# Properties

 Label 1323.4.a.bj.1.7 Level $1323$ Weight $4$ Character 1323.1 Self dual yes Analytic conductor $78.060$ Analytic rank $0$ Dimension $7$ 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: $$7$$ Coefficient field: $$\mathbb{Q}[x]/(x^{7} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{7} - x^{6} - 43x^{5} + 10x^{4} + 513x^{3} + 258x^{2} - 936x - 504$$ x^7 - x^6 - 43*x^5 + 10*x^4 + 513*x^3 + 258*x^2 - 936*x - 504 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$3^{3}$$ Twist minimal: no (minimal twist has level 189) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.7 Root $$5.09184$$ 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)

 $$f(q)$$ $$=$$ $$q+5.09184 q^{2} +17.9269 q^{4} -18.5985 q^{5} +50.5460 q^{8} +O(q^{10})$$ $$q+5.09184 q^{2} +17.9269 q^{4} -18.5985 q^{5} +50.5460 q^{8} -94.7009 q^{10} -1.05631 q^{11} -58.7684 q^{13} +113.957 q^{16} +43.7810 q^{17} +131.880 q^{19} -333.413 q^{20} -5.37857 q^{22} +161.410 q^{23} +220.906 q^{25} -299.239 q^{26} +64.0250 q^{29} +55.9772 q^{31} +175.885 q^{32} +222.926 q^{34} +296.995 q^{37} +671.513 q^{38} -940.082 q^{40} +80.6724 q^{41} -134.280 q^{43} -18.9363 q^{44} +821.874 q^{46} +233.366 q^{47} +1124.82 q^{50} -1053.53 q^{52} +387.066 q^{53} +19.6458 q^{55} +326.005 q^{58} +722.871 q^{59} -388.144 q^{61} +285.027 q^{62} -16.0817 q^{64} +1093.01 q^{65} -730.041 q^{67} +784.856 q^{68} -685.470 q^{71} +275.371 q^{73} +1512.25 q^{74} +2364.20 q^{76} +854.544 q^{79} -2119.44 q^{80} +410.771 q^{82} +922.479 q^{83} -814.263 q^{85} -683.734 q^{86} -53.3922 q^{88} -301.729 q^{89} +2893.57 q^{92} +1188.27 q^{94} -2452.78 q^{95} -913.706 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$7 q + q^{2} + 31 q^{4} - q^{5} + 84 q^{8}+O(q^{10})$$ 7 * q + q^2 + 31 * q^4 - q^5 + 84 * q^8 $$7 q + q^{2} + 31 q^{4} - q^{5} + 84 q^{8} + 12 q^{10} + 98 q^{11} - 124 q^{13} + 139 q^{16} + 30 q^{17} + 182 q^{19} - 110 q^{20} + 276 q^{22} - 6 q^{23} + 388 q^{25} - 245 q^{26} + 323 q^{29} + 26 q^{31} + 398 q^{32} + 114 q^{34} - 112 q^{37} + 1015 q^{38} - 147 q^{40} - 524 q^{41} + 8 q^{43} + 937 q^{44} - 339 q^{46} + 288 q^{47} + 2576 q^{50} - 1075 q^{52} + 1353 q^{53} + 156 q^{55} - 81 q^{58} + 165 q^{59} + 56 q^{61} - 1215 q^{62} - 1706 q^{64} + 1694 q^{65} - 988 q^{67} + 2625 q^{68} + 792 q^{71} + 1487 q^{73} + 2736 q^{74} + 1952 q^{76} - 1273 q^{79} - 2501 q^{80} - 2049 q^{82} - 1170 q^{83} + 216 q^{85} - 160 q^{86} + 9 q^{88} + 1058 q^{89} + 3834 q^{92} + 1653 q^{94} + 3260 q^{95} - 3730 q^{97}+O(q^{100})$$ 7 * q + q^2 + 31 * q^4 - q^5 + 84 * q^8 + 12 * q^10 + 98 * q^11 - 124 * q^13 + 139 * q^16 + 30 * q^17 + 182 * q^19 - 110 * q^20 + 276 * q^22 - 6 * q^23 + 388 * q^25 - 245 * q^26 + 323 * q^29 + 26 * q^31 + 398 * q^32 + 114 * q^34 - 112 * q^37 + 1015 * q^38 - 147 * q^40 - 524 * q^41 + 8 * q^43 + 937 * q^44 - 339 * q^46 + 288 * q^47 + 2576 * q^50 - 1075 * q^52 + 1353 * q^53 + 156 * q^55 - 81 * q^58 + 165 * q^59 + 56 * q^61 - 1215 * q^62 - 1706 * q^64 + 1694 * q^65 - 988 * q^67 + 2625 * q^68 + 792 * q^71 + 1487 * q^73 + 2736 * q^74 + 1952 * q^76 - 1273 * q^79 - 2501 * q^80 - 2049 * q^82 - 1170 * q^83 + 216 * q^85 - 160 * q^86 + 9 * q^88 + 1058 * q^89 + 3834 * q^92 + 1653 * q^94 + 3260 * q^95 - 3730 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 5.09184 1.80024 0.900119 0.435644i $$-0.143479\pi$$
0.900119 + 0.435644i $$0.143479\pi$$
$$3$$ 0 0
$$4$$ 17.9269 2.24086
$$5$$ −18.5985 −1.66350 −0.831752 0.555147i $$-0.812662\pi$$
−0.831752 + 0.555147i $$0.812662\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 50.5460 2.23384
$$9$$ 0 0
$$10$$ −94.7009 −2.99470
$$11$$ −1.05631 −0.0289536 −0.0144768 0.999895i $$-0.504608\pi$$
−0.0144768 + 0.999895i $$0.504608\pi$$
$$12$$ 0 0
$$13$$ −58.7684 −1.25380 −0.626900 0.779099i $$-0.715677\pi$$
−0.626900 + 0.779099i $$0.715677\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 113.957 1.78058
$$17$$ 43.7810 0.624615 0.312308 0.949981i $$-0.398898\pi$$
0.312308 + 0.949981i $$0.398898\pi$$
$$18$$ 0 0
$$19$$ 131.880 1.59239 0.796194 0.605041i $$-0.206843\pi$$
0.796194 + 0.605041i $$0.206843\pi$$
$$20$$ −333.413 −3.72768
$$21$$ 0 0
$$22$$ −5.37857 −0.0521234
$$23$$ 161.410 1.46332 0.731659 0.681671i $$-0.238747\pi$$
0.731659 + 0.681671i $$0.238747\pi$$
$$24$$ 0 0
$$25$$ 220.906 1.76725
$$26$$ −299.239 −2.25714
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 64.0250 0.409970 0.204985 0.978765i $$-0.434285\pi$$
0.204985 + 0.978765i $$0.434285\pi$$
$$30$$ 0 0
$$31$$ 55.9772 0.324316 0.162158 0.986765i $$-0.448155\pi$$
0.162158 + 0.986765i $$0.448155\pi$$
$$32$$ 175.885 0.971634
$$33$$ 0 0
$$34$$ 222.926 1.12446
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 296.995 1.31961 0.659807 0.751435i $$-0.270638\pi$$
0.659807 + 0.751435i $$0.270638\pi$$
$$38$$ 671.513 2.86668
$$39$$ 0 0
$$40$$ −940.082 −3.71600
$$41$$ 80.6724 0.307290 0.153645 0.988126i $$-0.450899\pi$$
0.153645 + 0.988126i $$0.450899\pi$$
$$42$$ 0 0
$$43$$ −134.280 −0.476222 −0.238111 0.971238i $$-0.576528\pi$$
−0.238111 + 0.971238i $$0.576528\pi$$
$$44$$ −18.9363 −0.0648809
$$45$$ 0 0
$$46$$ 821.874 2.63432
$$47$$ 233.366 0.724255 0.362128 0.932129i $$-0.382050\pi$$
0.362128 + 0.932129i $$0.382050\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 1124.82 3.18147
$$51$$ 0 0
$$52$$ −1053.53 −2.80959
$$53$$ 387.066 1.00316 0.501581 0.865111i $$-0.332752\pi$$
0.501581 + 0.865111i $$0.332752\pi$$
$$54$$ 0 0
$$55$$ 19.6458 0.0481645
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 326.005 0.738044
$$59$$ 722.871 1.59508 0.797541 0.603265i $$-0.206134\pi$$
0.797541 + 0.603265i $$0.206134\pi$$
$$60$$ 0 0
$$61$$ −388.144 −0.814701 −0.407351 0.913272i $$-0.633547\pi$$
−0.407351 + 0.913272i $$0.633547\pi$$
$$62$$ 285.027 0.583846
$$63$$ 0 0
$$64$$ −16.0817 −0.0314095
$$65$$ 1093.01 2.08570
$$66$$ 0 0
$$67$$ −730.041 −1.33117 −0.665587 0.746320i $$-0.731819\pi$$
−0.665587 + 0.746320i $$0.731819\pi$$
$$68$$ 784.856 1.39967
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −685.470 −1.14578 −0.572890 0.819632i $$-0.694178\pi$$
−0.572890 + 0.819632i $$0.694178\pi$$
$$72$$ 0 0
$$73$$ 275.371 0.441503 0.220751 0.975330i $$-0.429149\pi$$
0.220751 + 0.975330i $$0.429149\pi$$
$$74$$ 1512.25 2.37562
$$75$$ 0 0
$$76$$ 2364.20 3.56831
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 854.544 1.21701 0.608504 0.793551i $$-0.291770\pi$$
0.608504 + 0.793551i $$0.291770\pi$$
$$80$$ −2119.44 −2.96201
$$81$$ 0 0
$$82$$ 410.771 0.553196
$$83$$ 922.479 1.21994 0.609971 0.792424i $$-0.291181\pi$$
0.609971 + 0.792424i $$0.291181\pi$$
$$84$$ 0 0
$$85$$ −814.263 −1.03905
$$86$$ −683.734 −0.857313
$$87$$ 0 0
$$88$$ −53.3922 −0.0646776
$$89$$ −301.729 −0.359362 −0.179681 0.983725i $$-0.557507\pi$$
−0.179681 + 0.983725i $$0.557507\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 2893.57 3.27908
$$93$$ 0 0
$$94$$ 1188.27 1.30383
$$95$$ −2452.78 −2.64895
$$96$$ 0 0
$$97$$ −913.706 −0.956421 −0.478210 0.878245i $$-0.658714\pi$$
−0.478210 + 0.878245i $$0.658714\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 3960.15 3.96015
$$101$$ −354.715 −0.349460 −0.174730 0.984616i $$-0.555905\pi$$
−0.174730 + 0.984616i $$0.555905\pi$$
$$102$$ 0 0
$$103$$ −142.105 −0.135942 −0.0679708 0.997687i $$-0.521652\pi$$
−0.0679708 + 0.997687i $$0.521652\pi$$
$$104$$ −2970.50 −2.80079
$$105$$ 0 0
$$106$$ 1970.88 1.80593
$$107$$ 559.789 0.505765 0.252882 0.967497i $$-0.418621\pi$$
0.252882 + 0.967497i $$0.418621\pi$$
$$108$$ 0 0
$$109$$ −457.761 −0.402252 −0.201126 0.979565i $$-0.564460\pi$$
−0.201126 + 0.979565i $$0.564460\pi$$
$$110$$ 100.034 0.0867075
$$111$$ 0 0
$$112$$ 0 0
$$113$$ −647.946 −0.539412 −0.269706 0.962943i $$-0.586927\pi$$
−0.269706 + 0.962943i $$0.586927\pi$$
$$114$$ 0 0
$$115$$ −3001.99 −2.43424
$$116$$ 1147.77 0.918685
$$117$$ 0 0
$$118$$ 3680.75 2.87153
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −1329.88 −0.999162
$$122$$ −1976.37 −1.46666
$$123$$ 0 0
$$124$$ 1003.50 0.726746
$$125$$ −1783.71 −1.27632
$$126$$ 0 0
$$127$$ 1108.02 0.774180 0.387090 0.922042i $$-0.373480\pi$$
0.387090 + 0.922042i $$0.373480\pi$$
$$128$$ −1488.96 −1.02818
$$129$$ 0 0
$$130$$ 5565.42 3.75476
$$131$$ 706.818 0.471412 0.235706 0.971824i $$-0.424260\pi$$
0.235706 + 0.971824i $$0.424260\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ −3717.25 −2.39643
$$135$$ 0 0
$$136$$ 2212.95 1.39529
$$137$$ −319.732 −0.199391 −0.0996955 0.995018i $$-0.531787\pi$$
−0.0996955 + 0.995018i $$0.531787\pi$$
$$138$$ 0 0
$$139$$ −439.298 −0.268063 −0.134032 0.990977i $$-0.542792\pi$$
−0.134032 + 0.990977i $$0.542792\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ −3490.31 −2.06268
$$143$$ 62.0777 0.0363021
$$144$$ 0 0
$$145$$ −1190.77 −0.681987
$$146$$ 1402.14 0.794810
$$147$$ 0 0
$$148$$ 5324.19 2.95706
$$149$$ 1362.30 0.749017 0.374509 0.927223i $$-0.377811\pi$$
0.374509 + 0.927223i $$0.377811\pi$$
$$150$$ 0 0
$$151$$ 2209.41 1.19072 0.595362 0.803458i $$-0.297009\pi$$
0.595362 + 0.803458i $$0.297009\pi$$
$$152$$ 6666.01 3.55714
$$153$$ 0 0
$$154$$ 0 0
$$155$$ −1041.09 −0.539502
$$156$$ 0 0
$$157$$ −1109.10 −0.563795 −0.281898 0.959444i $$-0.590964\pi$$
−0.281898 + 0.959444i $$0.590964\pi$$
$$158$$ 4351.20 2.19091
$$159$$ 0 0
$$160$$ −3271.20 −1.61632
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 1477.44 0.709953 0.354977 0.934875i $$-0.384489\pi$$
0.354977 + 0.934875i $$0.384489\pi$$
$$164$$ 1446.20 0.688594
$$165$$ 0 0
$$166$$ 4697.12 2.19619
$$167$$ 1848.67 0.856615 0.428307 0.903633i $$-0.359110\pi$$
0.428307 + 0.903633i $$0.359110\pi$$
$$168$$ 0 0
$$169$$ 1256.72 0.572017
$$170$$ −4146.10 −1.87054
$$171$$ 0 0
$$172$$ −2407.22 −1.06715
$$173$$ 2269.80 0.997513 0.498757 0.866742i $$-0.333790\pi$$
0.498757 + 0.866742i $$0.333790\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ −120.374 −0.0515543
$$177$$ 0 0
$$178$$ −1536.36 −0.646937
$$179$$ 3666.49 1.53098 0.765492 0.643445i $$-0.222495\pi$$
0.765492 + 0.643445i $$0.222495\pi$$
$$180$$ 0 0
$$181$$ 3237.15 1.32937 0.664685 0.747124i $$-0.268566\pi$$
0.664685 + 0.747124i $$0.268566\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 8158.62 3.26881
$$185$$ −5523.68 −2.19518
$$186$$ 0 0
$$187$$ −46.2464 −0.0180849
$$188$$ 4183.53 1.62295
$$189$$ 0 0
$$190$$ −12489.2 −4.76873
$$191$$ −3353.99 −1.27061 −0.635304 0.772262i $$-0.719125\pi$$
−0.635304 + 0.772262i $$0.719125\pi$$
$$192$$ 0 0
$$193$$ −5321.04 −1.98454 −0.992271 0.124086i $$-0.960400\pi$$
−0.992271 + 0.124086i $$0.960400\pi$$
$$194$$ −4652.45 −1.72178
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 1750.53 0.633098 0.316549 0.948576i $$-0.397476\pi$$
0.316549 + 0.948576i $$0.397476\pi$$
$$198$$ 0 0
$$199$$ 1572.10 0.560017 0.280009 0.959997i $$-0.409663\pi$$
0.280009 + 0.959997i $$0.409663\pi$$
$$200$$ 11165.9 3.94774
$$201$$ 0 0
$$202$$ −1806.15 −0.629110
$$203$$ 0 0
$$204$$ 0 0
$$205$$ −1500.39 −0.511179
$$206$$ −723.574 −0.244727
$$207$$ 0 0
$$208$$ −6697.08 −2.23250
$$209$$ −139.306 −0.0461054
$$210$$ 0 0
$$211$$ −2884.12 −0.941000 −0.470500 0.882400i $$-0.655926\pi$$
−0.470500 + 0.882400i $$0.655926\pi$$
$$212$$ 6938.87 2.24794
$$213$$ 0 0
$$214$$ 2850.36 0.910497
$$215$$ 2497.42 0.792198
$$216$$ 0 0
$$217$$ 0 0
$$218$$ −2330.84 −0.724150
$$219$$ 0 0
$$220$$ 352.188 0.107930
$$221$$ −2572.94 −0.783143
$$222$$ 0 0
$$223$$ 1267.27 0.380551 0.190276 0.981731i $$-0.439062\pi$$
0.190276 + 0.981731i $$0.439062\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ −3299.24 −0.971071
$$227$$ 910.785 0.266304 0.133152 0.991096i $$-0.457490\pi$$
0.133152 + 0.991096i $$0.457490\pi$$
$$228$$ 0 0
$$229$$ 677.564 0.195523 0.0977613 0.995210i $$-0.468832\pi$$
0.0977613 + 0.995210i $$0.468832\pi$$
$$230$$ −15285.7 −4.38220
$$231$$ 0 0
$$232$$ 3236.20 0.915807
$$233$$ −3312.78 −0.931449 −0.465724 0.884930i $$-0.654206\pi$$
−0.465724 + 0.884930i $$0.654206\pi$$
$$234$$ 0 0
$$235$$ −4340.28 −1.20480
$$236$$ 12958.8 3.57435
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 4933.62 1.33527 0.667634 0.744490i $$-0.267307\pi$$
0.667634 + 0.744490i $$0.267307\pi$$
$$240$$ 0 0
$$241$$ −3931.98 −1.05096 −0.525479 0.850807i $$-0.676114\pi$$
−0.525479 + 0.850807i $$0.676114\pi$$
$$242$$ −6771.56 −1.79873
$$243$$ 0 0
$$244$$ −6958.20 −1.82563
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −7750.38 −1.99654
$$248$$ 2829.42 0.724470
$$249$$ 0 0
$$250$$ −9082.38 −2.29768
$$251$$ 1637.68 0.411831 0.205915 0.978570i $$-0.433983\pi$$
0.205915 + 0.978570i $$0.433983\pi$$
$$252$$ 0 0
$$253$$ −170.499 −0.0423683
$$254$$ 5641.86 1.39371
$$255$$ 0 0
$$256$$ −7452.90 −1.81956
$$257$$ −7868.11 −1.90973 −0.954863 0.297047i $$-0.903998\pi$$
−0.954863 + 0.297047i $$0.903998\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 19594.2 4.67376
$$261$$ 0 0
$$262$$ 3599.01 0.848654
$$263$$ 3047.36 0.714480 0.357240 0.934013i $$-0.383718\pi$$
0.357240 + 0.934013i $$0.383718\pi$$
$$264$$ 0 0
$$265$$ −7198.86 −1.66876
$$266$$ 0 0
$$267$$ 0 0
$$268$$ −13087.3 −2.98297
$$269$$ 2891.25 0.655326 0.327663 0.944795i $$-0.393739\pi$$
0.327663 + 0.944795i $$0.393739\pi$$
$$270$$ 0 0
$$271$$ −1804.39 −0.404461 −0.202230 0.979338i $$-0.564819\pi$$
−0.202230 + 0.979338i $$0.564819\pi$$
$$272$$ 4989.17 1.11218
$$273$$ 0 0
$$274$$ −1628.03 −0.358951
$$275$$ −233.345 −0.0511682
$$276$$ 0 0
$$277$$ −309.700 −0.0671772 −0.0335886 0.999436i $$-0.510694\pi$$
−0.0335886 + 0.999436i $$0.510694\pi$$
$$278$$ −2236.84 −0.482577
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −1192.16 −0.253090 −0.126545 0.991961i $$-0.540389\pi$$
−0.126545 + 0.991961i $$0.540389\pi$$
$$282$$ 0 0
$$283$$ 2248.09 0.472210 0.236105 0.971728i $$-0.424129\pi$$
0.236105 + 0.971728i $$0.424129\pi$$
$$284$$ −12288.3 −2.56753
$$285$$ 0 0
$$286$$ 316.090 0.0653523
$$287$$ 0 0
$$288$$ 0 0
$$289$$ −2996.22 −0.609856
$$290$$ −6063.22 −1.22774
$$291$$ 0 0
$$292$$ 4936.53 0.989344
$$293$$ −9472.83 −1.88877 −0.944383 0.328847i $$-0.893340\pi$$
−0.944383 + 0.328847i $$0.893340\pi$$
$$294$$ 0 0
$$295$$ −13444.4 −2.65343
$$296$$ 15011.9 2.94780
$$297$$ 0 0
$$298$$ 6936.59 1.34841
$$299$$ −9485.80 −1.83471
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 11250.0 2.14359
$$303$$ 0 0
$$304$$ 15028.7 2.83538
$$305$$ 7218.92 1.35526
$$306$$ 0 0
$$307$$ −2379.45 −0.442352 −0.221176 0.975234i $$-0.570990\pi$$
−0.221176 + 0.975234i $$0.570990\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ −5301.09 −0.971231
$$311$$ −1756.21 −0.320210 −0.160105 0.987100i $$-0.551183\pi$$
−0.160105 + 0.987100i $$0.551183\pi$$
$$312$$ 0 0
$$313$$ 8379.01 1.51313 0.756565 0.653918i $$-0.226876\pi$$
0.756565 + 0.653918i $$0.226876\pi$$
$$314$$ −5647.37 −1.01497
$$315$$ 0 0
$$316$$ 15319.3 2.72714
$$317$$ 6933.28 1.22843 0.614214 0.789139i $$-0.289473\pi$$
0.614214 + 0.789139i $$0.289473\pi$$
$$318$$ 0 0
$$319$$ −67.6303 −0.0118701
$$320$$ 299.095 0.0522498
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 5773.85 0.994630
$$324$$ 0 0
$$325$$ −12982.3 −2.21578
$$326$$ 7522.91 1.27808
$$327$$ 0 0
$$328$$ 4077.66 0.686437
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 11171.1 1.85504 0.927520 0.373773i $$-0.121936\pi$$
0.927520 + 0.373773i $$0.121936\pi$$
$$332$$ 16537.1 2.73372
$$333$$ 0 0
$$334$$ 9413.16 1.54211
$$335$$ 13577.7 2.21442
$$336$$ 0 0
$$337$$ −6360.51 −1.02813 −0.514064 0.857752i $$-0.671861\pi$$
−0.514064 + 0.857752i $$0.671861\pi$$
$$338$$ 6399.03 1.02977
$$339$$ 0 0
$$340$$ −14597.2 −2.32836
$$341$$ −59.1293 −0.00939012
$$342$$ 0 0
$$343$$ 0 0
$$344$$ −6787.33 −1.06380
$$345$$ 0 0
$$346$$ 11557.5 1.79576
$$347$$ 1727.97 0.267326 0.133663 0.991027i $$-0.457326\pi$$
0.133663 + 0.991027i $$0.457326\pi$$
$$348$$ 0 0
$$349$$ −2514.58 −0.385680 −0.192840 0.981230i $$-0.561770\pi$$
−0.192840 + 0.981230i $$0.561770\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ −185.789 −0.0281323
$$353$$ 5642.22 0.850722 0.425361 0.905024i $$-0.360147\pi$$
0.425361 + 0.905024i $$0.360147\pi$$
$$354$$ 0 0
$$355$$ 12748.7 1.90601
$$356$$ −5409.05 −0.805278
$$357$$ 0 0
$$358$$ 18669.2 2.75614
$$359$$ 1725.87 0.253727 0.126863 0.991920i $$-0.459509\pi$$
0.126863 + 0.991920i $$0.459509\pi$$
$$360$$ 0 0
$$361$$ 10533.4 1.53570
$$362$$ 16483.1 2.39318
$$363$$ 0 0
$$364$$ 0 0
$$365$$ −5121.50 −0.734442
$$366$$ 0 0
$$367$$ 8800.56 1.25173 0.625866 0.779931i $$-0.284746\pi$$
0.625866 + 0.779931i $$0.284746\pi$$
$$368$$ 18393.8 2.60556
$$369$$ 0 0
$$370$$ −28125.7 −3.95185
$$371$$ 0 0
$$372$$ 0 0
$$373$$ −10805.8 −1.50001 −0.750005 0.661432i $$-0.769949\pi$$
−0.750005 + 0.661432i $$0.769949\pi$$
$$374$$ −235.479 −0.0325570
$$375$$ 0 0
$$376$$ 11795.7 1.61787
$$377$$ −3762.64 −0.514021
$$378$$ 0 0
$$379$$ −7808.27 −1.05827 −0.529134 0.848538i $$-0.677483\pi$$
−0.529134 + 0.848538i $$0.677483\pi$$
$$380$$ −43970.6 −5.93591
$$381$$ 0 0
$$382$$ −17078.0 −2.28740
$$383$$ −6071.57 −0.810033 −0.405016 0.914309i $$-0.632734\pi$$
−0.405016 + 0.914309i $$0.632734\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −27093.9 −3.57265
$$387$$ 0 0
$$388$$ −16379.9 −2.14320
$$389$$ −2769.36 −0.360956 −0.180478 0.983579i $$-0.557764\pi$$
−0.180478 + 0.983579i $$0.557764\pi$$
$$390$$ 0 0
$$391$$ 7066.69 0.914010
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 8913.44 1.13973
$$395$$ −15893.3 −2.02450
$$396$$ 0 0
$$397$$ 8064.82 1.01955 0.509775 0.860308i $$-0.329729\pi$$
0.509775 + 0.860308i $$0.329729\pi$$
$$398$$ 8004.90 1.00816
$$399$$ 0 0
$$400$$ 25173.8 3.14673
$$401$$ −4.73201 −0.000589290 0 −0.000294645 1.00000i $$-0.500094\pi$$
−0.000294645 1.00000i $$0.500094\pi$$
$$402$$ 0 0
$$403$$ −3289.69 −0.406628
$$404$$ −6358.92 −0.783089
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −313.719 −0.0382076
$$408$$ 0 0
$$409$$ −12101.6 −1.46304 −0.731522 0.681818i $$-0.761190\pi$$
−0.731522 + 0.681818i $$0.761190\pi$$
$$410$$ −7639.74 −0.920244
$$411$$ 0 0
$$412$$ −2547.49 −0.304626
$$413$$ 0 0
$$414$$ 0 0
$$415$$ −17156.8 −2.02938
$$416$$ −10336.4 −1.21824
$$417$$ 0 0
$$418$$ −709.326 −0.0830007
$$419$$ 13439.5 1.56698 0.783488 0.621407i $$-0.213438\pi$$
0.783488 + 0.621407i $$0.213438\pi$$
$$420$$ 0 0
$$421$$ −9748.84 −1.12857 −0.564287 0.825579i $$-0.690849\pi$$
−0.564287 + 0.825579i $$0.690849\pi$$
$$422$$ −14685.5 −1.69402
$$423$$ 0 0
$$424$$ 19564.6 2.24090
$$425$$ 9671.49 1.10385
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 10035.3 1.13335
$$429$$ 0 0
$$430$$ 12716.5 1.42614
$$431$$ −90.9615 −0.0101658 −0.00508290 0.999987i $$-0.501618\pi$$
−0.00508290 + 0.999987i $$0.501618\pi$$
$$432$$ 0 0
$$433$$ 13636.5 1.51346 0.756728 0.653730i $$-0.226797\pi$$
0.756728 + 0.653730i $$0.226797\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ −8206.21 −0.901390
$$437$$ 21286.8 2.33017
$$438$$ 0 0
$$439$$ −2295.45 −0.249558 −0.124779 0.992185i $$-0.539822\pi$$
−0.124779 + 0.992185i $$0.539822\pi$$
$$440$$ 993.018 0.107592
$$441$$ 0 0
$$442$$ −13101.0 −1.40984
$$443$$ 2112.15 0.226526 0.113263 0.993565i $$-0.463870\pi$$
0.113263 + 0.993565i $$0.463870\pi$$
$$444$$ 0 0
$$445$$ 5611.72 0.597800
$$446$$ 6452.76 0.685083
$$447$$ 0 0
$$448$$ 0 0
$$449$$ −6924.57 −0.727818 −0.363909 0.931434i $$-0.618558\pi$$
−0.363909 + 0.931434i $$0.618558\pi$$
$$450$$ 0 0
$$451$$ −85.2151 −0.00889716
$$452$$ −11615.6 −1.20875
$$453$$ 0 0
$$454$$ 4637.57 0.479410
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −11733.6 −1.20104 −0.600519 0.799610i $$-0.705039\pi$$
−0.600519 + 0.799610i $$0.705039\pi$$
$$458$$ 3450.05 0.351987
$$459$$ 0 0
$$460$$ −53816.2 −5.45477
$$461$$ 4623.07 0.467067 0.233533 0.972349i $$-0.424971\pi$$
0.233533 + 0.972349i $$0.424971\pi$$
$$462$$ 0 0
$$463$$ 932.150 0.0935652 0.0467826 0.998905i $$-0.485103\pi$$
0.0467826 + 0.998905i $$0.485103\pi$$
$$464$$ 7296.11 0.729986
$$465$$ 0 0
$$466$$ −16868.2 −1.67683
$$467$$ −2223.55 −0.220329 −0.110164 0.993913i $$-0.535138\pi$$
−0.110164 + 0.993913i $$0.535138\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ −22100.0 −2.16893
$$471$$ 0 0
$$472$$ 36538.2 3.56315
$$473$$ 141.842 0.0137883
$$474$$ 0 0
$$475$$ 29133.1 2.81414
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 25121.2 2.40380
$$479$$ 1738.85 0.165867 0.0829333 0.996555i $$-0.473571\pi$$
0.0829333 + 0.996555i $$0.473571\pi$$
$$480$$ 0 0
$$481$$ −17453.9 −1.65453
$$482$$ −20021.0 −1.89197
$$483$$ 0 0
$$484$$ −23840.6 −2.23898
$$485$$ 16993.6 1.59101
$$486$$ 0 0
$$487$$ −13284.4 −1.23608 −0.618041 0.786146i $$-0.712074\pi$$
−0.618041 + 0.786146i $$0.712074\pi$$
$$488$$ −19619.1 −1.81991
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −7762.78 −0.713501 −0.356751 0.934200i $$-0.616115\pi$$
−0.356751 + 0.934200i $$0.616115\pi$$
$$492$$ 0 0
$$493$$ 2803.08 0.256074
$$494$$ −39463.7 −3.59424
$$495$$ 0 0
$$496$$ 6379.01 0.577472
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 18195.4 1.63234 0.816170 0.577811i $$-0.196093\pi$$
0.816170 + 0.577811i $$0.196093\pi$$
$$500$$ −31976.3 −2.86005
$$501$$ 0 0
$$502$$ 8338.81 0.741393
$$503$$ 3509.89 0.311130 0.155565 0.987826i $$-0.450280\pi$$
0.155565 + 0.987826i $$0.450280\pi$$
$$504$$ 0 0
$$505$$ 6597.17 0.581328
$$506$$ −868.154 −0.0762730
$$507$$ 0 0
$$508$$ 19863.3 1.73483
$$509$$ 618.568 0.0538655 0.0269328 0.999637i $$-0.491426\pi$$
0.0269328 + 0.999637i $$0.491426\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ −26037.3 −2.24746
$$513$$ 0 0
$$514$$ −40063.2 −3.43796
$$515$$ 2642.94 0.226139
$$516$$ 0 0
$$517$$ −246.507 −0.0209698
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 55247.1 4.65912
$$521$$ −19934.5 −1.67629 −0.838143 0.545450i $$-0.816359\pi$$
−0.838143 + 0.545450i $$0.816359\pi$$
$$522$$ 0 0
$$523$$ 4547.48 0.380206 0.190103 0.981764i $$-0.439118\pi$$
0.190103 + 0.981764i $$0.439118\pi$$
$$524$$ 12671.0 1.05637
$$525$$ 0 0
$$526$$ 15516.7 1.28623
$$527$$ 2450.74 0.202573
$$528$$ 0 0
$$529$$ 13886.2 1.14130
$$530$$ −36655.5 −3.00417
$$531$$ 0 0
$$532$$ 0 0
$$533$$ −4740.98 −0.385281
$$534$$ 0 0
$$535$$ −10411.3 −0.841342
$$536$$ −36900.6 −2.97363
$$537$$ 0 0
$$538$$ 14721.8 1.17974
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −15500.0 −1.23179 −0.615895 0.787828i $$-0.711206\pi$$
−0.615895 + 0.787828i $$0.711206\pi$$
$$542$$ −9187.66 −0.728125
$$543$$ 0 0
$$544$$ 7700.41 0.606897
$$545$$ 8513.68 0.669149
$$546$$ 0 0
$$547$$ −4515.69 −0.352974 −0.176487 0.984303i $$-0.556473\pi$$
−0.176487 + 0.984303i $$0.556473\pi$$
$$548$$ −5731.79 −0.446807
$$549$$ 0 0
$$550$$ −1188.16 −0.0921149
$$551$$ 8443.62 0.652832
$$552$$ 0 0
$$553$$ 0 0
$$554$$ −1576.95 −0.120935
$$555$$ 0 0
$$556$$ −7875.23 −0.600691
$$557$$ 11635.4 0.885112 0.442556 0.896741i $$-0.354072\pi$$
0.442556 + 0.896741i $$0.354072\pi$$
$$558$$ 0 0
$$559$$ 7891.43 0.597088
$$560$$ 0 0
$$561$$ 0 0
$$562$$ −6070.29 −0.455623
$$563$$ −2088.05 −0.156307 −0.0781536 0.996941i $$-0.524902\pi$$
−0.0781536 + 0.996941i $$0.524902\pi$$
$$564$$ 0 0
$$565$$ 12050.8 0.897315
$$566$$ 11446.9 0.850090
$$567$$ 0 0
$$568$$ −34647.8 −2.55948
$$569$$ 22610.1 1.66585 0.832923 0.553388i $$-0.186665\pi$$
0.832923 + 0.553388i $$0.186665\pi$$
$$570$$ 0 0
$$571$$ 7300.32 0.535042 0.267521 0.963552i $$-0.413796\pi$$
0.267521 + 0.963552i $$0.413796\pi$$
$$572$$ 1112.86 0.0813477
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 35656.4 2.58604
$$576$$ 0 0
$$577$$ −23134.9 −1.66918 −0.834592 0.550869i $$-0.814296\pi$$
−0.834592 + 0.550869i $$0.814296\pi$$
$$578$$ −15256.3 −1.09789
$$579$$ 0 0
$$580$$ −21346.8 −1.52824
$$581$$ 0 0
$$582$$ 0 0
$$583$$ −408.862 −0.0290452
$$584$$ 13918.9 0.986245
$$585$$ 0 0
$$586$$ −48234.1 −3.40023
$$587$$ 263.501 0.0185278 0.00926392 0.999957i $$-0.497051\pi$$
0.00926392 + 0.999957i $$0.497051\pi$$
$$588$$ 0 0
$$589$$ 7382.28 0.516437
$$590$$ −68456.5 −4.77680
$$591$$ 0 0
$$592$$ 33844.7 2.34968
$$593$$ 26652.2 1.84566 0.922829 0.385211i $$-0.125871\pi$$
0.922829 + 0.385211i $$0.125871\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 24421.7 1.67844
$$597$$ 0 0
$$598$$ −48300.2 −3.30291
$$599$$ −14447.5 −0.985491 −0.492746 0.870173i $$-0.664007\pi$$
−0.492746 + 0.870173i $$0.664007\pi$$
$$600$$ 0 0
$$601$$ −26357.9 −1.78896 −0.894478 0.447112i $$-0.852452\pi$$
−0.894478 + 0.447112i $$0.852452\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 39607.8 2.66824
$$605$$ 24733.9 1.66211
$$606$$ 0 0
$$607$$ 7693.50 0.514447 0.257224 0.966352i $$-0.417192\pi$$
0.257224 + 0.966352i $$0.417192\pi$$
$$608$$ 23195.7 1.54722
$$609$$ 0 0
$$610$$ 36757.6 2.43979
$$611$$ −13714.6 −0.908072
$$612$$ 0 0
$$613$$ 766.271 0.0504884 0.0252442 0.999681i $$-0.491964\pi$$
0.0252442 + 0.999681i $$0.491964\pi$$
$$614$$ −12115.8 −0.796340
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −17976.0 −1.17291 −0.586455 0.809982i $$-0.699477\pi$$
−0.586455 + 0.809982i $$0.699477\pi$$
$$618$$ 0 0
$$619$$ −7552.67 −0.490416 −0.245208 0.969471i $$-0.578856\pi$$
−0.245208 + 0.969471i $$0.578856\pi$$
$$620$$ −18663.6 −1.20895
$$621$$ 0 0
$$622$$ −8942.32 −0.576454
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 5561.19 0.355916
$$626$$ 42664.6 2.72399
$$627$$ 0 0
$$628$$ −19882.7 −1.26338
$$629$$ 13002.8 0.824251
$$630$$ 0 0
$$631$$ −13073.2 −0.824779 −0.412390 0.911008i $$-0.635306\pi$$
−0.412390 + 0.911008i $$0.635306\pi$$
$$632$$ 43193.7 2.71860
$$633$$ 0 0
$$634$$ 35303.2 2.21146
$$635$$ −20607.6 −1.28785
$$636$$ 0 0
$$637$$ 0 0
$$638$$ −344.363 −0.0213690
$$639$$ 0 0
$$640$$ 27692.5 1.71038
$$641$$ 11900.9 0.733318 0.366659 0.930355i $$-0.380502\pi$$
0.366659 + 0.930355i $$0.380502\pi$$
$$642$$ 0 0
$$643$$ 27777.9 1.70366 0.851829 0.523820i $$-0.175494\pi$$
0.851829 + 0.523820i $$0.175494\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 29399.5 1.79057
$$647$$ −14688.0 −0.892496 −0.446248 0.894909i $$-0.647240\pi$$
−0.446248 + 0.894909i $$0.647240\pi$$
$$648$$ 0 0
$$649$$ −763.577 −0.0461834
$$650$$ −66103.7 −3.98893
$$651$$ 0 0
$$652$$ 26485.9 1.59090
$$653$$ 17485.7 1.04788 0.523941 0.851754i $$-0.324461\pi$$
0.523941 + 0.851754i $$0.324461\pi$$
$$654$$ 0 0
$$655$$ −13145.8 −0.784196
$$656$$ 9193.20 0.547156
$$657$$ 0 0
$$658$$ 0 0
$$659$$ −18436.4 −1.08980 −0.544902 0.838500i $$-0.683433\pi$$
−0.544902 + 0.838500i $$0.683433\pi$$
$$660$$ 0 0
$$661$$ 9836.24 0.578798 0.289399 0.957209i $$-0.406545\pi$$
0.289399 + 0.957209i $$0.406545\pi$$
$$662$$ 56881.4 3.33951
$$663$$ 0 0
$$664$$ 46627.6 2.72515
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 10334.3 0.599917
$$668$$ 33140.9 1.91955
$$669$$ 0 0
$$670$$ 69135.5 3.98647
$$671$$ 410.001 0.0235885
$$672$$ 0 0
$$673$$ −9955.62 −0.570224 −0.285112 0.958494i $$-0.592031\pi$$
−0.285112 + 0.958494i $$0.592031\pi$$
$$674$$ −32386.7 −1.85088
$$675$$ 0 0
$$676$$ 22529.1 1.28181
$$677$$ −12851.6 −0.729581 −0.364791 0.931090i $$-0.618859\pi$$
−0.364791 + 0.931090i $$0.618859\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ −41157.7 −2.32107
$$681$$ 0 0
$$682$$ −301.077 −0.0169045
$$683$$ 17472.2 0.978853 0.489426 0.872045i $$-0.337206\pi$$
0.489426 + 0.872045i $$0.337206\pi$$
$$684$$ 0 0
$$685$$ 5946.56 0.331688
$$686$$ 0 0
$$687$$ 0 0
$$688$$ −15302.2 −0.847952
$$689$$ −22747.2 −1.25777
$$690$$ 0 0
$$691$$ 8275.06 0.455569 0.227784 0.973712i $$-0.426852\pi$$
0.227784 + 0.973712i $$0.426852\pi$$
$$692$$ 40690.4 2.23528
$$693$$ 0 0
$$694$$ 8798.55 0.481251
$$695$$ 8170.31 0.445924
$$696$$ 0 0
$$697$$ 3531.92 0.191938
$$698$$ −12803.8 −0.694315
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 30709.9 1.65463 0.827315 0.561738i $$-0.189867\pi$$
0.827315 + 0.561738i $$0.189867\pi$$
$$702$$ 0 0
$$703$$ 39167.8 2.10134
$$704$$ 16.9872 0.000909418 0
$$705$$ 0 0
$$706$$ 28729.3 1.53150
$$707$$ 0 0
$$708$$ 0 0
$$709$$ −18261.0 −0.967287 −0.483643 0.875265i $$-0.660687\pi$$
−0.483643 + 0.875265i $$0.660687\pi$$
$$710$$ 64914.6 3.43127
$$711$$ 0 0
$$712$$ −15251.2 −0.802756
$$713$$ 9035.28 0.474578
$$714$$ 0 0
$$715$$ −1154.55 −0.0603886
$$716$$ 65728.6 3.43072
$$717$$ 0 0
$$718$$ 8787.86 0.456769
$$719$$ 77.4789 0.00401874 0.00200937 0.999998i $$-0.499360\pi$$
0.00200937 + 0.999998i $$0.499360\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 53634.3 2.76463
$$723$$ 0 0
$$724$$ 58032.0 2.97893
$$725$$ 14143.5 0.724519
$$726$$ 0 0
$$727$$ −7339.56 −0.374428 −0.187214 0.982319i $$-0.559946\pi$$
−0.187214 + 0.982319i $$0.559946\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ −26077.8 −1.32217
$$731$$ −5878.93 −0.297455
$$732$$ 0 0
$$733$$ −21479.9 −1.08237 −0.541185 0.840903i $$-0.682024\pi$$
−0.541185 + 0.840903i $$0.682024\pi$$
$$734$$ 44811.0 2.25341
$$735$$ 0 0
$$736$$ 28389.5 1.42181
$$737$$ 771.150 0.0385423
$$738$$ 0 0
$$739$$ 19042.0 0.947863 0.473932 0.880562i $$-0.342834\pi$$
0.473932 + 0.880562i $$0.342834\pi$$
$$740$$ −99022.2 −4.91909
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 31563.4 1.55848 0.779238 0.626728i $$-0.215606\pi$$
0.779238 + 0.626728i $$0.215606\pi$$
$$744$$ 0 0
$$745$$ −25336.7 −1.24599
$$746$$ −55021.5 −2.70038
$$747$$ 0 0
$$748$$ −829.052 −0.0405256
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 30254.1 1.47003 0.735013 0.678053i $$-0.237176\pi$$
0.735013 + 0.678053i $$0.237176\pi$$
$$752$$ 26593.8 1.28960
$$753$$ 0 0
$$754$$ −19158.8 −0.925360
$$755$$ −41091.8 −1.98077
$$756$$ 0 0
$$757$$ 3384.65 0.162506 0.0812531 0.996693i $$-0.474108\pi$$
0.0812531 + 0.996693i $$0.474108\pi$$
$$758$$ −39758.5 −1.90514
$$759$$ 0 0
$$760$$ −123978. −5.91731
$$761$$ −552.737 −0.0263294 −0.0131647 0.999913i $$-0.504191\pi$$
−0.0131647 + 0.999913i $$0.504191\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ −60126.4 −2.84725
$$765$$ 0 0
$$766$$ −30915.5 −1.45825
$$767$$ −42482.0 −1.99992
$$768$$ 0 0
$$769$$ 6199.24 0.290703 0.145351 0.989380i $$-0.453569\pi$$
0.145351 + 0.989380i $$0.453569\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ −95389.5 −4.44708
$$773$$ −30857.1 −1.43577 −0.717887 0.696160i $$-0.754891\pi$$
−0.717887 + 0.696160i $$0.754891\pi$$
$$774$$ 0 0
$$775$$ 12365.7 0.573147
$$776$$ −46184.2 −2.13649
$$777$$ 0 0
$$778$$ −14101.1 −0.649807
$$779$$ 10639.1 0.489326
$$780$$ 0 0
$$781$$ 724.069 0.0331744
$$782$$ 35982.5 1.64544
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 20627.7 0.937876
$$786$$ 0 0
$$787$$ 3181.28 0.144092 0.0720459 0.997401i $$-0.477047\pi$$
0.0720459 + 0.997401i $$0.477047\pi$$
$$788$$ 31381.6 1.41868
$$789$$ 0 0
$$790$$ −80926.0 −3.64458
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 22810.6 1.02147
$$794$$ 41064.8 1.83543
$$795$$ 0 0
$$796$$ 28182.9 1.25492
$$797$$ −10742.3 −0.477429 −0.238714 0.971090i $$-0.576726\pi$$
−0.238714 + 0.971090i $$0.576726\pi$$
$$798$$ 0 0
$$799$$ 10217.0 0.452381
$$800$$ 38853.9 1.71712
$$801$$ 0 0
$$802$$ −24.0947 −0.00106086
$$803$$ −290.877 −0.0127831
$$804$$ 0 0
$$805$$ 0 0
$$806$$ −16750.6 −0.732027
$$807$$ 0 0
$$808$$ −17929.4 −0.780636
$$809$$ 17671.9 0.767998 0.383999 0.923334i $$-0.374547\pi$$
0.383999 + 0.923334i $$0.374547\pi$$
$$810$$ 0 0
$$811$$ −34620.2 −1.49899 −0.749494 0.662012i $$-0.769703\pi$$
−0.749494 + 0.662012i $$0.769703\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ −1597.41 −0.0687827
$$815$$ −27478.3 −1.18101
$$816$$ 0 0
$$817$$ −17708.9 −0.758330
$$818$$ −61619.4 −2.63383
$$819$$ 0 0
$$820$$ −26897.2 −1.14548
$$821$$ 38889.0 1.65315 0.826574 0.562829i $$-0.190287\pi$$
0.826574 + 0.562829i $$0.190287\pi$$
$$822$$ 0 0
$$823$$ −18404.8 −0.779529 −0.389764 0.920915i $$-0.627444\pi$$
−0.389764 + 0.920915i $$0.627444\pi$$
$$824$$ −7182.81 −0.303671
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −20341.4 −0.855307 −0.427654 0.903943i $$-0.640660\pi$$
−0.427654 + 0.903943i $$0.640660\pi$$
$$828$$ 0 0
$$829$$ −19988.8 −0.837442 −0.418721 0.908115i $$-0.637522\pi$$
−0.418721 + 0.908115i $$0.637522\pi$$
$$830$$ −87359.6 −3.65337
$$831$$ 0 0
$$832$$ 945.093 0.0393812
$$833$$ 0 0
$$834$$ 0 0
$$835$$ −34382.7 −1.42498
$$836$$ −2497.33 −0.103316
$$837$$ 0 0
$$838$$ 68431.9 2.82093
$$839$$ 556.650 0.0229055 0.0114527 0.999934i $$-0.496354\pi$$
0.0114527 + 0.999934i $$0.496354\pi$$
$$840$$ 0 0
$$841$$ −20289.8 −0.831924
$$842$$ −49639.5 −2.03170
$$843$$ 0 0
$$844$$ −51703.2 −2.10865
$$845$$ −23373.2 −0.951553
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 44109.0 1.78621
$$849$$ 0 0
$$850$$ 49245.7 1.98719
$$851$$ 47938.0 1.93101
$$852$$ 0 0
$$853$$ −16702.9 −0.670455 −0.335227 0.942137i $$-0.608813\pi$$
−0.335227 + 0.942137i $$0.608813\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 28295.1 1.12980
$$857$$ −8216.56 −0.327506 −0.163753 0.986501i $$-0.552360\pi$$
−0.163753 + 0.986501i $$0.552360\pi$$
$$858$$ 0 0
$$859$$ −19311.8 −0.767068 −0.383534 0.923527i $$-0.625293\pi$$
−0.383534 + 0.923527i $$0.625293\pi$$
$$860$$ 44770.8 1.77520
$$861$$ 0 0
$$862$$ −463.161 −0.0183009
$$863$$ −4145.65 −0.163522 −0.0817611 0.996652i $$-0.526054\pi$$
−0.0817611 + 0.996652i $$0.526054\pi$$
$$864$$ 0 0
$$865$$ −42215.0 −1.65937
$$866$$ 69434.7 2.72458
$$867$$ 0 0
$$868$$ 0 0
$$869$$ −902.664 −0.0352368
$$870$$ 0 0
$$871$$ 42903.3 1.66903
$$872$$ −23137.9 −0.898566
$$873$$ 0 0
$$874$$ 108389. 4.19486
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 6801.41 0.261878 0.130939 0.991390i $$-0.458201\pi$$
0.130939 + 0.991390i $$0.458201\pi$$
$$878$$ −11688.1 −0.449263
$$879$$ 0 0
$$880$$ 2238.79 0.0857608
$$881$$ −36057.6 −1.37890 −0.689451 0.724333i $$-0.742148\pi$$
−0.689451 + 0.724333i $$0.742148\pi$$
$$882$$ 0 0
$$883$$ −23743.3 −0.904898 −0.452449 0.891790i $$-0.649450\pi$$
−0.452449 + 0.891790i $$0.649450\pi$$
$$884$$ −46124.7 −1.75491
$$885$$ 0 0
$$886$$ 10754.7 0.407801
$$887$$ 34610.0 1.31014 0.655068 0.755570i $$-0.272640\pi$$
0.655068 + 0.755570i $$0.272640\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 28574.0 1.07618
$$891$$ 0 0
$$892$$ 22718.2 0.852761
$$893$$ 30776.4 1.15330
$$894$$ 0 0
$$895$$ −68191.4 −2.54680
$$896$$ 0 0
$$897$$ 0 0
$$898$$ −35258.8 −1.31025
$$899$$ 3583.94 0.132960
$$900$$ 0 0
$$901$$ 16946.1 0.626590
$$902$$ −433.902 −0.0160170
$$903$$ 0 0
$$904$$ −32751.0 −1.20496
$$905$$ −60206.4 −2.21141
$$906$$ 0 0
$$907$$ 6863.70 0.251274 0.125637 0.992076i $$-0.459903\pi$$
0.125637 + 0.992076i $$0.459903\pi$$
$$908$$ 16327.5 0.596748
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −13817.7 −0.502524 −0.251262 0.967919i $$-0.580846\pi$$
−0.251262 + 0.967919i $$0.580846\pi$$
$$912$$ 0 0
$$913$$ −974.424 −0.0353217
$$914$$ −59745.6 −2.16216
$$915$$ 0 0
$$916$$ 12146.6 0.438138
$$917$$ 0 0
$$918$$ 0 0
$$919$$ 30221.0 1.08476 0.542382 0.840132i $$-0.317523\pi$$
0.542382 + 0.840132i $$0.317523\pi$$
$$920$$ −151739. −5.43769
$$921$$ 0 0
$$922$$ 23539.9 0.840831
$$923$$ 40284.0 1.43658
$$924$$ 0 0
$$925$$ 65608.0 2.33208
$$926$$ 4746.36 0.168440
$$927$$ 0 0
$$928$$ 11261.0 0.398341
$$929$$ −52669.0 −1.86008 −0.930040 0.367459i $$-0.880228\pi$$
−0.930040 + 0.367459i $$0.880228\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ −59387.8 −2.08724
$$933$$ 0 0
$$934$$ −11322.0 −0.396644
$$935$$ 860.115 0.0300842
$$936$$ 0 0
$$937$$ −54528.9 −1.90116 −0.950578 0.310486i $$-0.899508\pi$$
−0.950578 + 0.310486i $$0.899508\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ −77807.5 −2.69979
$$941$$ 1414.91 0.0490169 0.0245085 0.999700i $$-0.492198\pi$$
0.0245085 + 0.999700i $$0.492198\pi$$
$$942$$ 0 0
$$943$$ 13021.3 0.449663
$$944$$ 82376.4 2.84017
$$945$$ 0 0
$$946$$ 722.235 0.0248223
$$947$$ 45287.6 1.55401 0.777005 0.629494i $$-0.216738\pi$$
0.777005 + 0.629494i $$0.216738\pi$$
$$948$$ 0 0
$$949$$ −16183.1 −0.553557
$$950$$ 148341. 5.06613
$$951$$ 0 0
$$952$$ 0 0
$$953$$ −21718.7 −0.738235 −0.369117 0.929383i $$-0.620340\pi$$
−0.369117 + 0.929383i $$0.620340\pi$$
$$954$$ 0 0
$$955$$ 62379.3 2.11366
$$956$$ 88444.2 2.99214
$$957$$ 0 0
$$958$$ 8853.96 0.298599
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −26657.6 −0.894819
$$962$$ −88872.6 −2.97855
$$963$$ 0 0
$$964$$ −70487.9 −2.35505
$$965$$ 98963.6 3.30130
$$966$$ 0 0
$$967$$ 38754.4 1.28879 0.644393 0.764694i $$-0.277110\pi$$
0.644393 + 0.764694i $$0.277110\pi$$
$$968$$ −67220.3 −2.23196
$$969$$ 0 0
$$970$$ 86528.8 2.86420
$$971$$ −53591.6 −1.77120 −0.885601 0.464446i $$-0.846253\pi$$
−0.885601 + 0.464446i $$0.846253\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ −67641.9 −2.22524
$$975$$ 0 0
$$976$$ −44231.8 −1.45064
$$977$$ −18556.3 −0.607645 −0.303822 0.952729i $$-0.598263\pi$$
−0.303822 + 0.952729i $$0.598263\pi$$
$$978$$ 0 0
$$979$$ 318.719 0.0104048
$$980$$ 0 0
$$981$$ 0 0
$$982$$ −39526.8 −1.28447
$$983$$ 3557.17 0.115418 0.0577091 0.998333i $$-0.481620\pi$$
0.0577091 + 0.998333i $$0.481620\pi$$
$$984$$ 0 0
$$985$$ −32557.4 −1.05316
$$986$$ 14272.8 0.460994
$$987$$ 0 0
$$988$$ −138940. −4.47396
$$989$$ −21674.2 −0.696864
$$990$$ 0 0
$$991$$ 44666.7 1.43177 0.715886 0.698218i $$-0.246023\pi$$
0.715886 + 0.698218i $$0.246023\pi$$
$$992$$ 9845.53 0.315117
$$993$$ 0 0
$$994$$ 0 0
$$995$$ −29238.8 −0.931592
$$996$$ 0 0
$$997$$ −1142.64 −0.0362967 −0.0181484 0.999835i $$-0.505777\pi$$
−0.0181484 + 0.999835i $$0.505777\pi$$
$$998$$ 92648.1 2.93860
$$999$$ 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.bj.1.7 7
3.2 odd 2 1323.4.a.bi.1.1 7
7.2 even 3 189.4.e.f.109.1 14
7.4 even 3 189.4.e.f.163.1 yes 14
7.6 odd 2 1323.4.a.bk.1.7 7
21.2 odd 6 189.4.e.g.109.7 yes 14
21.11 odd 6 189.4.e.g.163.7 yes 14
21.20 even 2 1323.4.a.bh.1.1 7

By twisted newform
Twist Min Dim Char Parity Ord Type
189.4.e.f.109.1 14 7.2 even 3
189.4.e.f.163.1 yes 14 7.4 even 3
189.4.e.g.109.7 yes 14 21.2 odd 6
189.4.e.g.163.7 yes 14 21.11 odd 6
1323.4.a.bh.1.1 7 21.20 even 2
1323.4.a.bi.1.1 7 3.2 odd 2
1323.4.a.bj.1.7 7 1.1 even 1 trivial
1323.4.a.bk.1.7 7 7.6 odd 2