0
$$847$$ 0 0
$$848$$ −8863.22 −0.358920
$$849$$ 0 0
$$850$$ 8821.58 0.355974
$$851$$ −1434.59 −0.0577875
$$852$$ 0 0
$$853$$ 18316.2 0.735212 0.367606 0.929982i $$-0.380178\pi$$
0.367606 + 0.929982i $$0.380178\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 21469.0 0.857237
$$857$$ 657.584 0.0262108 0.0131054 0.999914i $$-0.495828\pi$$
0.0131054 + 0.999914i $$0.495828\pi$$
$$858$$ 0 0
$$859$$ 16467.4 0.654085 0.327042 0.945010i $$-0.393948\pi$$
0.327042 + 0.945010i $$0.393948\pi$$
$$860$$ −12298.6 −0.487651
$$861$$ 0 0
$$862$$ 6002.34 0.237170
$$863$$ 24470.4 0.965217 0.482609 0.875836i $$-0.339689\pi$$
0.482609 + 0.875836i $$0.339689\pi$$
$$864$$ 0 0
$$865$$ −3889.69 −0.152894
$$866$$ −19598.3 −0.769027
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 42213.6 1.64787
$$870$$ 0 0
$$871$$ −43658.0 −1.69839
$$872$$ −21384.4 −0.830467
$$873$$ 0 0
$$874$$ 13476.7 0.521575
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −23651.8 −0.910678 −0.455339 0.890318i $$-0.650482\pi$$
−0.455339 + 0.890318i $$0.650482\pi$$
$$878$$ −12400.1 −0.476632
$$879$$ 0 0
$$880$$ 5687.88 0.217884
$$881$$ 2325.38 0.0889264 0.0444632 0.999011i $$-0.485842\pi$$
0.0444632 + 0.999011i $$0.485842\pi$$
$$882$$ 0 0
$$883$$ 2193.02 0.0835798 0.0417899 0.999126i $$-0.486694\pi$$
0.0417899 + 0.999126i $$0.486694\pi$$
$$884$$ 27426.0 1.04348
$$885$$ 0 0
$$886$$ 18602.1 0.705362
$$887$$ −38.3431 −0.00145145 −0.000725725 1.00000i $$-0.500231\pi$$
−0.000725725 1.00000i $$0.500231\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ −1273.49 −0.0479635
$$891$$ 0 0
$$892$$ 27002.1 1.01356
$$893$$ 44149.3 1.65442
$$894$$ 0 0
$$895$$ 2919.70 0.109045
$$896$$ 0 0
$$897$$ 0 0
$$898$$ −638.150 −0.0237142
$$899$$ 6436.15 0.238774
$$900$$ 0 0
$$901$$ −20099.7 −0.743194
$$902$$ −15496.5 −0.572037
$$903$$ 0 0
$$904$$ −7778.16 −0.286170
$$905$$ −9514.65 −0.349478
$$906$$ 0 0
$$907$$ 18032.2 0.660142 0.330071 0.943956i $$-0.392927\pi$$
0.330071 + 0.943956i $$0.392927\pi$$
$$908$$ −39988.0 −1.46151
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 12877.7 0.468338 0.234169 0.972196i $$-0.424763\pi$$
0.234169 + 0.972196i $$0.424763\pi$$
$$912$$ 0 0
$$913$$ 52190.9 1.89186
$$914$$ 22870.3 0.827661
$$915$$ 0 0
$$916$$ −33482.2 −1.20773
$$917$$ 0 0
$$918$$ 0 0
$$919$$ −16267.1 −0.583899 −0.291950 0.956434i $$-0.594304\pi$$
−0.291950 + 0.956434i $$0.594304\pi$$
$$920$$ 10593.5 0.379628
$$921$$ 0 0
$$922$$ −13228.0 −0.472497
$$923$$ −31071.4 −1.10805
$$924$$ 0 0
$$925$$ −1063.66 −0.0378086
$$926$$ −262.367 −0.00931092
$$927$$ 0 0
$$928$$ 21351.9 0.755290
$$929$$ −14942.6 −0.527718 −0.263859 0.964561i $$-0.584995\pi$$
−0.263859 + 0.964561i $$0.584995\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ −34733.9 −1.22076
$$933$$ 0 0
$$934$$ 2293.45 0.0803469
$$935$$ 12898.8 0.451160
$$936$$ 0 0
$$937$$ −16005.5 −0.558032 −0.279016 0.960287i $$-0.590008\pi$$
−0.279016 + 0.960287i $$0.590008\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 15413.7 0.534830
$$941$$ 35515.1 1.23035 0.615175 0.788390i $$-0.289085\pi$$
0.615175 + 0.788390i $$0.289085\pi$$
$$942$$ 0 0
$$943$$ 34926.0 1.20609
$$944$$ 68.2062 0.00235161
$$945$$ 0 0
$$946$$ −32114.8 −1.10374
$$947$$ 15151.4 0.519911 0.259955 0.965621i $$-0.416292\pi$$
0.259955 + 0.965621i $$0.416292\pi$$
$$948$$ 0 0
$$949$$ 9298.55 0.318065
$$950$$ 9992.14 0.341250
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 38421.2 1.30596 0.652982 0.757374i $$-0.273518\pi$$
0.652982 + 0.757374i $$0.273518\pi$$
$$954$$ 0 0
$$955$$ −1421.60 −0.0481697
$$956$$ −18711.3 −0.633018
$$957$$ 0 0
$$958$$ 4145.25 0.139799
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −26798.1 −0.899536
$$962$$ 831.697 0.0278742
$$963$$ 0 0
$$964$$ 25070.8 0.837631
$$965$$ −4432.53 −0.147863
$$966$$ 0 0
$$967$$ 32575.3 1.08330 0.541650 0.840604i $$-0.317800\pi$$
0.541650 + 0.840604i $$0.317800\pi$$
$$968$$ 24511.9 0.813888
$$969$$ 0 0
$$970$$ 313.328 0.0103715
$$971$$ 4374.24 0.144569 0.0722843 0.997384i $$-0.476971\pi$$
0.0722843 + 0.997384i $$0.476971\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 19766.6 0.650270
$$975$$ 0 0
$$976$$ −13519.3 −0.443382
$$977$$ 37493.4 1.22776 0.613880 0.789399i $$-0.289608\pi$$
0.613880 + 0.789399i $$0.289608\pi$$
$$978$$ 0 0
$$979$$ 13222.0 0.431642
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 15256.1 0.495765
$$983$$ −38197.3 −1.23938 −0.619688 0.784848i $$-0.712741\pi$$
−0.619688 + 0.784848i $$0.712741\pi$$
$$984$$ 0 0
$$985$$ −7748.53 −0.250649
$$986$$ 9471.87 0.305929
$$987$$ 0 0
$$988$$ 31065.2 1.00032
$$989$$ 72380.1 2.32715
$$990$$ 0 0
$$991$$ −39675.4 −1.27178 −0.635889 0.771781i $$-0.719366\pi$$
−0.635889 + 0.771781i $$0.719366\pi$$
$$992$$ 9929.02 0.317789
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 4323.02 0.137737
$$996$$ 0 0
$$997$$ 21444.3 0.681193 0.340596 0.940210i $$-0.389371\pi$$
0.340596 + 0.940210i $$0.389371\pi$$
$$998$$ 12627.1 0.400503
$$999$$ 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.o.1.2 2
3.2 odd 2 1323.4.a.x.1.1 2
7.6 odd 2 189.4.a.e.1.2 2
21.20 even 2 189.4.a.i.1.1 yes 2

By twisted newform
Twist Min Dim Char Parity Ord Type
189.4.a.e.1.2 2 7.6 odd 2
189.4.a.i.1.1 yes 2 21.20 even 2
1323.4.a.o.1.2 2 1.1 even 1 trivial
1323.4.a.x.1.1 2 3.2 odd 2