# Properties

 Label 1859.4.a.a.1.1 Level $1859$ Weight $4$ Character 1859.1 Self dual yes Analytic conductor $109.685$ 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] = [1859,4,Mod(1,1859)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1859, 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("1859.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1859 = 11 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1859.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: $$109.684550701$$ 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 1.1 Root $$-1.73205$$ of defining polynomial Character $$\chi$$ $$=$$ 1859.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-2.73205 q^{2} -7.92820 q^{3} -0.535898 q^{4} -14.8564 q^{5} +21.6603 q^{6} -3.07180 q^{7} +23.3205 q^{8} +35.8564 q^{9} +O(q^{10})$$ $$q-2.73205 q^{2} -7.92820 q^{3} -0.535898 q^{4} -14.8564 q^{5} +21.6603 q^{6} -3.07180 q^{7} +23.3205 q^{8} +35.8564 q^{9} +40.5885 q^{10} +11.0000 q^{11} +4.24871 q^{12} +8.39230 q^{14} +117.785 q^{15} -59.4256 q^{16} -41.2154 q^{17} -97.9615 q^{18} -139.923 q^{19} +7.96152 q^{20} +24.3538 q^{21} -30.0526 q^{22} -111.354 q^{23} -184.890 q^{24} +95.7128 q^{25} -70.2154 q^{27} +1.64617 q^{28} -24.9948 q^{29} -321.794 q^{30} -31.4974 q^{31} -24.2102 q^{32} -87.2102 q^{33} +112.603 q^{34} +45.6359 q^{35} -19.2154 q^{36} -13.1436 q^{37} +382.277 q^{38} -346.459 q^{40} -261.072 q^{41} -66.5359 q^{42} -57.7128 q^{43} -5.89488 q^{44} -532.697 q^{45} +304.224 q^{46} +343.846 q^{47} +471.138 q^{48} -333.564 q^{49} -261.492 q^{50} +326.764 q^{51} -342.995 q^{53} +191.832 q^{54} -163.420 q^{55} -71.6359 q^{56} +1109.34 q^{57} +68.2872 q^{58} -88.3693 q^{59} -63.1206 q^{60} +738.697 q^{61} +86.0526 q^{62} -110.144 q^{63} +541.549 q^{64} +238.263 q^{66} -342.359 q^{67} +22.0873 q^{68} +882.836 q^{69} -124.679 q^{70} +207.364 q^{71} +836.190 q^{72} +1010.60 q^{73} +35.9090 q^{74} -758.831 q^{75} +74.9845 q^{76} -33.7898 q^{77} +1294.23 q^{79} +882.851 q^{80} -411.441 q^{81} +713.261 q^{82} -441.846 q^{83} -13.0512 q^{84} +612.313 q^{85} +157.674 q^{86} +198.164 q^{87} +256.526 q^{88} +1489.11 q^{89} +1455.36 q^{90} +59.6743 q^{92} +249.718 q^{93} -939.405 q^{94} +2078.75 q^{95} +191.944 q^{96} -1346.42 q^{97} +911.314 q^{98} +394.420 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} - 2 q^{3} - 8 q^{4} - 2 q^{5} + 26 q^{6} - 20 q^{7} + 12 q^{8} + 44 q^{9}+O(q^{10})$$ 2 * q - 2 * q^2 - 2 * q^3 - 8 * q^4 - 2 * q^5 + 26 * q^6 - 20 * q^7 + 12 * q^8 + 44 * q^9 $$2 q - 2 q^{2} - 2 q^{3} - 8 q^{4} - 2 q^{5} + 26 q^{6} - 20 q^{7} + 12 q^{8} + 44 q^{9} + 50 q^{10} + 22 q^{11} - 40 q^{12} - 4 q^{14} + 194 q^{15} - 8 q^{16} - 124 q^{17} - 92 q^{18} - 72 q^{19} - 88 q^{20} - 76 q^{21} - 22 q^{22} - 98 q^{23} - 252 q^{24} + 136 q^{25} - 182 q^{27} + 128 q^{28} + 144 q^{29} - 266 q^{30} + 34 q^{31} + 104 q^{32} - 22 q^{33} + 52 q^{34} - 172 q^{35} - 80 q^{36} - 54 q^{37} + 432 q^{38} - 492 q^{40} - 536 q^{41} - 140 q^{42} - 60 q^{43} - 88 q^{44} - 428 q^{45} + 314 q^{46} + 272 q^{47} + 776 q^{48} - 390 q^{49} - 232 q^{50} - 164 q^{51} - 492 q^{53} + 110 q^{54} - 22 q^{55} + 120 q^{56} + 1512 q^{57} + 192 q^{58} - 634 q^{59} - 632 q^{60} + 840 q^{61} + 134 q^{62} - 248 q^{63} + 224 q^{64} + 286 q^{66} - 754 q^{67} + 640 q^{68} + 962 q^{69} - 284 q^{70} + 678 q^{71} + 744 q^{72} + 400 q^{73} + 6 q^{74} - 520 q^{75} - 432 q^{76} - 220 q^{77} + 316 q^{79} + 1544 q^{80} - 1294 q^{81} + 512 q^{82} - 468 q^{83} + 736 q^{84} - 452 q^{85} + 156 q^{86} + 1200 q^{87} + 132 q^{88} + 1842 q^{89} + 1532 q^{90} - 40 q^{92} + 638 q^{93} - 992 q^{94} + 2952 q^{95} + 952 q^{96} - 2194 q^{97} + 870 q^{98} + 484 q^{99}+O(q^{100})$$ 2 * q - 2 * q^2 - 2 * q^3 - 8 * q^4 - 2 * q^5 + 26 * q^6 - 20 * q^7 + 12 * q^8 + 44 * q^9 + 50 * q^10 + 22 * q^11 - 40 * q^12 - 4 * q^14 + 194 * q^15 - 8 * q^16 - 124 * q^17 - 92 * q^18 - 72 * q^19 - 88 * q^20 - 76 * q^21 - 22 * q^22 - 98 * q^23 - 252 * q^24 + 136 * q^25 - 182 * q^27 + 128 * q^28 + 144 * q^29 - 266 * q^30 + 34 * q^31 + 104 * q^32 - 22 * q^33 + 52 * q^34 - 172 * q^35 - 80 * q^36 - 54 * q^37 + 432 * q^38 - 492 * q^40 - 536 * q^41 - 140 * q^42 - 60 * q^43 - 88 * q^44 - 428 * q^45 + 314 * q^46 + 272 * q^47 + 776 * q^48 - 390 * q^49 - 232 * q^50 - 164 * q^51 - 492 * q^53 + 110 * q^54 - 22 * q^55 + 120 * q^56 + 1512 * q^57 + 192 * q^58 - 634 * q^59 - 632 * q^60 + 840 * q^61 + 134 * q^62 - 248 * q^63 + 224 * q^64 + 286 * q^66 - 754 * q^67 + 640 * q^68 + 962 * q^69 - 284 * q^70 + 678 * q^71 + 744 * q^72 + 400 * q^73 + 6 * q^74 - 520 * q^75 - 432 * q^76 - 220 * q^77 + 316 * q^79 + 1544 * q^80 - 1294 * q^81 + 512 * q^82 - 468 * q^83 + 736 * q^84 - 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 + 870 * q^98 + 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 $$-0.660439\pi$$
−0.482963 + 0.875641i $$0.660439\pi$$
$$3$$ −7.92820 −1.52578 −0.762892 0.646526i $$-0.776221\pi$$
−0.762892 + 0.646526i $$0.776221\pi$$
$$4$$ −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
$$7$$ −3.07180 −0.165861 −0.0829307 0.996555i $$-0.526428\pi$$
−0.0829307 + 0.996555i $$0.526428\pi$$
$$8$$ 23.3205 1.03063
$$9$$ 35.8564 1.32802
$$10$$ 40.5885 1.28352
$$11$$ 11.0000 0.301511
$$12$$ 4.24871 0.102208
$$13$$ 0 0
$$14$$ 8.39230 0.160210
$$15$$ 117.785 2.02746
$$16$$ −59.4256 −0.928525
$$17$$ −41.2154 −0.588012 −0.294006 0.955804i $$-0.594989\pi$$
−0.294006 + 0.955804i $$0.594989\pi$$
$$18$$ −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
$$21$$ 24.3538 0.253069
$$22$$ −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
$$26$$ 0 0
$$27$$ −70.2154 −0.500480
$$28$$ 1.64617 0.0111106
$$29$$ −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
$$35$$ 45.6359 0.220396
$$36$$ −19.2154 −0.0889601
$$37$$ −13.1436 −0.0583998 −0.0291999 0.999574i $$-0.509296\pi$$
−0.0291999 + 0.999574i $$0.509296\pi$$
$$38$$ 382.277 1.63193
$$39$$ 0 0
$$40$$ −346.459 −1.36950
$$41$$ −261.072 −0.994453 −0.497226 0.867621i $$-0.665648\pi$$
−0.497226 + 0.867621i $$0.665648\pi$$
$$42$$ −66.5359 −0.244446
$$43$$ −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
$$49$$ −333.564 −0.972490
$$50$$ −261.492 −0.739612
$$51$$ 326.764 0.897179
$$52$$ 0 0
$$53$$ −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
$$56$$ −71.6359 −0.170942
$$57$$ 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 $$-0.217624\pi$$
0.775250 + 0.631654i $$0.217624\pi$$
$$62$$ 86.0526 0.176269
$$63$$ −110.144 −0.220266
$$64$$ 541.549 1.05771
$$65$$ 0 0
$$66$$ 238.263 0.444365
$$67$$ −342.359 −0.624266 −0.312133 0.950038i $$-0.601043\pi$$
−0.312133 + 0.950038i $$0.601043\pi$$
$$68$$ 22.0873 0.0393893
$$69$$ 882.836 1.54030
$$70$$ −124.679 −0.212886
$$71$$ 207.364 0.346614 0.173307 0.984868i $$-0.444555\pi$$
0.173307 + 0.984868i $$0.444555\pi$$
$$72$$ 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
$$77$$ −33.7898 −0.0500091
$$78$$ 0 0
$$79$$ 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 $$0.594375\pi$$
$$84$$ −13.0512 −0.0169524
$$85$$ 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 $$0.748909\pi$$
$$98$$ 911.314 0.939353
$$99$$ 394.420 0.400412
$$100$$ −51.2923 −0.0512923
$$101$$ −161.461 −0.159069 −0.0795347 0.996832i $$-0.525343\pi$$
−0.0795347 + 0.996832i $$0.525343\pi$$
$$102$$ −892.736 −0.866608
$$103$$ −34.7592 −0.0332517 −0.0166259 0.999862i $$-0.505292\pi$$
−0.0166259 + 0.999862i $$0.505292\pi$$
$$104$$ 0 0
$$105$$ −361.810 −0.336277
$$106$$ 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 $$-0.651726\pi$$
−0.458815 + 0.888532i $$0.651726\pi$$
$$110$$ 446.473 0.386996
$$111$$ 104.205 0.0891055
$$112$$ 182.543 0.154007
$$113$$ 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
$$117$$ 0 0
$$118$$ 241.429 0.188351
$$119$$ 126.605 0.0975285
$$120$$ 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
$$126$$ 300.918 0.212761
$$127$$ −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
$$130$$ 0 0
$$131$$ −1600.71 −1.06759 −0.533797 0.845612i $$-0.679235\pi$$
−0.533797 + 0.845612i $$0.679235\pi$$
$$132$$ 46.7358 0.0308169
$$133$$ 429.815 0.280223
$$134$$ 935.342 0.602994
$$135$$ 1043.15 0.665036
$$136$$ −961.164 −0.606023
$$137$$ −1611.68 −1.00507 −0.502536 0.864556i $$-0.667600\pi$$
−0.502536 + 0.864556i $$0.667600\pi$$
$$138$$ −2411.95 −1.48782
$$139$$ −31.8619 −0.0194424 −0.00972120 0.999953i $$-0.503094\pi$$
−0.00972120 + 0.999953i $$0.503094\pi$$
$$140$$ −24.4562 −0.0147637
$$141$$ −2726.08 −1.62821
$$142$$ −566.529 −0.334803
$$143$$ 0 0
$$144$$ −2130.79 −1.23310
$$145$$ 371.334 0.212673
$$146$$ −2761.01 −1.56509
$$147$$ 2644.56 1.48381
$$148$$ 7.04363 0.00391205
$$149$$ 2428.34 1.33515 0.667576 0.744542i $$-0.267332\pi$$
0.667576 + 0.744542i $$0.267332\pi$$
$$150$$ 2073.16 1.12849
$$151$$ 2576.68 1.38866 0.694328 0.719659i $$-0.255702\pi$$
0.694328 + 0.719659i $$0.255702\pi$$
$$152$$ −3263.08 −1.74125
$$153$$ −1477.84 −0.780889
$$154$$ 92.3154 0.0483051
$$155$$ 467.939 0.242489
$$156$$ 0 0
$$157$$ 2475.94 1.25861 0.629305 0.777158i $$-0.283340\pi$$
0.629305 + 0.777158i $$0.283340\pi$$
$$158$$ −3535.89 −1.78038
$$159$$ 2719.33 1.35633
$$160$$ 359.677 0.177719
$$161$$ 342.056 0.167440
$$162$$ 1124.08 0.545160
$$163$$ 2725.11 1.30949 0.654745 0.755850i $$-0.272776\pi$$
0.654745 + 0.755850i $$0.272776\pi$$
$$164$$ 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 $$0.718665\pi$$
$$168$$ 567.944 0.260820
$$169$$ 0 0
$$170$$ −1672.87 −0.754725
$$171$$ −5017.14 −2.24368
$$172$$ 30.9282 0.0137108
$$173$$ 2307.42 1.01404 0.507022 0.861933i $$-0.330746\pi$$
0.507022 + 0.861933i $$0.330746\pi$$
$$174$$ −541.395 −0.235879
$$175$$ −294.010 −0.127001
$$176$$ −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 $$-0.552735\pi$$
−0.164916 + 0.986308i $$0.552735\pi$$
$$182$$ 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
$$189$$ 215.687 0.0830103
$$190$$ −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 $$-0.580404\pi$$
−0.249919 + 0.968267i $$0.580404\pi$$
$$194$$ 3678.48 1.36134
$$195$$ 0 0
$$196$$ 178.756 0.0651445
$$197$$ 3518.33 1.27244 0.636220 0.771508i $$-0.280497\pi$$
0.636220 + 0.771508i $$0.280497\pi$$
$$198$$ −1077.58 −0.386768
$$199$$ 823.692 0.293417 0.146709 0.989180i $$-0.453132\pi$$
0.146709 + 0.989180i $$0.453132\pi$$
$$200$$ 2232.07 0.789156
$$201$$ 2714.29 0.952494
$$202$$ 441.121 0.153649
$$203$$ 76.7791 0.0265460
$$204$$ −175.112 −0.0600996
$$205$$ 3878.59 1.32143
$$206$$ 94.9639 0.0321187
$$207$$ −3992.75 −1.34065
$$208$$ 0 0
$$209$$ −1539.15 −0.509404
$$210$$ 988.484 0.324819
$$211$$ −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
$$217$$ 96.7537 0.0302676
$$218$$ 2852.96 0.886362
$$219$$ −8012.24 −2.47222
$$220$$ 87.5768 0.0268383
$$221$$ 0 0
$$222$$ −284.694 −0.0860693
$$223$$ 3933.68 1.18125 0.590625 0.806946i $$-0.298881\pi$$
0.590625 + 0.806946i $$0.298881\pi$$
$$224$$ 74.3689 0.0221830
$$225$$ 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
$$231$$ 267.892 0.0763031
$$232$$ −582.892 −0.164952
$$233$$ 4396.32 1.23610 0.618052 0.786137i $$-0.287922\pi$$
0.618052 + 0.786137i $$0.287922\pi$$
$$234$$ 0 0
$$235$$ −5108.32 −1.41800
$$236$$ 47.3570 0.0130622
$$237$$ −10260.9 −2.81230
$$238$$ −345.892 −0.0942053
$$239$$ 4084.49 1.10546 0.552728 0.833362i $$-0.313587\pi$$
0.552728 + 0.833362i $$0.313587\pi$$
$$240$$ −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
$$245$$ 4955.56 1.29224
$$246$$ −5654.88 −1.46562
$$247$$ 0 0
$$248$$ −734.536 −0.188077
$$249$$ 3503.05 0.891552
$$250$$ −1188.72 −0.300725
$$251$$ 1094.89 0.275335 0.137667 0.990479i $$-0.456040\pi$$
0.137667 + 0.990479i $$0.456040\pi$$
$$252$$ 59.0258 0.0147551
$$253$$ −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 $$-0.469700\pi$$
0.0950454 + 0.995473i $$0.469700\pi$$
$$258$$ −1250.07 −0.301652
$$259$$ 40.3744 0.00968628
$$260$$ 0 0
$$261$$ −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
$$266$$ −1174.28 −0.270675
$$267$$ −11806.0 −2.70605
$$268$$ 183.470 0.0418179
$$269$$ 986.965 0.223704 0.111852 0.993725i $$-0.464322\pi$$
0.111852 + 0.993725i $$0.464322\pi$$
$$270$$ −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
$$280$$ 1064.25 0.227147
$$281$$ −5311.01 −1.12750 −0.563752 0.825944i $$-0.690643\pi$$
−0.563752 + 0.825944i $$0.690643\pi$$
$$282$$ 7447.79 1.57273
$$283$$ −4728.44 −0.993204 −0.496602 0.867978i $$-0.665419\pi$$
−0.496602 + 0.867978i $$0.665419\pi$$
$$284$$ −111.126 −0.0232187
$$285$$ −16480.8 −3.42539
$$286$$ 0 0
$$287$$ 801.960 0.164941
$$288$$ −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 $$0.574585\pi$$
$$294$$ −7225.08 −1.43325
$$295$$ 1312.85 0.259109
$$296$$ −306.515 −0.0601886
$$297$$ −772.369 −0.150900
$$298$$ −6634.36 −1.28966
$$299$$ 0 0
$$300$$ 406.656 0.0782610
$$301$$ 177.282 0.0339481
$$302$$ −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 $$0.450146\pi$$
$$308$$ 18.1079 0.00334997
$$309$$ 275.578 0.0507349
$$310$$ −1278.43 −0.234226
$$311$$ 3572.71 0.651413 0.325707 0.945471i $$-0.394398\pi$$
0.325707 + 0.945471i $$0.394398\pi$$
$$312$$ 0 0
$$313$$ 7184.36 1.29739 0.648697 0.761047i $$-0.275314\pi$$
0.648697 + 0.761047i $$0.275314\pi$$
$$314$$ −6764.40 −1.21572
$$315$$ 1636.34 0.292690
$$316$$ −693.573 −0.123470
$$317$$ 15.7077 0.00278306 0.00139153 0.999999i $$-0.499557\pi$$
0.00139153 + 0.999999i $$0.499557\pi$$
$$318$$ −7429.36 −1.31012
$$319$$ −274.943 −0.0482566
$$320$$ −8045.47 −1.40549
$$321$$ −6597.69 −1.14719
$$322$$ −934.515 −0.161734
$$323$$ 5766.98 0.993447
$$324$$ 220.491 0.0378070
$$325$$ 0 0
$$326$$ −7445.13 −1.26487
$$327$$ 8279.08 1.40010
$$328$$ −6088.33 −1.02491
$$329$$ −1056.23 −0.176996
$$330$$ −3539.73 −0.590472
$$331$$ 1318.95 0.219022 0.109511 0.993986i $$-0.465072\pi$$
0.109511 + 0.993986i $$0.465072\pi$$
$$332$$ 236.785 0.0391423
$$333$$ −471.282 −0.0775558
$$334$$ 7478.43 1.22515
$$335$$ 5086.22 0.829523
$$336$$ −1447.24 −0.234981
$$337$$ −239.183 −0.0386621 −0.0193310 0.999813i $$-0.506154\pi$$
−0.0193310 + 0.999813i $$0.506154\pi$$
$$338$$ 0 0
$$339$$ −2339.47 −0.374816
$$340$$ −328.137 −0.0523404
$$341$$ −346.472 −0.0550220
$$342$$ 13707.1 2.16723
$$343$$ 2078.27 0.327160
$$344$$ −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 $$0.586289\pi$$
$$350$$ 803.251 0.122673
$$351$$ 0 0
$$352$$ −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
$$357$$ −1003.75 −0.148807
$$358$$ 3584.87 0.529236
$$359$$ 11500.7 1.69077 0.845384 0.534160i $$-0.179372\pi$$
0.845384 + 0.534160i $$0.179372\pi$$
$$360$$ −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 $$-0.340184\pi$$
0.481246 + 0.876585i $$0.340184\pi$$
$$368$$ 6617.27 0.937362
$$369$$ −9361.10 −1.32065
$$370$$ −533.478 −0.0749573
$$371$$ 1053.61 0.147441
$$372$$ −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
$$377$$ 0 0
$$378$$ −589.269 −0.0801818
$$379$$ 838.267 0.113612 0.0568059 0.998385i $$-0.481908\pi$$
0.0568059 + 0.998385i $$0.481908\pi$$
$$380$$ −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
$$385$$ 501.994 0.0664520
$$386$$ 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 $$0.435009\pi$$
$$390$$ 0 0
$$391$$ 4589.49 0.593608
$$392$$ −7778.88 −1.00228
$$393$$ 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
$$399$$ −3407.66 −0.427560
$$400$$ −5687.79 −0.710974
$$401$$ 6261.68 0.779784 0.389892 0.920861i $$-0.372512\pi$$
0.389892 + 0.920861i $$0.372512\pi$$
$$402$$ −7415.58 −0.920039
$$403$$ 0 0
$$404$$ 86.5269 0.0106556
$$405$$ 6112.54 0.749961
$$406$$ −209.764 −0.0256415
$$407$$ −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
$$413$$ 271.453 0.0323421
$$414$$ 10908.4 1.29497
$$415$$ 6564.25 0.776448
$$416$$ 0 0
$$417$$ 252.608 0.0296649
$$418$$ 4205.05 0.492047
$$419$$ −9287.15 −1.08283 −0.541416 0.840755i $$-0.682112\pi$$
−0.541416 + 0.840755i $$0.682112\pi$$
$$420$$ 193.894 0.0225263
$$421$$ −13146.0 −1.52185 −0.760923 0.648842i $$-0.775254\pi$$
−0.760923 + 0.648842i $$0.775254\pi$$
$$422$$ 293.267 0.0338294
$$423$$ 12329.1 1.41716
$$424$$ −7998.81 −0.916172
$$425$$ −3944.84 −0.450242
$$426$$ 4491.56 0.510838
$$427$$ −2269.13 −0.257168
$$428$$ −445.964 −0.0503656
$$429$$ 0 0
$$430$$ −2342.47 −0.262707
$$431$$ −4909.67 −0.548701 −0.274351 0.961630i $$-0.588463\pi$$
−0.274351 + 0.961630i $$0.588463\pi$$
$$432$$ 4172.59 0.464708
$$433$$ −11743.3 −1.30334 −0.651671 0.758502i $$-0.725932\pi$$
−0.651671 + 0.758502i $$0.725932\pi$$
$$434$$ −264.336 −0.0292363
$$435$$ −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 $$-0.722208\pi$$
−0.642754 + 0.766073i $$0.722208\pi$$
$$440$$ −3811.05 −0.412920
$$441$$ −11960.4 −1.29148
$$442$$ 0 0
$$443$$ 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
$$448$$ −1663.53 −0.175434
$$449$$ 345.254 0.0362885 0.0181443 0.999835i $$-0.494224\pi$$
0.0181443 + 0.999835i $$0.494224\pi$$
$$450$$ −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 $$-0.318114\pi$$
0.540821 + 0.841138i $$0.318114\pi$$
$$458$$ 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 $$0.576864\pi$$
$$462$$ −731.895 −0.0737031
$$463$$ −3431.20 −0.344409 −0.172204 0.985061i $$-0.555089\pi$$
−0.172204 + 0.985061i $$0.555089\pi$$
$$464$$ 1485.33 0.148610
$$465$$ −3709.91 −0.369985
$$466$$ −12011.0 −1.19399
$$467$$ 5116.96 0.507034 0.253517 0.967331i $$-0.418413\pi$$
0.253517 + 0.967331i $$0.418413\pi$$
$$468$$ 0 0
$$469$$ 1051.66 0.103542
$$470$$ 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
$$476$$ −67.8476 −0.00653317
$$477$$ −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
$$481$$ 0 0
$$482$$ 10678.4 1.00911
$$483$$ −2711.89 −0.255477
$$484$$ −64.8437 −0.00608975
$$485$$ 20002.9 1.87275
$$486$$ −14091.4 −1.31522
$$487$$ 18326.5 1.70525 0.852623 0.522527i $$-0.175010\pi$$
0.852623 + 0.522527i $$0.175010\pi$$
$$488$$ 17226.8 1.59799
$$489$$ −21605.2 −1.99800
$$490$$ −13538.9 −1.24821
$$491$$ −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
$$494$$ 0 0
$$495$$ −5859.67 −0.532066
$$496$$ 1871.75 0.169444
$$497$$ −636.980 −0.0574899
$$498$$ −9570.50 −0.861173
$$499$$ −12909.1 −1.15810 −0.579050 0.815292i $$-0.696576\pi$$
−0.579050 + 0.815292i $$0.696576\pi$$
$$500$$ −233.171 −0.0208554
$$501$$ 21701.8 1.93526
$$502$$ −2991.30 −0.265953
$$503$$ 10165.7 0.901121 0.450561 0.892746i $$-0.351224\pi$$
0.450561 + 0.892746i $$0.351224\pi$$
$$504$$ −2568.60 −0.227013
$$505$$ 2398.74 0.211371
$$506$$ 3346.47 0.294009
$$507$$ 0 0
$$508$$ 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
$$511$$ −3104.36 −0.268745
$$512$$ 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
$$518$$ −110.305 −0.00935623
$$519$$ −18293.7 −1.54721
$$520$$ 0 0
$$521$$ −19327.4 −1.62524 −0.812620 0.582794i $$-0.801959\pi$$
−0.812620 + 0.582794i $$0.801959\pi$$
$$522$$ 2448.53 0.205305
$$523$$ 6259.09 0.523310 0.261655 0.965161i $$-0.415732\pi$$
0.261655 + 0.965161i $$0.415732\pi$$
$$524$$ 857.819 0.0715153
$$525$$ 2330.97 0.193775
$$526$$ −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
$$532$$ −230.337 −0.0187714
$$533$$ 0 0
$$534$$ 32254.6 2.61384
$$535$$ −12363.2 −0.999079
$$536$$ −7983.99 −0.643387
$$537$$ 10403.0 0.835985
$$538$$ −2696.44 −0.216081
$$539$$ −3669.20 −0.293217
$$540$$ −559.022 −0.0445490
$$541$$ 14008.2 1.11323 0.556616 0.830770i $$-0.312100\pi$$
0.556616 + 0.830770i $$0.312100\pi$$
$$542$$ 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
$$553$$ −3975.60 −0.305714
$$554$$ −1551.36 −0.118973
$$555$$ −1548.11 −0.118403
$$556$$ 17.0748 0.00130239
$$557$$ 3801.58 0.289188 0.144594 0.989491i $$-0.453812\pi$$
0.144594 + 0.989491i $$0.453812\pi$$
$$558$$ 3085.54 0.234088
$$559$$ 0 0
$$560$$ −2711.94 −0.204644
$$561$$ 3594.40 0.270510
$$562$$ 14510.0 1.08908
$$563$$ −9900.11 −0.741101 −0.370551 0.928812i $$-0.620831\pi$$
−0.370551 + 0.928812i $$0.620831\pi$$
$$564$$ 1460.90 0.109069
$$565$$ −4383.85 −0.326425
$$566$$ 12918.3 0.959361
$$567$$ 1263.86 0.0936107
$$568$$ 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 $$0.713516\pi$$
$$572$$ 0 0
$$573$$ −13622.6 −0.993181
$$574$$ −2190.99 −0.159321
$$575$$ −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
$$581$$ 1357.26 0.0969169
$$582$$ −29163.7 −2.07710
$$583$$ −3772.94 −0.268026
$$584$$ 23567.7 1.66993
$$585$$ 0 0
$$586$$ 6362.72 0.448535
$$587$$ −11084.2 −0.779373 −0.389686 0.920948i $$-0.627417\pi$$
−0.389686 + 0.920948i $$0.627417\pi$$
$$588$$ −1417.22 −0.0993964
$$589$$ 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
$$595$$ −1880.90 −0.129596
$$596$$ −1301.34 −0.0894382
$$597$$ −6530.40 −0.447691
$$598$$ 0 0
$$599$$ 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 $$-0.719745\pi$$
−0.636806 + 0.771024i $$0.719745\pi$$
$$602$$ −484.344 −0.0327913
$$603$$ −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 $$-0.238940\pi$$
0.731244 + 0.682116i $$0.238940\pi$$
$$608$$ 3387.57 0.225961
$$609$$ −608.720 −0.0405034
$$610$$ 29982.6 1.99010
$$611$$ 0 0
$$612$$ 791.970 0.0523096
$$613$$ 3527.85 0.232445 0.116222 0.993223i $$-0.462921\pi$$
0.116222 + 0.993223i $$0.462921\pi$$
$$614$$ −4584.56 −0.301332
$$615$$ −30750.2 −2.01621
$$616$$ −787.994 −0.0515409
$$617$$ 22728.1 1.48298 0.741490 0.670963i $$-0.234119\pi$$
0.741490 + 0.670963i $$0.234119\pi$$
$$618$$ −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
$$623$$ −4574.25 −0.294163
$$624$$ 0 0
$$625$$ −18428.2 −1.17940
$$626$$ −19628.0 −1.25319
$$627$$ 12202.7 0.777240
$$628$$ −1326.85 −0.0843109
$$629$$ 541.718 0.0343398
$$630$$ −4470.56 −0.282716
$$631$$ −21532.0 −1.35844 −0.679219 0.733936i $$-0.737681\pi$$
−0.679219 + 0.733936i $$0.737681\pi$$
$$632$$ 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 $$0.846054\pi$$
$$644$$ −183.307 −0.0112163
$$645$$ −6797.68 −0.414974
$$646$$ −15755.7 −0.959597
$$647$$ −1590.02 −0.0966155 −0.0483077 0.998833i $$-0.515383\pi$$
−0.0483077 + 0.998833i $$0.515383\pi$$
$$648$$ −9595.02 −0.581679
$$649$$ −972.062 −0.0587932
$$650$$ 0 0
$$651$$ −767.083 −0.0461818
$$652$$ −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
$$658$$ 2885.66 0.170965
$$659$$ −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
$$663$$ 0 0
$$664$$ −10304.1 −0.602222
$$665$$ −6385.51 −0.372360
$$666$$ 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
$$672$$ −589.612 −0.0338464
$$673$$ −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
$$676$$ 0 0
$$677$$ 13221.4 0.750574 0.375287 0.926909i $$-0.377544\pi$$
0.375287 + 0.926909i $$0.377544\pi$$
$$678$$ 6391.55 0.362044
$$679$$ 4135.91 0.233758
$$680$$ 14279.4 0.805282
$$681$$ −14048.0 −0.790485
$$682$$ 946.578 0.0531471
$$683$$ 13831.4 0.774882 0.387441 0.921894i $$-0.373359\pi$$
0.387441 + 0.921894i $$0.373359\pi$$
$$684$$ 2688.68 0.150298
$$685$$ 23943.7 1.33554
$$686$$ −5677.93 −0.316012
$$687$$ 15185.4 0.843320
$$688$$ 3429.62 0.190048
$$689$$ 0 0
$$690$$ 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
$$693$$ −1211.58 −0.0664128
$$694$$ 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
$$700$$ 157.560 0.00850742
$$701$$ 29949.8 1.61368 0.806838 0.590773i $$-0.201177\pi$$
0.806838 + 0.590773i $$0.201177\pi$$
$$702$$ 0 0
$$703$$ 1839.09 0.0986667
$$704$$ 5957.03 0.318912
$$705$$ 40499.8 2.16356
$$706$$ −29825.0 −1.58991
$$707$$ 495.976 0.0263835
$$708$$ −375.456 −0.0199301
$$709$$ −11307.5 −0.598959 −0.299479 0.954103i $$-0.596813\pi$$
−0.299479 + 0.954103i $$0.596813\pi$$
$$710$$ 8416.59 0.444886
$$711$$ 46406.3 2.44778
$$712$$ 34726.9 1.82787
$$713$$ 3507.36 0.184224
$$714$$ 2742.30 0.143737
$$715$$ 0 0
$$716$$ 703.181 0.0367027
$$717$$ −32382.7 −1.68669
$$718$$ −31420.6 −1.63316
$$719$$ −32623.4 −1.69214 −0.846070 0.533071i $$-0.821038\pi$$
−0.846070 + 0.533071i $$0.821038\pi$$
$$720$$ 31655.9 1.63853
$$721$$ 106.773 0.00551518
$$722$$ −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 $$-0.504080\pi$$
−0.0128187 + 0.999918i $$0.504080\pi$$
$$728$$ 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
$$735$$ −39288.7 −1.97168
$$736$$ 2695.90 0.135017
$$737$$ −3765.95 −0.188223
$$738$$ 25575.0 1.27565
$$739$$ 18357.5 0.913792 0.456896 0.889520i $$-0.348961\pi$$
0.456896 + 0.889520i $$0.348961\pi$$
$$740$$ −104.643 −0.00519832
$$741$$ 0 0
$$742$$ −2878.52 −0.142417
$$743$$ −11182.6 −0.552155 −0.276078 0.961135i $$-0.589035\pi$$
−0.276078 + 0.961135i $$0.589035\pi$$
$$744$$ 5823.55 0.286965
$$745$$ −36076.4 −1.77415
$$746$$ 14507.8 0.712021
$$747$$ −15843.0 −0.775991
$$748$$ 242.960 0.0118763
$$749$$ −2556.29 −0.124706
$$750$$ 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
$$754$$ 0 0
$$755$$ −38280.2 −1.84524
$$756$$ −115.587 −0.00556064
$$757$$ −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
$$763$$ 3207.74 0.152199
$$764$$ −920.805 −0.0436042
$$765$$ 21955.3 1.03764
$$766$$ −7737.62 −0.364976
$$767$$ 0 0
$$768$$ 6496.08 0.305218
$$769$$ −32834.7 −1.53973 −0.769864 0.638208i $$-0.779676\pi$$
−0.769864 + 0.638208i $$0.779676\pi$$
$$770$$ −1371.47 −0.0641877
$$771$$ −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
$$777$$ −320.097 −0.0147792
$$778$$ −8500.11 −0.391701
$$779$$ 36530.0 1.68013
$$780$$ 0 0
$$781$$ 2281.01 0.104508
$$782$$ −12538.7 −0.573381
$$783$$ 1755.02 0.0801014
$$784$$ 19822.3 0.902982
$$785$$ −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
$$791$$ −906.431 −0.0407446
$$792$$ 9198.09 0.412676
$$793$$ 0 0
$$794$$ 38819.0 1.73506
$$795$$ −40399.5 −1.80229
$$796$$ −441.415 −0.0196552
$$797$$ 7888.07 0.350577 0.175288 0.984517i $$-0.443914\pi$$
0.175288 + 0.984517i $$0.443914\pi$$
$$798$$ 9309.91 0.412991
$$799$$ −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
$$805$$ −5081.73 −0.222494
$$806$$ 0 0
$$807$$ −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 $$0.599450\pi$$
$$812$$ −41.1458 −0.00177824
$$813$$ 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 $$-0.361423\pi$$
0.421729 + 0.906722i $$0.361423\pi$$
$$822$$ −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
$$826$$ −741.622 −0.0312401
$$827$$ −34031.0 −1.43092 −0.715462 0.698651i $$-0.753784\pi$$
−0.715462 + 0.698651i $$0.753784\pi$$
$$828$$ 2139.71 0.0898067
$$829$$ 4931.55 0.206610 0.103305 0.994650i $$-0.467058\pi$$
0.103305 + 0.994650i $$0.467058\pi$$
$$830$$ −17933.9 −0.749992
$$831$$ −4501.92 −0.187930
$$832$$ 0 0
$$833$$ 13748.0 0.571836
$$834$$ −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 $$0.212287\pi$$
$$840$$ −8437.60 −0.346577
$$841$$ −23764.3 −0.974384
$$842$$ 35915.6 1.46999
$$843$$ 42106.8 1.72033
$$844$$ 57.5250 0.00234608
$$845$$ 0 0
$$846$$ −33683.7 −1.36888
$$847$$ −371.687 −0.0150783
$$848$$ 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 $$0.831003\pi$$
$$854$$ 6199.37 0.248405
$$855$$ 74536.6 2.98140
$$856$$ 19406.8 0.774898
$$857$$ −17281.5 −0.688828 −0.344414 0.938818i $$-0.611922\pi$$
−0.344414 + 0.938818i $$0.611922\pi$$
$$858$$ 0 0
$$859$$ 9316.75 0.370062 0.185031 0.982733i $$-0.440761\pi$$
0.185031 + 0.982733i $$0.440761\pi$$
$$860$$ −459.482 −0.0182188
$$861$$ −6358.10 −0.251665
$$862$$ 13413.5 0.530005
$$863$$ 9647.65 0.380544 0.190272 0.981731i $$-0.439063\pi$$
0.190272 + 0.981731i $$0.439063\pi$$
$$864$$ 1699.93 0.0669361
$$865$$ −34279.9 −1.34746
$$866$$ 32083.3 1.25893
$$867$$ 25483.6 0.998232
$$868$$ −51.8501 −0.00202754
$$869$$ 14236.5 0.555742
$$870$$ 8043.18 0.313436
$$871$$ 0 0
$$872$$ −24352.6 −0.945737
$$873$$ −48277.6 −1.87165
$$874$$ −42568.0 −1.64746
$$875$$ −1336.55 −0.0516383
$$876$$ 4293.75 0.165608
$$877$$ −19728.7 −0.759624 −0.379812 0.925064i $$-0.624011\pi$$
−0.379812 + 0.925064i $$0.624011\pi$$
$$878$$ 32304.3 1.24171
$$879$$ 18464.1 0.708509
$$880$$ 9711.36 0.372011
$$881$$ 19473.9 0.744712 0.372356 0.928090i $$-0.378550\pi$$
0.372356 + 0.928090i $$0.378550\pi$$
$$882$$ 32676.4 1.24748
$$883$$ 49092.4 1.87100 0.935499 0.353329i $$-0.114950\pi$$
0.935499 + 0.353329i $$0.114950\pi$$
$$884$$ 0 0
$$885$$ −10408.5 −0.395344
$$886$$ −27599.5 −1.04653
$$887$$ 9292.86 0.351774 0.175887 0.984410i $$-0.443721\pi$$
0.175887 + 0.984410i $$0.443721\pi$$
$$888$$ 2430.12 0.0918348
$$889$$ 4047.41 0.152695
$$890$$ 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
$$896$$ 3949.89 0.147273
$$897$$ 0 0
$$898$$ −943.252 −0.0350520
$$899$$ 787.273 0.0292069
$$900$$ −1839.16 −0.0681170
$$901$$ 14136.7 0.522709
$$902$$ 7845.88 0.289622
$$903$$ −1405.53 −0.0517974
$$904$$ 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
$$917$$ 4917.06 0.177073
$$918$$ −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
$$923$$ 0 0
$$924$$ −143.563 −0.00511134
$$925$$ −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
$$931$$ 46673.3 1.64302
$$932$$ −2355.98 −0.0828033
$$933$$ −28325.1 −0.993916
$$934$$ −13979.8 −0.489757
$$935$$ 6735.44 0.235585
$$936$$ 0 0
$$937$$ −34574.7 −1.20545 −0.602724 0.797950i $$-0.705918\pi$$
−0.602724 + 0.797950i $$0.705918\pi$$
$$938$$ −2873.18 −0.100014
$$939$$ −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
$$945$$ −3204.34 −0.110304
$$946$$ 1734.42 0.0596097
$$947$$ −27231.2 −0.934419 −0.467209 0.884147i $$-0.654741\pi$$
−0.467209 + 0.884147i $$0.654741\pi$$
$$948$$ 5498.79 0.188389
$$949$$ 0 0
$$950$$ 36588.8 1.24958
$$951$$ −124.534 −0.00424635
$$952$$ 2952.50 0.100516
$$953$$ 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
$$959$$ 4950.74 0.166703
$$960$$ 63786.1 2.14447
$$961$$ −28798.9 −0.966698
$$962$$ 0 0
$$963$$ 29839.0 0.998491
$$964$$ 2094.60 0.0699820
$$965$$ 19910.3 0.664182
$$966$$ 7409.03 0.246772
$$967$$ 46173.1 1.53550 0.767750 0.640750i $$-0.221376\pi$$
0.767750 + 0.640750i $$0.221376\pi$$
$$968$$ 2821.78 0.0936937
$$969$$ −45721.8 −1.51579
$$970$$ −54648.9 −1.80894
$$971$$ −5153.91 −0.170337 −0.0851683 0.996367i $$-0.527143\pi$$
−0.0851683 + 0.996367i $$0.527143\pi$$
$$972$$ −2764.06 −0.0912111
$$973$$ 97.8734 0.00322474
$$974$$ −50069.0 −1.64714
$$975$$ 0 0
$$976$$ −43897.6 −1.43968
$$977$$ −9692.13 −0.317378 −0.158689 0.987329i $$-0.550727\pi$$
−0.158689 + 0.987329i $$0.550727\pi$$
$$978$$ 59026.5 1.92992
$$979$$ 16380.2 0.534744
$$980$$ −2655.68 −0.0865638
$$981$$ −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
$$987$$ 8373.97 0.270057
$$988$$ 0 0
$$989$$ 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
$$994$$ 1740.26 0.0555310
$$995$$ −12237.1 −0.389892
$$996$$ −1877.28 −0.0597227
$$997$$ −31944.4 −1.01473 −0.507366 0.861731i $$-0.669381\pi$$
−0.507366 + 0.861731i $$0.669381\pi$$
$$998$$ 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 1859.4.a.a.1.1 2
13.12 even 2 11.4.a.a.1.2 2
39.38 odd 2 99.4.a.c.1.1 2
52.51 odd 2 176.4.a.i.1.2 2
65.12 odd 4 275.4.b.c.199.4 4
65.38 odd 4 275.4.b.c.199.1 4
65.64 even 2 275.4.a.b.1.1 2
91.90 odd 2 539.4.a.e.1.2 2
104.51 odd 2 704.4.a.n.1.1 2
104.77 even 2 704.4.a.p.1.2 2
143.25 even 10 121.4.c.c.9.2 8
143.38 even 10 121.4.c.c.3.1 8
143.51 odd 10 121.4.c.f.27.1 8
143.64 even 10 121.4.c.c.81.1 8
143.90 odd 10 121.4.c.f.81.2 8
143.103 even 10 121.4.c.c.27.2 8
143.116 odd 10 121.4.c.f.3.2 8
143.129 odd 10 121.4.c.f.9.1 8
143.142 odd 2 121.4.a.c.1.1 2
156.155 even 2 1584.4.a.bc.1.1 2
195.194 odd 2 2475.4.a.q.1.2 2
429.428 even 2 1089.4.a.v.1.2 2
572.571 even 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 13.12 even 2
99.4.a.c.1.1 2 39.38 odd 2
121.4.a.c.1.1 2 143.142 odd 2
121.4.c.c.3.1 8 143.38 even 10
121.4.c.c.9.2 8 143.25 even 10
121.4.c.c.27.2 8 143.103 even 10
121.4.c.c.81.1 8 143.64 even 10
121.4.c.f.3.2 8 143.116 odd 10
121.4.c.f.9.1 8 143.129 odd 10
121.4.c.f.27.1 8 143.51 odd 10
121.4.c.f.81.2 8 143.90 odd 10
176.4.a.i.1.2 2 52.51 odd 2
275.4.a.b.1.1 2 65.64 even 2
275.4.b.c.199.1 4 65.38 odd 4
275.4.b.c.199.4 4 65.12 odd 4
539.4.a.e.1.2 2 91.90 odd 2
704.4.a.n.1.1 2 104.51 odd 2
704.4.a.p.1.2 2 104.77 even 2
1089.4.a.v.1.2 2 429.428 even 2
1584.4.a.bc.1.1 2 156.155 even 2
1859.4.a.a.1.1 2 1.1 even 1 trivial
1936.4.a.w.1.2 2 572.571 even 2
2475.4.a.q.1.2 2 195.194 odd 2