# Properties

 Label 539.4.a.e.1.2 Level $539$ Weight $4$ Character 539.1 Self dual yes Analytic conductor $31.802$ Analytic rank $1$ 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] = [539,4,Mod(1,539)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(539, 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("539.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$539 = 7^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 539.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: $$31.8020294931$$ Analytic rank: $$1$$ 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 11) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$1.73205$$ of defining polynomial Character $$\chi$$ $$=$$ 539.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+2.73205 q^{2} +7.92820 q^{3} -0.535898 q^{4} -14.8564 q^{5} +21.6603 q^{6} -23.3205 q^{8} +35.8564 q^{9} +O(q^{10})$$ $$q+2.73205 q^{2} +7.92820 q^{3} -0.535898 q^{4} -14.8564 q^{5} +21.6603 q^{6} -23.3205 q^{8} +35.8564 q^{9} -40.5885 q^{10} -11.0000 q^{11} -4.24871 q^{12} -5.35898 q^{13} -117.785 q^{15} -59.4256 q^{16} +41.2154 q^{17} +97.9615 q^{18} -139.923 q^{19} +7.96152 q^{20} -30.0526 q^{22} -111.354 q^{23} -184.890 q^{24} +95.7128 q^{25} -14.6410 q^{26} +70.2154 q^{27} -24.9948 q^{29} -321.794 q^{30} -31.4974 q^{31} +24.2102 q^{32} -87.2102 q^{33} +112.603 q^{34} -19.2154 q^{36} +13.1436 q^{37} -382.277 q^{38} -42.4871 q^{39} +346.459 q^{40} -261.072 q^{41} -57.7128 q^{43} +5.89488 q^{44} -532.697 q^{45} -304.224 q^{46} +343.846 q^{47} -471.138 q^{48} +261.492 q^{50} +326.764 q^{51} +2.87187 q^{52} -342.995 q^{53} +191.832 q^{54} +163.420 q^{55} -1109.34 q^{57} -68.2872 q^{58} -88.3693 q^{59} +63.1206 q^{60} -738.697 q^{61} -86.0526 q^{62} +541.549 q^{64} +79.6152 q^{65} -238.263 q^{66} +342.359 q^{67} -22.0873 q^{68} -882.836 q^{69} -207.364 q^{71} -836.190 q^{72} +1010.60 q^{73} +35.9090 q^{74} +758.831 q^{75} +74.9845 q^{76} -116.077 q^{78} +1294.23 q^{79} +882.851 q^{80} -411.441 q^{81} -713.261 q^{82} -441.846 q^{83} -612.313 q^{85} -157.674 q^{86} -198.164 q^{87} +256.526 q^{88} +1489.11 q^{89} -1455.36 q^{90} +59.6743 q^{92} -249.718 q^{93} +939.405 q^{94} +2078.75 q^{95} +191.944 q^{96} -1346.42 q^{97} -394.420 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} + 2 q^{3} - 8 q^{4} - 2 q^{5} + 26 q^{6} - 12 q^{8} + 44 q^{9}+O(q^{10})$$ 2 * q + 2 * q^2 + 2 * q^3 - 8 * q^4 - 2 * q^5 + 26 * q^6 - 12 * q^8 + 44 * q^9 $$2 q + 2 q^{2} + 2 q^{3} - 8 q^{4} - 2 q^{5} + 26 q^{6} - 12 q^{8} + 44 q^{9} - 50 q^{10} - 22 q^{11} + 40 q^{12} - 80 q^{13} - 194 q^{15} - 8 q^{16} + 124 q^{17} + 92 q^{18} - 72 q^{19} - 88 q^{20} - 22 q^{22} - 98 q^{23} - 252 q^{24} + 136 q^{25} + 40 q^{26} + 182 q^{27} + 144 q^{29} - 266 q^{30} + 34 q^{31} - 104 q^{32} - 22 q^{33} + 52 q^{34} - 80 q^{36} + 54 q^{37} - 432 q^{38} + 400 q^{39} + 492 q^{40} - 536 q^{41} - 60 q^{43} + 88 q^{44} - 428 q^{45} - 314 q^{46} + 272 q^{47} - 776 q^{48} + 232 q^{50} - 164 q^{51} + 560 q^{52} - 492 q^{53} + 110 q^{54} + 22 q^{55} - 1512 q^{57} - 192 q^{58} - 634 q^{59} + 632 q^{60} - 840 q^{61} - 134 q^{62} + 224 q^{64} - 880 q^{65} - 286 q^{66} + 754 q^{67} - 640 q^{68} - 962 q^{69} - 678 q^{71} - 744 q^{72} + 400 q^{73} + 6 q^{74} + 520 q^{75} - 432 q^{76} - 440 q^{78} + 316 q^{79} + 1544 q^{80} - 1294 q^{81} - 512 q^{82} - 468 q^{83} + 452 q^{85} - 156 q^{86} - 1200 q^{87} + 132 q^{88} + 1842 q^{89} - 1532 q^{90} - 40 q^{92} - 638 q^{93} + 992 q^{94} + 2952 q^{95} + 952 q^{96} - 2194 q^{97} - 484 q^{99}+O(q^{100})$$ 2 * q + 2 * q^2 + 2 * q^3 - 8 * q^4 - 2 * q^5 + 26 * q^6 - 12 * q^8 + 44 * q^9 - 50 * q^10 - 22 * q^11 + 40 * q^12 - 80 * q^13 - 194 * q^15 - 8 * q^16 + 124 * q^17 + 92 * q^18 - 72 * q^19 - 88 * q^20 - 22 * q^22 - 98 * q^23 - 252 * q^24 + 136 * q^25 + 40 * q^26 + 182 * q^27 + 144 * q^29 - 266 * q^30 + 34 * q^31 - 104 * q^32 - 22 * q^33 + 52 * q^34 - 80 * q^36 + 54 * q^37 - 432 * q^38 + 400 * q^39 + 492 * q^40 - 536 * q^41 - 60 * q^43 + 88 * q^44 - 428 * q^45 - 314 * q^46 + 272 * q^47 - 776 * q^48 + 232 * q^50 - 164 * q^51 + 560 * q^52 - 492 * q^53 + 110 * q^54 + 22 * q^55 - 1512 * q^57 - 192 * q^58 - 634 * q^59 + 632 * q^60 - 840 * q^61 - 134 * q^62 + 224 * q^64 - 880 * q^65 - 286 * q^66 + 754 * q^67 - 640 * q^68 - 962 * q^69 - 678 * q^71 - 744 * q^72 + 400 * q^73 + 6 * q^74 + 520 * q^75 - 432 * q^76 - 440 * q^78 + 316 * q^79 + 1544 * q^80 - 1294 * q^81 - 512 * q^82 - 468 * q^83 + 452 * q^85 - 156 * q^86 - 1200 * q^87 + 132 * q^88 + 1842 * q^89 - 1532 * q^90 - 40 * q^92 - 638 * q^93 + 992 * q^94 + 2952 * q^95 + 952 * q^96 - 2194 * q^97 - 484 * q^99

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 2.73205 0.965926 0.482963 0.875641i $$-0.339561\pi$$
0.482963 + 0.875641i $$0.339561\pi$$
$$3$$ 7.92820 1.52578 0.762892 0.646526i $$-0.223779\pi$$
0.762892 + 0.646526i $$0.223779\pi$$
$$4$$ −0.535898 −0.0669873
$$5$$ −14.8564 −1.32880 −0.664399 0.747378i $$-0.731312\pi$$
−0.664399 + 0.747378i $$0.731312\pi$$
$$6$$ 21.6603 1.47379
$$7$$ 0 0
$$8$$ −23.3205 −1.03063
$$9$$ 35.8564 1.32802
$$10$$ −40.5885 −1.28352
$$11$$ −11.0000 −0.301511
$$12$$ −4.24871 −0.102208
$$13$$ −5.35898 −0.114332 −0.0571659 0.998365i $$-0.518206\pi$$
−0.0571659 + 0.998365i $$0.518206\pi$$
$$14$$ 0 0
$$15$$ −117.785 −2.02746
$$16$$ −59.4256 −0.928525
$$17$$ 41.2154 0.588012 0.294006 0.955804i $$-0.405011\pi$$
0.294006 + 0.955804i $$0.405011\pi$$
$$18$$ 97.9615 1.28276
$$19$$ −139.923 −1.68950 −0.844751 0.535159i $$-0.820252\pi$$
−0.844751 + 0.535159i $$0.820252\pi$$
$$20$$ 7.96152 0.0890125
$$21$$ 0 0
$$22$$ −30.0526 −0.291238
$$23$$ −111.354 −1.00952 −0.504758 0.863261i $$-0.668418\pi$$
−0.504758 + 0.863261i $$0.668418\pi$$
$$24$$ −184.890 −1.57252
$$25$$ 95.7128 0.765703
$$26$$ −14.6410 −0.110436
$$27$$ 70.2154 0.500480
$$28$$ 0 0
$$29$$ −24.9948 −0.160049 −0.0800246 0.996793i $$-0.525500\pi$$
−0.0800246 + 0.996793i $$0.525500\pi$$
$$30$$ −321.794 −1.95837
$$31$$ −31.4974 −0.182487 −0.0912436 0.995829i $$-0.529084\pi$$
−0.0912436 + 0.995829i $$0.529084\pi$$
$$32$$ 24.2102 0.133744
$$33$$ −87.2102 −0.460041
$$34$$ 112.603 0.567976
$$35$$ 0 0
$$36$$ −19.2154 −0.0889601
$$37$$ 13.1436 0.0583998 0.0291999 0.999574i $$-0.490704\pi$$
0.0291999 + 0.999574i $$0.490704\pi$$
$$38$$ −382.277 −1.63193
$$39$$ −42.4871 −0.174446
$$40$$ 346.459 1.36950
$$41$$ −261.072 −0.994453 −0.497226 0.867621i $$-0.665648\pi$$
−0.497226 + 0.867621i $$0.665648\pi$$
$$42$$ 0 0
$$43$$ −57.7128 −0.204677 −0.102339 0.994750i $$-0.532633\pi$$
−0.102339 + 0.994750i $$0.532633\pi$$
$$44$$ 5.89488 0.0201974
$$45$$ −532.697 −1.76466
$$46$$ −304.224 −0.975118
$$47$$ 343.846 1.06713 0.533565 0.845759i $$-0.320852\pi$$
0.533565 + 0.845759i $$0.320852\pi$$
$$48$$ −471.138 −1.41673
$$49$$ 0 0
$$50$$ 261.492 0.739612
$$51$$ 326.764 0.897179
$$52$$ 2.87187 0.00765879
$$53$$ −342.995 −0.888943 −0.444471 0.895793i $$-0.646608\pi$$
−0.444471 + 0.895793i $$0.646608\pi$$
$$54$$ 191.832 0.483426
$$55$$ 163.420 0.400647
$$56$$ 0 0
$$57$$ −1109.34 −2.57782
$$58$$ −68.2872 −0.154596
$$59$$ −88.3693 −0.194995 −0.0974975 0.995236i $$-0.531084\pi$$
−0.0974975 + 0.995236i $$0.531084\pi$$
$$60$$ 63.1206 0.135814
$$61$$ −738.697 −1.55050 −0.775250 0.631654i $$-0.782376\pi$$
−0.775250 + 0.631654i $$0.782376\pi$$
$$62$$ −86.0526 −0.176269
$$63$$ 0 0
$$64$$ 541.549 1.05771
$$65$$ 79.6152 0.151924
$$66$$ −238.263 −0.444365
$$67$$ 342.359 0.624266 0.312133 0.950038i $$-0.398957\pi$$
0.312133 + 0.950038i $$0.398957\pi$$
$$68$$ −22.0873 −0.0393893
$$69$$ −882.836 −1.54030
$$70$$ 0 0
$$71$$ −207.364 −0.346614 −0.173307 0.984868i $$-0.555445\pi$$
−0.173307 + 0.984868i $$0.555445\pi$$
$$72$$ −836.190 −1.36869
$$73$$ 1010.60 1.62030 0.810149 0.586224i $$-0.199386\pi$$
0.810149 + 0.586224i $$0.199386\pi$$
$$74$$ 35.9090 0.0564099
$$75$$ 758.831 1.16830
$$76$$ 74.9845 0.113175
$$77$$ 0 0
$$78$$ −116.077 −0.168502
$$79$$ 1294.23 1.84319 0.921593 0.388157i $$-0.126888\pi$$
0.921593 + 0.388157i $$0.126888\pi$$
$$80$$ 882.851 1.23382
$$81$$ −411.441 −0.564391
$$82$$ −713.261 −0.960568
$$83$$ −441.846 −0.584324 −0.292162 0.956369i $$-0.594375\pi$$
−0.292162 + 0.956369i $$0.594375\pi$$
$$84$$ 0 0
$$85$$ −612.313 −0.781349
$$86$$ −157.674 −0.197703
$$87$$ −198.164 −0.244200
$$88$$ 256.526 0.310747
$$89$$ 1489.11 1.77355 0.886773 0.462205i $$-0.152942\pi$$
0.886773 + 0.462205i $$0.152942\pi$$
$$90$$ −1455.36 −1.70453
$$91$$ 0 0
$$92$$ 59.6743 0.0676248
$$93$$ −249.718 −0.278436
$$94$$ 939.405 1.03077
$$95$$ 2078.75 2.24501
$$96$$ 191.944 0.204064
$$97$$ −1346.42 −1.40936 −0.704679 0.709526i $$-0.748909\pi$$
−0.704679 + 0.709526i $$0.748909\pi$$
$$98$$ 0 0
$$99$$ −394.420 −0.400412
$$100$$ −51.2923 −0.0512923
$$101$$ 161.461 0.159069 0.0795347 0.996832i $$-0.474657\pi$$
0.0795347 + 0.996832i $$0.474657\pi$$
$$102$$ 892.736 0.866608
$$103$$ 34.7592 0.0332517 0.0166259 0.999862i $$-0.494708\pi$$
0.0166259 + 0.999862i $$0.494708\pi$$
$$104$$ 124.974 0.117834
$$105$$ 0 0
$$106$$ −937.079 −0.858653
$$107$$ 832.179 0.751867 0.375934 0.926647i $$-0.377322\pi$$
0.375934 + 0.926647i $$0.377322\pi$$
$$108$$ −37.6283 −0.0335258
$$109$$ 1044.26 0.917629 0.458815 0.888532i $$-0.348274\pi$$
0.458815 + 0.888532i $$0.348274\pi$$
$$110$$ 446.473 0.386996
$$111$$ 104.205 0.0891055
$$112$$ 0 0
$$113$$ 295.082 0.245654 0.122827 0.992428i $$-0.460804\pi$$
0.122827 + 0.992428i $$0.460804\pi$$
$$114$$ −3030.77 −2.48998
$$115$$ 1654.32 1.34144
$$116$$ 13.3947 0.0107213
$$117$$ −192.154 −0.151834
$$118$$ −241.429 −0.188351
$$119$$ 0 0
$$120$$ 2746.80 2.08956
$$121$$ 121.000 0.0909091
$$122$$ −2018.16 −1.49767
$$123$$ −2069.83 −1.51732
$$124$$ 16.8794 0.0122243
$$125$$ 435.102 0.311334
$$126$$ 0 0
$$127$$ −1317.60 −0.920618 −0.460309 0.887759i $$-0.652261\pi$$
−0.460309 + 0.887759i $$0.652261\pi$$
$$128$$ 1285.86 0.887928
$$129$$ −457.559 −0.312293
$$130$$ 217.513 0.146747
$$131$$ 1600.71 1.06759 0.533797 0.845612i $$-0.320765\pi$$
0.533797 + 0.845612i $$0.320765\pi$$
$$132$$ 46.7358 0.0308169
$$133$$ 0 0
$$134$$ 935.342 0.602994
$$135$$ −1043.15 −0.665036
$$136$$ −961.164 −0.606023
$$137$$ 1611.68 1.00507 0.502536 0.864556i $$-0.332400\pi$$
0.502536 + 0.864556i $$0.332400\pi$$
$$138$$ −2411.95 −1.48782
$$139$$ 31.8619 0.0194424 0.00972120 0.999953i $$-0.496906\pi$$
0.00972120 + 0.999953i $$0.496906\pi$$
$$140$$ 0 0
$$141$$ 2726.08 1.62821
$$142$$ −566.529 −0.334803
$$143$$ 58.9488 0.0344724
$$144$$ −2130.79 −1.23310
$$145$$ 371.334 0.212673
$$146$$ 2761.01 1.56509
$$147$$ 0 0
$$148$$ −7.04363 −0.00391205
$$149$$ −2428.34 −1.33515 −0.667576 0.744542i $$-0.732668\pi$$
−0.667576 + 0.744542i $$0.732668\pi$$
$$150$$ 2073.16 1.12849
$$151$$ −2576.68 −1.38866 −0.694328 0.719659i $$-0.744298\pi$$
−0.694328 + 0.719659i $$0.744298\pi$$
$$152$$ 3263.08 1.74125
$$153$$ 1477.84 0.780889
$$154$$ 0 0
$$155$$ 467.939 0.242489
$$156$$ 22.7688 0.0116856
$$157$$ −2475.94 −1.25861 −0.629305 0.777158i $$-0.716660\pi$$
−0.629305 + 0.777158i $$0.716660\pi$$
$$158$$ 3535.89 1.78038
$$159$$ −2719.33 −1.35633
$$160$$ −359.677 −0.177719
$$161$$ 0 0
$$162$$ −1124.08 −0.545160
$$163$$ −2725.11 −1.30949 −0.654745 0.755850i $$-0.727224\pi$$
−0.654745 + 0.755850i $$0.727224\pi$$
$$164$$ 139.908 0.0666157
$$165$$ 1295.63 0.611301
$$166$$ −1207.15 −0.564414
$$167$$ −2737.30 −1.26837 −0.634187 0.773180i $$-0.718665\pi$$
−0.634187 + 0.773180i $$0.718665\pi$$
$$168$$ 0 0
$$169$$ −2168.28 −0.986928
$$170$$ −1672.87 −0.754725
$$171$$ −5017.14 −2.24368
$$172$$ 30.9282 0.0137108
$$173$$ −2307.42 −1.01404 −0.507022 0.861933i $$-0.669254\pi$$
−0.507022 + 0.861933i $$0.669254\pi$$
$$174$$ −541.395 −0.235879
$$175$$ 0 0
$$176$$ 653.682 0.279961
$$177$$ −700.610 −0.297520
$$178$$ 4068.33 1.71311
$$179$$ −1312.15 −0.547905 −0.273953 0.961743i $$-0.588331\pi$$
−0.273953 + 0.961743i $$0.588331\pi$$
$$180$$ 285.472 0.118210
$$181$$ 803.174 0.329831 0.164916 0.986308i $$-0.447265\pi$$
0.164916 + 0.986308i $$0.447265\pi$$
$$182$$ 0 0
$$183$$ −5856.54 −2.36573
$$184$$ 2596.83 1.04044
$$185$$ −195.267 −0.0776015
$$186$$ −682.242 −0.268949
$$187$$ −453.369 −0.177292
$$188$$ −184.267 −0.0714842
$$189$$ 0 0
$$190$$ 5679.26 2.16851
$$191$$ 1718.25 0.650932 0.325466 0.945554i $$-0.394479\pi$$
0.325466 + 0.945554i $$0.394479\pi$$
$$192$$ 4293.51 1.61384
$$193$$ 1340.18 0.499837 0.249919 0.968267i $$-0.419596\pi$$
0.249919 + 0.968267i $$0.419596\pi$$
$$194$$ −3678.48 −1.36134
$$195$$ 631.206 0.231803
$$196$$ 0 0
$$197$$ −3518.33 −1.27244 −0.636220 0.771508i $$-0.719503\pi$$
−0.636220 + 0.771508i $$0.719503\pi$$
$$198$$ −1077.58 −0.386768
$$199$$ −823.692 −0.293417 −0.146709 0.989180i $$-0.546868\pi$$
−0.146709 + 0.989180i $$0.546868\pi$$
$$200$$ −2232.07 −0.789156
$$201$$ 2714.29 0.952494
$$202$$ 441.121 0.153649
$$203$$ 0 0
$$204$$ −175.112 −0.0600996
$$205$$ 3878.59 1.32143
$$206$$ 94.9639 0.0321187
$$207$$ −3992.75 −1.34065
$$208$$ 318.461 0.106160
$$209$$ 1539.15 0.509404
$$210$$ 0 0
$$211$$ −107.343 −0.0350228 −0.0175114 0.999847i $$-0.505574\pi$$
−0.0175114 + 0.999847i $$0.505574\pi$$
$$212$$ 183.810 0.0595479
$$213$$ −1644.03 −0.528858
$$214$$ 2273.56 0.726248
$$215$$ 857.405 0.271975
$$216$$ −1637.46 −0.515810
$$217$$ 0 0
$$218$$ 2852.96 0.886362
$$219$$ 8012.24 2.47222
$$220$$ −87.5768 −0.0268383
$$221$$ −220.873 −0.0672285
$$222$$ 284.694 0.0860693
$$223$$ 3933.68 1.18125 0.590625 0.806946i $$-0.298881\pi$$
0.590625 + 0.806946i $$0.298881\pi$$
$$224$$ 0 0
$$225$$ 3431.92 1.01686
$$226$$ 806.178 0.237284
$$227$$ 1771.90 0.518085 0.259042 0.965866i $$-0.416593\pi$$
0.259042 + 0.965866i $$0.416593\pi$$
$$228$$ 594.493 0.172681
$$229$$ −1915.37 −0.552713 −0.276356 0.961055i $$-0.589127\pi$$
−0.276356 + 0.961055i $$0.589127\pi$$
$$230$$ 4519.68 1.29573
$$231$$ 0 0
$$232$$ 582.892 0.164952
$$233$$ 4396.32 1.23610 0.618052 0.786137i $$-0.287922\pi$$
0.618052 + 0.786137i $$0.287922\pi$$
$$234$$ −524.974 −0.146661
$$235$$ −5108.32 −1.41800
$$236$$ 47.3570 0.0130622
$$237$$ 10260.9 2.81230
$$238$$ 0 0
$$239$$ −4084.49 −1.10546 −0.552728 0.833362i $$-0.686413\pi$$
−0.552728 + 0.833362i $$0.686413\pi$$
$$240$$ 6999.42 1.88255
$$241$$ −3908.58 −1.04471 −0.522353 0.852730i $$-0.674946\pi$$
−0.522353 + 0.852730i $$0.674946\pi$$
$$242$$ 330.578 0.0878114
$$243$$ −5157.80 −1.36162
$$244$$ 395.867 0.103864
$$245$$ 0 0
$$246$$ −5654.88 −1.46562
$$247$$ 749.845 0.193164
$$248$$ 734.536 0.188077
$$249$$ −3503.05 −0.891552
$$250$$ 1188.72 0.300725
$$251$$ −1094.89 −0.275335 −0.137667 0.990479i $$-0.543960\pi$$
−0.137667 + 0.990479i $$0.543960\pi$$
$$252$$ 0 0
$$253$$ 1224.89 0.304381
$$254$$ −3599.76 −0.889249
$$255$$ −4854.54 −1.19217
$$256$$ −819.364 −0.200040
$$257$$ −783.179 −0.190091 −0.0950454 0.995473i $$-0.530300\pi$$
−0.0950454 + 0.995473i $$0.530300\pi$$
$$258$$ −1250.07 −0.301652
$$259$$ 0 0
$$260$$ −42.6657 −0.0101770
$$261$$ −896.225 −0.212548
$$262$$ 4373.23 1.03122
$$263$$ 6180.06 1.44897 0.724484 0.689292i $$-0.242078\pi$$
0.724484 + 0.689292i $$0.242078\pi$$
$$264$$ 2033.79 0.474132
$$265$$ 5095.67 1.18122
$$266$$ 0 0
$$267$$ 11806.0 2.70605
$$268$$ −183.470 −0.0418179
$$269$$ −986.965 −0.223704 −0.111852 0.993725i $$-0.535678\pi$$
−0.111852 + 0.993725i $$0.535678\pi$$
$$270$$ −2849.93 −0.642376
$$271$$ −4576.99 −1.02595 −0.512975 0.858404i $$-0.671457\pi$$
−0.512975 + 0.858404i $$0.671457\pi$$
$$272$$ −2449.25 −0.545984
$$273$$ 0 0
$$274$$ 4403.18 0.970825
$$275$$ −1052.84 −0.230868
$$276$$ 473.110 0.103181
$$277$$ 567.836 0.123169 0.0615847 0.998102i $$-0.480385\pi$$
0.0615847 + 0.998102i $$0.480385\pi$$
$$278$$ 87.0484 0.0187799
$$279$$ −1129.38 −0.242346
$$280$$ 0 0
$$281$$ 5311.01 1.12750 0.563752 0.825944i $$-0.309357\pi$$
0.563752 + 0.825944i $$0.309357\pi$$
$$282$$ 7447.79 1.57273
$$283$$ 4728.44 0.993204 0.496602 0.867978i $$-0.334581\pi$$
0.496602 + 0.867978i $$0.334581\pi$$
$$284$$ 111.126 0.0232187
$$285$$ 16480.8 3.42539
$$286$$ 161.051 0.0332977
$$287$$ 0 0
$$288$$ 868.092 0.177614
$$289$$ −3214.29 −0.654242
$$290$$ 1014.50 0.205426
$$291$$ −10674.7 −2.15038
$$292$$ −541.579 −0.108539
$$293$$ −2328.92 −0.464358 −0.232179 0.972673i $$-0.574585\pi$$
−0.232179 + 0.972673i $$0.574585\pi$$
$$294$$ 0 0
$$295$$ 1312.85 0.259109
$$296$$ −306.515 −0.0601886
$$297$$ −772.369 −0.150900
$$298$$ −6634.36 −1.28966
$$299$$ 596.743 0.115420
$$300$$ −406.656 −0.0782610
$$301$$ 0 0
$$302$$ −7039.61 −1.34134
$$303$$ 1280.10 0.242705
$$304$$ 8315.01 1.56875
$$305$$ 10974.4 2.06030
$$306$$ 4037.52 0.754280
$$307$$ 1678.07 0.311962 0.155981 0.987760i $$-0.450146\pi$$
0.155981 + 0.987760i $$0.450146\pi$$
$$308$$ 0 0
$$309$$ 275.578 0.0507349
$$310$$ 1278.43 0.234226
$$311$$ −3572.71 −0.651413 −0.325707 0.945471i $$-0.605602\pi$$
−0.325707 + 0.945471i $$0.605602\pi$$
$$312$$ 990.821 0.179789
$$313$$ −7184.36 −1.29739 −0.648697 0.761047i $$-0.724686\pi$$
−0.648697 + 0.761047i $$0.724686\pi$$
$$314$$ −6764.40 −1.21572
$$315$$ 0 0
$$316$$ −693.573 −0.123470
$$317$$ −15.7077 −0.00278306 −0.00139153 0.999999i $$-0.500443\pi$$
−0.00139153 + 0.999999i $$0.500443\pi$$
$$318$$ −7429.36 −1.31012
$$319$$ 274.943 0.0482566
$$320$$ −8045.47 −1.40549
$$321$$ 6597.69 1.14719
$$322$$ 0 0
$$323$$ −5766.98 −0.993447
$$324$$ 220.491 0.0378070
$$325$$ −512.923 −0.0875442
$$326$$ −7445.13 −1.26487
$$327$$ 8279.08 1.40010
$$328$$ 6088.33 1.02491
$$329$$ 0 0
$$330$$ 3539.73 0.590472
$$331$$ −1318.95 −0.219022 −0.109511 0.993986i $$-0.534928\pi$$
−0.109511 + 0.993986i $$0.534928\pi$$
$$332$$ 236.785 0.0391423
$$333$$ 471.282 0.0775558
$$334$$ −7478.43 −1.22515
$$335$$ −5086.22 −0.829523
$$336$$ 0 0
$$337$$ −239.183 −0.0386621 −0.0193310 0.999813i $$-0.506154\pi$$
−0.0193310 + 0.999813i $$0.506154\pi$$
$$338$$ −5923.85 −0.953299
$$339$$ 2339.47 0.374816
$$340$$ 328.137 0.0523404
$$341$$ 346.472 0.0550220
$$342$$ −13707.1 −2.16723
$$343$$ 0 0
$$344$$ 1345.89 0.210947
$$345$$ 13115.8 2.04675
$$346$$ −6303.98 −0.979491
$$347$$ −5862.79 −0.907006 −0.453503 0.891255i $$-0.649826\pi$$
−0.453503 + 0.891255i $$0.649826\pi$$
$$348$$ 106.196 0.0163583
$$349$$ −3491.73 −0.535553 −0.267776 0.963481i $$-0.586289\pi$$
−0.267776 + 0.963481i $$0.586289\pi$$
$$350$$ 0 0
$$351$$ −376.283 −0.0572208
$$352$$ −266.313 −0.0403253
$$353$$ 10916.7 1.64600 0.822999 0.568043i $$-0.192299\pi$$
0.822999 + 0.568043i $$0.192299\pi$$
$$354$$ −1914.10 −0.287382
$$355$$ 3080.69 0.460580
$$356$$ −798.013 −0.118805
$$357$$ 0 0
$$358$$ −3584.87 −0.529236
$$359$$ −11500.7 −1.69077 −0.845384 0.534160i $$-0.820628\pi$$
−0.845384 + 0.534160i $$0.820628\pi$$
$$360$$ 12422.8 1.81872
$$361$$ 12719.5 1.85442
$$362$$ 2194.31 0.318592
$$363$$ 959.313 0.138708
$$364$$ 0 0
$$365$$ −15013.9 −2.15305
$$366$$ −16000.4 −2.28512
$$367$$ −6767.01 −0.962493 −0.481246 0.876585i $$-0.659816\pi$$
−0.481246 + 0.876585i $$0.659816\pi$$
$$368$$ 6617.27 0.937362
$$369$$ −9361.10 −1.32065
$$370$$ −533.478 −0.0749573
$$371$$ 0 0
$$372$$ 133.823 0.0186517
$$373$$ −5310.22 −0.737139 −0.368569 0.929600i $$-0.620152\pi$$
−0.368569 + 0.929600i $$0.620152\pi$$
$$374$$ −1238.63 −0.171251
$$375$$ 3449.58 0.475028
$$376$$ −8018.67 −1.09982
$$377$$ 133.947 0.0182987
$$378$$ 0 0
$$379$$ −838.267 −0.113612 −0.0568059 0.998385i $$-0.518092\pi$$
−0.0568059 + 0.998385i $$0.518092\pi$$
$$380$$ −1114.00 −0.150387
$$381$$ −10446.2 −1.40466
$$382$$ 4694.34 0.628752
$$383$$ 2832.16 0.377851 0.188925 0.981991i $$-0.439500\pi$$
0.188925 + 0.981991i $$0.439500\pi$$
$$384$$ 10194.5 1.35479
$$385$$ 0 0
$$386$$ 3661.45 0.482806
$$387$$ −2069.37 −0.271814
$$388$$ 721.542 0.0944091
$$389$$ 3111.25 0.405519 0.202759 0.979229i $$-0.435009\pi$$
0.202759 + 0.979229i $$0.435009\pi$$
$$390$$ 1724.49 0.223905
$$391$$ −4589.49 −0.593608
$$392$$ 0 0
$$393$$ 12690.8 1.62892
$$394$$ −9612.25 −1.22908
$$395$$ −19227.5 −2.44922
$$396$$ 211.369 0.0268225
$$397$$ −14208.7 −1.79626 −0.898131 0.439728i $$-0.855075\pi$$
−0.898131 + 0.439728i $$0.855075\pi$$
$$398$$ −2250.37 −0.283419
$$399$$ 0 0
$$400$$ −5687.79 −0.710974
$$401$$ −6261.68 −0.779784 −0.389892 0.920861i $$-0.627488\pi$$
−0.389892 + 0.920861i $$0.627488\pi$$
$$402$$ 7415.58 0.920039
$$403$$ 168.794 0.0208641
$$404$$ −86.5269 −0.0106556
$$405$$ 6112.54 0.749961
$$406$$ 0 0
$$407$$ −144.580 −0.0176082
$$408$$ −7620.30 −0.924660
$$409$$ 4192.50 0.506860 0.253430 0.967354i $$-0.418441\pi$$
0.253430 + 0.967354i $$0.418441\pi$$
$$410$$ 10596.5 1.27640
$$411$$ 12777.7 1.53352
$$412$$ −18.6274 −0.00222744
$$413$$ 0 0
$$414$$ −10908.4 −1.29497
$$415$$ 6564.25 0.776448
$$416$$ −129.742 −0.0152912
$$417$$ 252.608 0.0296649
$$418$$ 4205.05 0.492047
$$419$$ 9287.15 1.08283 0.541416 0.840755i $$-0.317888\pi$$
0.541416 + 0.840755i $$0.317888\pi$$
$$420$$ 0 0
$$421$$ 13146.0 1.52185 0.760923 0.648842i $$-0.224746\pi$$
0.760923 + 0.648842i $$0.224746\pi$$
$$422$$ −293.267 −0.0338294
$$423$$ 12329.1 1.41716
$$424$$ 7998.81 0.916172
$$425$$ 3944.84 0.450242
$$426$$ −4491.56 −0.510838
$$427$$ 0 0
$$428$$ −445.964 −0.0503656
$$429$$ 467.358 0.0525974
$$430$$ 2342.47 0.262707
$$431$$ 4909.67 0.548701 0.274351 0.961630i $$-0.411537\pi$$
0.274351 + 0.961630i $$0.411537\pi$$
$$432$$ −4172.59 −0.464708
$$433$$ 11743.3 1.30334 0.651671 0.758502i $$-0.274068\pi$$
0.651671 + 0.758502i $$0.274068\pi$$
$$434$$ 0 0
$$435$$ 2944.01 0.324493
$$436$$ −559.615 −0.0614695
$$437$$ 15581.0 1.70558
$$438$$ 21889.8 2.38798
$$439$$ 11824.2 1.28551 0.642754 0.766073i $$-0.277792\pi$$
0.642754 + 0.766073i $$0.277792\pi$$
$$440$$ −3811.05 −0.412920
$$441$$ 0 0
$$442$$ −603.435 −0.0649377
$$443$$ 10102.1 1.08344 0.541722 0.840558i $$-0.317772\pi$$
0.541722 + 0.840558i $$0.317772\pi$$
$$444$$ −55.8433 −0.00596894
$$445$$ −22122.9 −2.35668
$$446$$ 10747.0 1.14100
$$447$$ −19252.4 −2.03715
$$448$$ 0 0
$$449$$ −345.254 −0.0362885 −0.0181443 0.999835i $$-0.505776\pi$$
−0.0181443 + 0.999835i $$0.505776\pi$$
$$450$$ 9376.17 0.982216
$$451$$ 2871.79 0.299839
$$452$$ −158.134 −0.0164557
$$453$$ −20428.4 −2.11879
$$454$$ 4840.93 0.500431
$$455$$ 0 0
$$456$$ 25870.3 2.65678
$$457$$ −10567.1 −1.08164 −0.540821 0.841138i $$-0.681886\pi$$
−0.540821 + 0.841138i $$0.681886\pi$$
$$458$$ −5232.89 −0.533879
$$459$$ 2893.95 0.294288
$$460$$ −886.546 −0.0898596
$$461$$ −4733.96 −0.478270 −0.239135 0.970986i $$-0.576864\pi$$
−0.239135 + 0.970986i $$0.576864\pi$$
$$462$$ 0 0
$$463$$ 3431.20 0.344409 0.172204 0.985061i $$-0.444911\pi$$
0.172204 + 0.985061i $$0.444911\pi$$
$$464$$ 1485.33 0.148610
$$465$$ 3709.91 0.369985
$$466$$ 12011.0 1.19399
$$467$$ −5116.96 −0.507034 −0.253517 0.967331i $$-0.581587\pi$$
−0.253517 + 0.967331i $$0.581587\pi$$
$$468$$ 102.975 0.0101710
$$469$$ 0 0
$$470$$ −13956.2 −1.36968
$$471$$ −19629.8 −1.92037
$$472$$ 2060.82 0.200968
$$473$$ 634.841 0.0617125
$$474$$ 28033.2 2.71648
$$475$$ −13392.4 −1.29366
$$476$$ 0 0
$$477$$ −12298.6 −1.18053
$$478$$ −11159.0 −1.06779
$$479$$ −11566.9 −1.10335 −0.551675 0.834059i $$-0.686011\pi$$
−0.551675 + 0.834059i $$0.686011\pi$$
$$480$$ −2851.59 −0.271160
$$481$$ −70.4363 −0.00667696
$$482$$ −10678.4 −1.00911
$$483$$ 0 0
$$484$$ −64.8437 −0.00608975
$$485$$ 20002.9 1.87275
$$486$$ −14091.4 −1.31522
$$487$$ −18326.5 −1.70525 −0.852623 0.522527i $$-0.824990\pi$$
−0.852623 + 0.522527i $$0.824990\pi$$
$$488$$ 17226.8 1.59799
$$489$$ −21605.2 −1.99800
$$490$$ 0 0
$$491$$ −7617.58 −0.700156 −0.350078 0.936721i $$-0.613845\pi$$
−0.350078 + 0.936721i $$0.613845\pi$$
$$492$$ 1109.22 0.101641
$$493$$ −1030.17 −0.0941108
$$494$$ 2048.62 0.186582
$$495$$ 5859.67 0.532066
$$496$$ 1871.75 0.169444
$$497$$ 0 0
$$498$$ −9570.50 −0.861173
$$499$$ 12909.1 1.15810 0.579050 0.815292i $$-0.303424\pi$$
0.579050 + 0.815292i $$0.303424\pi$$
$$500$$ −233.171 −0.0208554
$$501$$ −21701.8 −1.93526
$$502$$ −2991.30 −0.265953
$$503$$ −10165.7 −0.901121 −0.450561 0.892746i $$-0.648776\pi$$
−0.450561 + 0.892746i $$0.648776\pi$$
$$504$$ 0 0
$$505$$ −2398.74 −0.211371
$$506$$ 3346.47 0.294009
$$507$$ −17190.6 −1.50584
$$508$$ 706.102 0.0616697
$$509$$ −6449.93 −0.561666 −0.280833 0.959757i $$-0.590611\pi$$
−0.280833 + 0.959757i $$0.590611\pi$$
$$510$$ −13262.8 −1.15155
$$511$$ 0 0
$$512$$ −12525.4 −1.08115
$$513$$ −9824.75 −0.845562
$$514$$ −2139.68 −0.183614
$$515$$ −516.397 −0.0441848
$$516$$ 245.205 0.0209197
$$517$$ −3782.31 −0.321752
$$518$$ 0 0
$$519$$ −18293.7 −1.54721
$$520$$ −1856.67 −0.156577
$$521$$ 19327.4 1.62524 0.812620 0.582794i $$-0.198041\pi$$
0.812620 + 0.582794i $$0.198041\pi$$
$$522$$ −2448.53 −0.205305
$$523$$ −6259.09 −0.523310 −0.261655 0.965161i $$-0.584268\pi$$
−0.261655 + 0.965161i $$0.584268\pi$$
$$524$$ −857.819 −0.0715153
$$525$$ 0 0
$$526$$ 16884.2 1.39960
$$527$$ −1298.18 −0.107305
$$528$$ 5182.52 0.427160
$$529$$ 232.675 0.0191235
$$530$$ 13921.6 1.14098
$$531$$ −3168.61 −0.258956
$$532$$ 0 0
$$533$$ 1399.08 0.113698
$$534$$ 32254.6 2.61384
$$535$$ −12363.2 −0.999079
$$536$$ −7983.99 −0.643387
$$537$$ −10403.0 −0.835985
$$538$$ −2696.44 −0.216081
$$539$$ 0 0
$$540$$ 559.022 0.0445490
$$541$$ −14008.2 −1.11323 −0.556616 0.830770i $$-0.687900\pi$$
−0.556616 + 0.830770i $$0.687900\pi$$
$$542$$ −12504.6 −0.990991
$$543$$ 6367.72 0.503251
$$544$$ 997.834 0.0786430
$$545$$ −15513.9 −1.21934
$$546$$ 0 0
$$547$$ −4949.45 −0.386879 −0.193440 0.981112i $$-0.561964\pi$$
−0.193440 + 0.981112i $$0.561964\pi$$
$$548$$ −863.695 −0.0673270
$$549$$ −26487.0 −2.05909
$$550$$ −2876.41 −0.223001
$$551$$ 3497.35 0.270404
$$552$$ 20588.2 1.58748
$$553$$ 0 0
$$554$$ 1551.36 0.118973
$$555$$ −1548.11 −0.118403
$$556$$ −17.0748 −0.00130239
$$557$$ −3801.58 −0.289188 −0.144594 0.989491i $$-0.546188\pi$$
−0.144594 + 0.989491i $$0.546188\pi$$
$$558$$ −3085.54 −0.234088
$$559$$ 309.282 0.0234011
$$560$$ 0 0
$$561$$ −3594.40 −0.270510
$$562$$ 14510.0 1.08908
$$563$$ 9900.11 0.741101 0.370551 0.928812i $$-0.379169\pi$$
0.370551 + 0.928812i $$0.379169\pi$$
$$564$$ −1460.90 −0.109069
$$565$$ −4383.85 −0.326425
$$566$$ 12918.3 0.959361
$$567$$ 0 0
$$568$$ 4835.84 0.357231
$$569$$ 5329.16 0.392636 0.196318 0.980540i $$-0.437102\pi$$
0.196318 + 0.980540i $$0.437102\pi$$
$$570$$ 45026.3 3.30868
$$571$$ −16962.6 −1.24319 −0.621597 0.783337i $$-0.713516\pi$$
−0.621597 + 0.783337i $$0.713516\pi$$
$$572$$ −31.5906 −0.00230921
$$573$$ 13622.6 0.993181
$$574$$ 0 0
$$575$$ −10658.0 −0.772989
$$576$$ 19418.0 1.40466
$$577$$ 15487.0 1.11738 0.558692 0.829375i $$-0.311303\pi$$
0.558692 + 0.829375i $$0.311303\pi$$
$$578$$ −8781.61 −0.631949
$$579$$ 10625.3 0.762643
$$580$$ −198.997 −0.0142464
$$581$$ 0 0
$$582$$ −29163.7 −2.07710
$$583$$ 3772.94 0.268026
$$584$$ −23567.7 −1.66993
$$585$$ 2854.72 0.201757
$$586$$ −6362.72 −0.448535
$$587$$ −11084.2 −0.779373 −0.389686 0.920948i $$-0.627417\pi$$
−0.389686 + 0.920948i $$0.627417\pi$$
$$588$$ 0 0
$$589$$ 4407.22 0.308313
$$590$$ 3586.77 0.250280
$$591$$ −27894.0 −1.94147
$$592$$ −781.066 −0.0542257
$$593$$ −4349.68 −0.301214 −0.150607 0.988594i $$-0.548123\pi$$
−0.150607 + 0.988594i $$0.548123\pi$$
$$594$$ −2110.15 −0.145759
$$595$$ 0 0
$$596$$ 1301.34 0.0894382
$$597$$ −6530.40 −0.447691
$$598$$ 1630.33 0.111487
$$599$$ 13183.9 0.899299 0.449650 0.893205i $$-0.351549\pi$$
0.449650 + 0.893205i $$0.351549\pi$$
$$600$$ −17696.3 −1.20408
$$601$$ 18765.0 1.27361 0.636806 0.771024i $$-0.280255\pi$$
0.636806 + 0.771024i $$0.280255\pi$$
$$602$$ 0 0
$$603$$ 12275.8 0.829034
$$604$$ 1380.84 0.0930223
$$605$$ −1797.63 −0.120800
$$606$$ 3497.29 0.234435
$$607$$ −21871.4 −1.46249 −0.731244 0.682116i $$-0.761060\pi$$
−0.731244 + 0.682116i $$0.761060\pi$$
$$608$$ −3387.57 −0.225961
$$609$$ 0 0
$$610$$ 29982.6 1.99010
$$611$$ −1842.67 −0.122007
$$612$$ −791.970 −0.0523096
$$613$$ −3527.85 −0.232445 −0.116222 0.993223i $$-0.537079\pi$$
−0.116222 + 0.993223i $$0.537079\pi$$
$$614$$ 4584.56 0.301332
$$615$$ 30750.2 2.01621
$$616$$ 0 0
$$617$$ −22728.1 −1.48298 −0.741490 0.670963i $$-0.765881\pi$$
−0.741490 + 0.670963i $$0.765881\pi$$
$$618$$ 752.893 0.0490062
$$619$$ 21443.3 1.39237 0.696187 0.717861i $$-0.254879\pi$$
0.696187 + 0.717861i $$0.254879\pi$$
$$620$$ −250.767 −0.0162437
$$621$$ −7818.75 −0.505243
$$622$$ −9760.81 −0.629217
$$623$$ 0 0
$$624$$ 2524.82 0.161977
$$625$$ −18428.2 −1.17940
$$626$$ −19628.0 −1.25319
$$627$$ 12202.7 0.777240
$$628$$ 1326.85 0.0843109
$$629$$ 541.718 0.0343398
$$630$$ 0 0
$$631$$ 21532.0 1.35844 0.679219 0.733936i $$-0.262319\pi$$
0.679219 + 0.733936i $$0.262319\pi$$
$$632$$ −30182.0 −1.89964
$$633$$ −851.038 −0.0534372
$$634$$ −42.9141 −0.00268823
$$635$$ 19574.9 1.22332
$$636$$ 1457.29 0.0908572
$$637$$ 0 0
$$638$$ 751.159 0.0466123
$$639$$ −7435.33 −0.460309
$$640$$ −19103.2 −1.17988
$$641$$ 20148.3 1.24151 0.620756 0.784004i $$-0.286826\pi$$
0.620756 + 0.784004i $$0.286826\pi$$
$$642$$ 18025.2 1.10810
$$643$$ −28869.7 −1.77062 −0.885310 0.465000i $$-0.846054\pi$$
−0.885310 + 0.465000i $$0.846054\pi$$
$$644$$ 0 0
$$645$$ 6797.68 0.414974
$$646$$ −15755.7 −0.959597
$$647$$ 1590.02 0.0966155 0.0483077 0.998833i $$-0.484617\pi$$
0.0483077 + 0.998833i $$0.484617\pi$$
$$648$$ 9595.02 0.581679
$$649$$ 972.062 0.0587932
$$650$$ −1401.33 −0.0845612
$$651$$ 0 0
$$652$$ 1460.38 0.0877192
$$653$$ 20028.1 1.20024 0.600122 0.799909i $$-0.295119\pi$$
0.600122 + 0.799909i $$0.295119\pi$$
$$654$$ 22618.9 1.35240
$$655$$ −23780.8 −1.41862
$$656$$ 15514.4 0.923375
$$657$$ 36236.5 2.15178
$$658$$ 0 0
$$659$$ −10520.7 −0.621897 −0.310948 0.950427i $$-0.600647\pi$$
−0.310948 + 0.950427i $$0.600647\pi$$
$$660$$ −694.326 −0.0409494
$$661$$ −3295.83 −0.193938 −0.0969690 0.995287i $$-0.530915\pi$$
−0.0969690 + 0.995287i $$0.530915\pi$$
$$662$$ −3603.45 −0.211559
$$663$$ −1751.12 −0.102576
$$664$$ 10304.1 0.602222
$$665$$ 0 0
$$666$$ 1287.57 0.0749132
$$667$$ 2783.27 0.161572
$$668$$ 1466.91 0.0849649
$$669$$ 31187.0 1.80233
$$670$$ −13895.8 −0.801257
$$671$$ 8125.67 0.467493
$$672$$ 0 0
$$673$$ −1187.64 −0.0680239 −0.0340119 0.999421i $$-0.510828\pi$$
−0.0340119 + 0.999421i $$0.510828\pi$$
$$674$$ −653.460 −0.0373447
$$675$$ 6720.51 0.383219
$$676$$ 1161.98 0.0661117
$$677$$ −13221.4 −0.750574 −0.375287 0.926909i $$-0.622456\pi$$
−0.375287 + 0.926909i $$0.622456\pi$$
$$678$$ 6391.55 0.362044
$$679$$ 0 0
$$680$$ 14279.4 0.805282
$$681$$ 14048.0 0.790485
$$682$$ 946.578 0.0531471
$$683$$ −13831.4 −0.774882 −0.387441 0.921894i $$-0.626641\pi$$
−0.387441 + 0.921894i $$0.626641\pi$$
$$684$$ 2688.68 0.150298
$$685$$ −23943.7 −1.33554
$$686$$ 0 0
$$687$$ −15185.4 −0.843320
$$688$$ 3429.62 0.190048
$$689$$ 1838.10 0.101635
$$690$$ 35832.9 1.97701
$$691$$ 9817.07 0.540462 0.270231 0.962796i $$-0.412900\pi$$
0.270231 + 0.962796i $$0.412900\pi$$
$$692$$ 1236.54 0.0679280
$$693$$ 0 0
$$694$$ −16017.4 −0.876101
$$695$$ −473.354 −0.0258350
$$696$$ 4621.29 0.251680
$$697$$ −10760.2 −0.584750
$$698$$ −9539.58 −0.517304
$$699$$ 34854.9 1.88603
$$700$$ 0 0
$$701$$ 29949.8 1.61368 0.806838 0.590773i $$-0.201177\pi$$
0.806838 + 0.590773i $$0.201177\pi$$
$$702$$ −1028.02 −0.0552711
$$703$$ −1839.09 −0.0986667
$$704$$ −5957.03 −0.318912
$$705$$ −40499.8 −2.16356
$$706$$ 29825.0 1.58991
$$707$$ 0 0
$$708$$ 375.456 0.0199301
$$709$$ 11307.5 0.598959 0.299479 0.954103i $$-0.403187\pi$$
0.299479 + 0.954103i $$0.403187\pi$$
$$710$$ 8416.59 0.444886
$$711$$ 46406.3 2.44778
$$712$$ −34726.9 −1.82787
$$713$$ 3507.36 0.184224
$$714$$ 0 0
$$715$$ −875.768 −0.0458068
$$716$$ 703.181 0.0367027
$$717$$ −32382.7 −1.68669
$$718$$ −31420.6 −1.63316
$$719$$ 32623.4 1.69214 0.846070 0.533071i $$-0.178962\pi$$
0.846070 + 0.533071i $$0.178962\pi$$
$$720$$ 31655.9 1.63853
$$721$$ 0 0
$$722$$ 34750.2 1.79123
$$723$$ −30988.0 −1.59399
$$724$$ −430.420 −0.0220945
$$725$$ −2392.33 −0.122550
$$726$$ 2620.89 0.133981
$$727$$ 502.545 0.0256373 0.0128187 0.999918i $$-0.495920\pi$$
0.0128187 + 0.999918i $$0.495920\pi$$
$$728$$ 0 0
$$729$$ −29783.2 −1.51314
$$730$$ −41018.7 −2.07968
$$731$$ −2378.66 −0.120353
$$732$$ 3138.51 0.158474
$$733$$ −8631.37 −0.434935 −0.217467 0.976068i $$-0.569780\pi$$
−0.217467 + 0.976068i $$0.569780\pi$$
$$734$$ −18487.8 −0.929697
$$735$$ 0 0
$$736$$ −2695.90 −0.135017
$$737$$ −3765.95 −0.188223
$$738$$ −25575.0 −1.27565
$$739$$ −18357.5 −0.913792 −0.456896 0.889520i $$-0.651039\pi$$
−0.456896 + 0.889520i $$0.651039\pi$$
$$740$$ 104.643 0.00519832
$$741$$ 5944.93 0.294726
$$742$$ 0 0
$$743$$ 11182.6 0.552155 0.276078 0.961135i $$-0.410965\pi$$
0.276078 + 0.961135i $$0.410965\pi$$
$$744$$ 5823.55 0.286965
$$745$$ 36076.4 1.77415
$$746$$ −14507.8 −0.712021
$$747$$ −15843.0 −0.775991
$$748$$ 242.960 0.0118763
$$749$$ 0 0
$$750$$ 9424.43 0.458842
$$751$$ 16733.4 0.813063 0.406531 0.913637i $$-0.366738\pi$$
0.406531 + 0.913637i $$0.366738\pi$$
$$752$$ −20433.3 −0.990857
$$753$$ −8680.53 −0.420101
$$754$$ 365.950 0.0176752
$$755$$ 38280.2 1.84524
$$756$$ 0 0
$$757$$ −24402.4 −1.17163 −0.585813 0.810446i $$-0.699225\pi$$
−0.585813 + 0.810446i $$0.699225\pi$$
$$758$$ −2290.19 −0.109741
$$759$$ 9711.19 0.464419
$$760$$ −48477.6 −2.31377
$$761$$ −8469.33 −0.403434 −0.201717 0.979444i $$-0.564652\pi$$
−0.201717 + 0.979444i $$0.564652\pi$$
$$762$$ −28539.7 −1.35680
$$763$$ 0 0
$$764$$ −920.805 −0.0436042
$$765$$ −21955.3 −1.03764
$$766$$ 7737.62 0.364976
$$767$$ 473.570 0.0222941
$$768$$ −6496.08 −0.305218
$$769$$ −32834.7 −1.53973 −0.769864 0.638208i $$-0.779676\pi$$
−0.769864 + 0.638208i $$0.779676\pi$$
$$770$$ 0 0
$$771$$ −6209.20 −0.290038
$$772$$ −718.202 −0.0334827
$$773$$ 35571.4 1.65513 0.827564 0.561371i $$-0.189726\pi$$
0.827564 + 0.561371i $$0.189726\pi$$
$$774$$ −5653.64 −0.262553
$$775$$ −3014.71 −0.139731
$$776$$ 31399.1 1.45253
$$777$$ 0 0
$$778$$ 8500.11 0.391701
$$779$$ 36530.0 1.68013
$$780$$ −338.262 −0.0155279
$$781$$ 2281.01 0.104508
$$782$$ −12538.7 −0.573381
$$783$$ −1755.02 −0.0801014
$$784$$ 0 0
$$785$$ 36783.6 1.67244
$$786$$ 34671.8 1.57341
$$787$$ −15729.6 −0.712452 −0.356226 0.934400i $$-0.615937\pi$$
−0.356226 + 0.934400i $$0.615937\pi$$
$$788$$ 1885.47 0.0852373
$$789$$ 48996.7 2.21081
$$790$$ −52530.6 −2.36577
$$791$$ 0 0
$$792$$ 9198.09 0.412676
$$793$$ 3958.67 0.177272
$$794$$ −38819.0 −1.73506
$$795$$ 40399.5 1.80229
$$796$$ 441.415 0.0196552
$$797$$ −7888.07 −0.350577 −0.175288 0.984517i $$-0.556086\pi$$
−0.175288 + 0.984517i $$0.556086\pi$$
$$798$$ 0 0
$$799$$ 14171.8 0.627485
$$800$$ 2317.23 0.102408
$$801$$ 53394.2 2.35530
$$802$$ −17107.2 −0.753214
$$803$$ −11116.6 −0.488538
$$804$$ −1454.58 −0.0638050
$$805$$ 0 0
$$806$$ 461.154 0.0201532
$$807$$ −7824.86 −0.341323
$$808$$ −3765.36 −0.163942
$$809$$ 5896.97 0.256275 0.128138 0.991756i $$-0.459100\pi$$
0.128138 + 0.991756i $$0.459100\pi$$
$$810$$ 16699.8 0.724407
$$811$$ −14197.9 −0.614744 −0.307372 0.951589i $$-0.599450\pi$$
−0.307372 + 0.951589i $$0.599450\pi$$
$$812$$ 0 0
$$813$$ −36287.3 −1.56538
$$814$$ −394.999 −0.0170082
$$815$$ 40485.3 1.74005
$$816$$ −19418.2 −0.833053
$$817$$ 8075.35 0.345803
$$818$$ 11454.1 0.489589
$$819$$ 0 0
$$820$$ −2078.53 −0.0885188
$$821$$ −19841.7 −0.843459 −0.421729 0.906722i $$-0.638577\pi$$
−0.421729 + 0.906722i $$0.638577\pi$$
$$822$$ 34909.3 1.48127
$$823$$ −28202.2 −1.19449 −0.597246 0.802058i $$-0.703738\pi$$
−0.597246 + 0.802058i $$0.703738\pi$$
$$824$$ −810.602 −0.0342702
$$825$$ −8347.14 −0.352255
$$826$$ 0 0
$$827$$ 34031.0 1.43092 0.715462 0.698651i $$-0.246216\pi$$
0.715462 + 0.698651i $$0.246216\pi$$
$$828$$ 2139.71 0.0898067
$$829$$ −4931.55 −0.206610 −0.103305 0.994650i $$-0.532942\pi$$
−0.103305 + 0.994650i $$0.532942\pi$$
$$830$$ 17933.9 0.749992
$$831$$ 4501.92 0.187930
$$832$$ −2902.15 −0.120930
$$833$$ 0 0
$$834$$ 690.138 0.0286541
$$835$$ 40666.4 1.68541
$$836$$ −824.830 −0.0341236
$$837$$ −2211.60 −0.0913312
$$838$$ 25373.0 1.04594
$$839$$ 38189.8 1.57146 0.785731 0.618568i $$-0.212287\pi$$
0.785731 + 0.618568i $$0.212287\pi$$
$$840$$ 0 0
$$841$$ −23764.3 −0.974384
$$842$$ 35915.6 1.46999
$$843$$ 42106.8 1.72033
$$844$$ 57.5250 0.00234608
$$845$$ 32212.9 1.31143
$$846$$ 33683.7 1.36888
$$847$$ 0 0
$$848$$ 20382.7 0.825406
$$849$$ 37488.0 1.51541
$$850$$ 10777.5 0.434900
$$851$$ −1463.59 −0.0589556
$$852$$ 881.030 0.0354268
$$853$$ −42966.8 −1.72469 −0.862343 0.506325i $$-0.831003\pi$$
−0.862343 + 0.506325i $$0.831003\pi$$
$$854$$ 0 0
$$855$$ 74536.6 2.98140
$$856$$ −19406.8 −0.774898
$$857$$ 17281.5 0.688828 0.344414 0.938818i $$-0.388078\pi$$
0.344414 + 0.938818i $$0.388078\pi$$
$$858$$ 1276.85 0.0508052
$$859$$ −9316.75 −0.370062 −0.185031 0.982733i $$-0.559239\pi$$
−0.185031 + 0.982733i $$0.559239\pi$$
$$860$$ −459.482 −0.0182188
$$861$$ 0 0
$$862$$ 13413.5 0.530005
$$863$$ −9647.65 −0.380544 −0.190272 0.981731i $$-0.560937\pi$$
−0.190272 + 0.981731i $$0.560937\pi$$
$$864$$ 1699.93 0.0669361
$$865$$ 34279.9 1.34746
$$866$$ 32083.3 1.25893
$$867$$ −25483.6 −0.998232
$$868$$ 0 0
$$869$$ −14236.5 −0.555742
$$870$$ 8043.18 0.313436
$$871$$ −1834.70 −0.0713735
$$872$$ −24352.6 −0.945737
$$873$$ −48277.6 −1.87165
$$874$$ 42568.0 1.64746
$$875$$ 0 0
$$876$$ −4293.75 −0.165608
$$877$$ 19728.7 0.759624 0.379812 0.925064i $$-0.375989\pi$$
0.379812 + 0.925064i $$0.375989\pi$$
$$878$$ 32304.3 1.24171
$$879$$ −18464.1 −0.708509
$$880$$ −9711.36 −0.372011
$$881$$ −19473.9 −0.744712 −0.372356 0.928090i $$-0.621450\pi$$
−0.372356 + 0.928090i $$0.621450\pi$$
$$882$$ 0 0
$$883$$ 49092.4 1.87100 0.935499 0.353329i $$-0.114950\pi$$
0.935499 + 0.353329i $$0.114950\pi$$
$$884$$ 118.365 0.00450346
$$885$$ 10408.5 0.395344
$$886$$ 27599.5 1.04653
$$887$$ −9292.86 −0.351774 −0.175887 0.984410i $$-0.556279\pi$$
−0.175887 + 0.984410i $$0.556279\pi$$
$$888$$ −2430.12 −0.0918348
$$889$$ 0 0
$$890$$ −60440.8 −2.27638
$$891$$ 4525.85 0.170170
$$892$$ −2108.05 −0.0791288
$$893$$ −48112.0 −1.80292
$$894$$ −52598.5 −1.96774
$$895$$ 19493.9 0.728055
$$896$$ 0 0
$$897$$ 4731.10 0.176106
$$898$$ −943.252 −0.0350520
$$899$$ 787.273 0.0292069
$$900$$ −1839.16 −0.0681170
$$901$$ −14136.7 −0.522709
$$902$$ 7845.88 0.289622
$$903$$ 0 0
$$904$$ −6881.46 −0.253179
$$905$$ −11932.3 −0.438279
$$906$$ −55811.5 −2.04659
$$907$$ 37688.7 1.37975 0.689875 0.723928i $$-0.257665\pi$$
0.689875 + 0.723928i $$0.257665\pi$$
$$908$$ −949.559 −0.0347051
$$909$$ 5789.42 0.211246
$$910$$ 0 0
$$911$$ 33049.6 1.20196 0.600979 0.799265i $$-0.294778\pi$$
0.600979 + 0.799265i $$0.294778\pi$$
$$912$$ 65923.1 2.39357
$$913$$ 4860.31 0.176180
$$914$$ −28870.0 −1.04479
$$915$$ 87007.2 3.14357
$$916$$ 1026.44 0.0370247
$$917$$ 0 0
$$918$$ 7906.43 0.284260
$$919$$ −23148.0 −0.830883 −0.415442 0.909620i $$-0.636373\pi$$
−0.415442 + 0.909620i $$0.636373\pi$$
$$920$$ −38579.5 −1.38253
$$921$$ 13304.1 0.475986
$$922$$ −12933.4 −0.461973
$$923$$ 1111.26 0.0396290
$$924$$ 0 0
$$925$$ 1258.01 0.0447169
$$926$$ 9374.21 0.332673
$$927$$ 1246.34 0.0441588
$$928$$ −605.131 −0.0214056
$$929$$ 23177.9 0.818561 0.409280 0.912409i $$-0.365780\pi$$
0.409280 + 0.912409i $$0.365780\pi$$
$$930$$ 10135.7 0.357378
$$931$$ 0 0
$$932$$ −2355.98 −0.0828033
$$933$$ −28325.1 −0.993916
$$934$$ −13979.8 −0.489757
$$935$$ 6735.44 0.235585
$$936$$ 4481.13 0.156485
$$937$$ 34574.7 1.20545 0.602724 0.797950i $$-0.294082\pi$$
0.602724 + 0.797950i $$0.294082\pi$$
$$938$$ 0 0
$$939$$ −56959.1 −1.97954
$$940$$ 2737.54 0.0949880
$$941$$ −41831.2 −1.44916 −0.724578 0.689192i $$-0.757966\pi$$
−0.724578 + 0.689192i $$0.757966\pi$$
$$942$$ −53629.6 −1.85493
$$943$$ 29071.3 1.00392
$$944$$ 5251.40 0.181058
$$945$$ 0 0
$$946$$ 1734.42 0.0596097
$$947$$ 27231.2 0.934419 0.467209 0.884147i $$-0.345259\pi$$
0.467209 + 0.884147i $$0.345259\pi$$
$$948$$ −5498.79 −0.188389
$$949$$ −5415.79 −0.185252
$$950$$ −36588.8 −1.24958
$$951$$ −124.534 −0.00424635
$$952$$ 0 0
$$953$$ 40939.4 1.39156 0.695781 0.718254i $$-0.255058\pi$$
0.695781 + 0.718254i $$0.255058\pi$$
$$954$$ −33600.3 −1.14030
$$955$$ −25527.0 −0.864956
$$956$$ 2188.87 0.0740515
$$957$$ 2179.81 0.0736292
$$958$$ −31601.3 −1.06575
$$959$$ 0 0
$$960$$ −63786.1 −2.14447
$$961$$ −28798.9 −0.966698
$$962$$ −192.436 −0.00644945
$$963$$ 29839.0 0.998491
$$964$$ 2094.60 0.0699820
$$965$$ −19910.3 −0.664182
$$966$$ 0 0
$$967$$ −46173.1 −1.53550 −0.767750 0.640750i $$-0.778624\pi$$
−0.767750 + 0.640750i $$0.778624\pi$$
$$968$$ −2821.78 −0.0936937
$$969$$ −45721.8 −1.51579
$$970$$ 54648.9 1.80894
$$971$$ 5153.91 0.170337 0.0851683 0.996367i $$-0.472857\pi$$
0.0851683 + 0.996367i $$0.472857\pi$$
$$972$$ 2764.06 0.0912111
$$973$$ 0 0
$$974$$ −50069.0 −1.64714
$$975$$ −4066.56 −0.133574
$$976$$ 43897.6 1.43968
$$977$$ 9692.13 0.317378 0.158689 0.987329i $$-0.449273\pi$$
0.158689 + 0.987329i $$0.449273\pi$$
$$978$$ −59026.5 −1.92992
$$979$$ −16380.2 −0.534744
$$980$$ 0 0
$$981$$ 37443.3 1.21863
$$982$$ −20811.6 −0.676299
$$983$$ −32915.7 −1.06800 −0.534002 0.845483i $$-0.679313\pi$$
−0.534002 + 0.845483i $$0.679313\pi$$
$$984$$ 48269.5 1.56380
$$985$$ 52269.7 1.69081
$$986$$ −2814.48 −0.0909041
$$987$$ 0 0
$$988$$ −401.841 −0.0129395
$$989$$ 6426.54 0.206625
$$990$$ 16008.9 0.513936
$$991$$ 29477.9 0.944901 0.472451 0.881357i $$-0.343370\pi$$
0.472451 + 0.881357i $$0.343370\pi$$
$$992$$ −762.560 −0.0244066
$$993$$ −10456.9 −0.334180
$$994$$ 0 0
$$995$$ 12237.1 0.389892
$$996$$ 1877.28 0.0597227
$$997$$ 31944.4 1.01473 0.507366 0.861731i $$-0.330619\pi$$
0.507366 + 0.861731i $$0.330619\pi$$
$$998$$ 35268.4 1.11864
$$999$$ 922.883 0.0292279
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 539.4.a.e.1.2 2
7.6 odd 2 11.4.a.a.1.2 2
21.20 even 2 99.4.a.c.1.1 2
28.27 even 2 176.4.a.i.1.2 2
35.13 even 4 275.4.b.c.199.1 4
35.27 even 4 275.4.b.c.199.4 4
35.34 odd 2 275.4.a.b.1.1 2
56.13 odd 2 704.4.a.p.1.2 2
56.27 even 2 704.4.a.n.1.1 2
77.6 even 10 121.4.c.f.3.2 8
77.13 even 10 121.4.c.f.81.2 8
77.20 odd 10 121.4.c.c.81.1 8
77.27 odd 10 121.4.c.c.3.1 8
77.41 even 10 121.4.c.f.9.1 8
77.48 odd 10 121.4.c.c.27.2 8
77.62 even 10 121.4.c.f.27.1 8
77.69 odd 10 121.4.c.c.9.2 8
77.76 even 2 121.4.a.c.1.1 2
84.83 odd 2 1584.4.a.bc.1.1 2
91.90 odd 2 1859.4.a.a.1.1 2
105.104 even 2 2475.4.a.q.1.2 2
231.230 odd 2 1089.4.a.v.1.2 2
308.307 odd 2 1936.4.a.w.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
11.4.a.a.1.2 2 7.6 odd 2
99.4.a.c.1.1 2 21.20 even 2
121.4.a.c.1.1 2 77.76 even 2
121.4.c.c.3.1 8 77.27 odd 10
121.4.c.c.9.2 8 77.69 odd 10
121.4.c.c.27.2 8 77.48 odd 10
121.4.c.c.81.1 8 77.20 odd 10
121.4.c.f.3.2 8 77.6 even 10
121.4.c.f.9.1 8 77.41 even 10
121.4.c.f.27.1 8 77.62 even 10
121.4.c.f.81.2 8 77.13 even 10
176.4.a.i.1.2 2 28.27 even 2
275.4.a.b.1.1 2 35.34 odd 2
275.4.b.c.199.1 4 35.13 even 4
275.4.b.c.199.4 4 35.27 even 4
539.4.a.e.1.2 2 1.1 even 1 trivial
704.4.a.n.1.1 2 56.27 even 2
704.4.a.p.1.2 2 56.13 odd 2
1089.4.a.v.1.2 2 231.230 odd 2
1584.4.a.bc.1.1 2 84.83 odd 2
1859.4.a.a.1.1 2 91.90 odd 2
1936.4.a.w.1.2 2 308.307 odd 2
2475.4.a.q.1.2 2 105.104 even 2