# Properties

 Label 121.4.a.c.1.1 Level $121$ Weight $4$ Character 121.1 Self dual yes Analytic conductor $7.139$ 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] = [121,4,Mod(1,121)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(121, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("121.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$121 = 11^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 121.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: $$7.13923111069$$ 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 $$\chi$$ $$=$$ 121.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-2.73205 q^{2} -7.92820 q^{3} -0.535898 q^{4} +14.8564 q^{5} +21.6603 q^{6} -3.07180 q^{7} +23.3205 q^{8} +35.8564 q^{9} +O(q^{10})$$ $$q-2.73205 q^{2} -7.92820 q^{3} -0.535898 q^{4} +14.8564 q^{5} +21.6603 q^{6} -3.07180 q^{7} +23.3205 q^{8} +35.8564 q^{9} -40.5885 q^{10} +4.24871 q^{12} -5.35898 q^{13} +8.39230 q^{14} -117.785 q^{15} -59.4256 q^{16} +41.2154 q^{17} -97.9615 q^{18} -139.923 q^{19} -7.96152 q^{20} +24.3538 q^{21} -111.354 q^{23} -184.890 q^{24} +95.7128 q^{25} +14.6410 q^{26} -70.2154 q^{27} +1.64617 q^{28} +24.9948 q^{29} +321.794 q^{30} +31.4974 q^{31} -24.2102 q^{32} -112.603 q^{34} -45.6359 q^{35} -19.2154 q^{36} +13.1436 q^{37} +382.277 q^{38} +42.4871 q^{39} +346.459 q^{40} -261.072 q^{41} -66.5359 q^{42} +57.7128 q^{43} +532.697 q^{45} +304.224 q^{46} -343.846 q^{47} +471.138 q^{48} -333.564 q^{49} -261.492 q^{50} -326.764 q^{51} +2.87187 q^{52} -342.995 q^{53} +191.832 q^{54} -71.6359 q^{56} +1109.34 q^{57} -68.2872 q^{58} +88.3693 q^{59} +63.1206 q^{60} -738.697 q^{61} -86.0526 q^{62} -110.144 q^{63} +541.549 q^{64} -79.6152 q^{65} +342.359 q^{67} -22.0873 q^{68} +882.836 q^{69} +124.679 q^{70} -207.364 q^{71} +836.190 q^{72} +1010.60 q^{73} -35.9090 q^{74} -758.831 q^{75} +74.9845 q^{76} -116.077 q^{78} -1294.23 q^{79} -882.851 q^{80} -411.441 q^{81} +713.261 q^{82} -441.846 q^{83} -13.0512 q^{84} +612.313 q^{85} -157.674 q^{86} -198.164 q^{87} -1489.11 q^{89} -1455.36 q^{90} +16.4617 q^{91} +59.6743 q^{92} -249.718 q^{93} +939.405 q^{94} -2078.75 q^{95} +191.944 q^{96} +1346.42 q^{97} +911.314 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} - 2 q^{3} - 8 q^{4} + 2 q^{5} + 26 q^{6} - 20 q^{7} + 12 q^{8} + 44 q^{9}+O(q^{10})$$ 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 $$2 q - 2 q^{2} - 2 q^{3} - 8 q^{4} + 2 q^{5} + 26 q^{6} - 20 q^{7} + 12 q^{8} + 44 q^{9} - 50 q^{10} - 40 q^{12} - 80 q^{13} - 4 q^{14} - 194 q^{15} - 8 q^{16} + 124 q^{17} - 92 q^{18} - 72 q^{19} + 88 q^{20} - 76 q^{21} - 98 q^{23} - 252 q^{24} + 136 q^{25} - 40 q^{26} - 182 q^{27} + 128 q^{28} - 144 q^{29} + 266 q^{30} - 34 q^{31} + 104 q^{32} - 52 q^{34} + 172 q^{35} - 80 q^{36} + 54 q^{37} + 432 q^{38} - 400 q^{39} + 492 q^{40} - 536 q^{41} - 140 q^{42} + 60 q^{43} + 428 q^{45} + 314 q^{46} - 272 q^{47} + 776 q^{48} - 390 q^{49} - 232 q^{50} + 164 q^{51} + 560 q^{52} - 492 q^{53} + 110 q^{54} + 120 q^{56} + 1512 q^{57} - 192 q^{58} + 634 q^{59} + 632 q^{60} - 840 q^{61} - 134 q^{62} - 248 q^{63} + 224 q^{64} + 880 q^{65} + 754 q^{67} - 640 q^{68} + 962 q^{69} + 284 q^{70} - 678 q^{71} + 744 q^{72} + 400 q^{73} - 6 q^{74} - 520 q^{75} - 432 q^{76} - 440 q^{78} - 316 q^{79} - 1544 q^{80} - 1294 q^{81} + 512 q^{82} - 468 q^{83} + 736 q^{84} - 452 q^{85} - 156 q^{86} - 1200 q^{87} - 1842 q^{89} - 1532 q^{90} + 1280 q^{91} - 40 q^{92} - 638 q^{93} + 992 q^{94} - 2952 q^{95} + 952 q^{96} + 2194 q^{97} + 870 q^{98}+O(q^{100})$$ 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 - 40 * q^12 - 80 * q^13 - 4 * q^14 - 194 * q^15 - 8 * q^16 + 124 * q^17 - 92 * q^18 - 72 * q^19 + 88 * q^20 - 76 * q^21 - 98 * q^23 - 252 * q^24 + 136 * q^25 - 40 * q^26 - 182 * q^27 + 128 * q^28 - 144 * q^29 + 266 * q^30 - 34 * q^31 + 104 * q^32 - 52 * q^34 + 172 * q^35 - 80 * q^36 + 54 * q^37 + 432 * q^38 - 400 * q^39 + 492 * q^40 - 536 * q^41 - 140 * q^42 + 60 * q^43 + 428 * q^45 + 314 * q^46 - 272 * q^47 + 776 * q^48 - 390 * q^49 - 232 * q^50 + 164 * q^51 + 560 * q^52 - 492 * q^53 + 110 * q^54 + 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 + 880 * q^65 + 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 - 440 * q^78 - 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 - 1842 * q^89 - 1532 * q^90 + 1280 * q^91 - 40 * q^92 - 638 * q^93 + 992 * q^94 - 2952 * q^95 + 952 * q^96 + 2194 * q^97 + 870 * q^98

## 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 $$-0.268688\pi$$
0.664399 + 0.747378i $$0.268688\pi$$
$$6$$ 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
$$11$$ 0 0
$$12$$ 4.24871 0.102208
$$13$$ −5.35898 −0.114332 −0.0571659 0.998365i $$-0.518206\pi$$
−0.0571659 + 0.998365i $$0.518206\pi$$
$$14$$ 8.39230 0.160210
$$15$$ −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
$$21$$ 24.3538 0.253069
$$22$$ 0 0
$$23$$ −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
$$28$$ 1.64617 0.0111106
$$29$$ 24.9948 0.160049 0.0800246 0.996793i $$-0.474500\pi$$
0.0800246 + 0.996793i $$0.474500\pi$$
$$30$$ 321.794 1.95837
$$31$$ 31.4974 0.182487 0.0912436 0.995829i $$-0.470916\pi$$
0.0912436 + 0.995829i $$0.470916\pi$$
$$32$$ −24.2102 −0.133744
$$33$$ 0 0
$$34$$ −112.603 −0.567976
$$35$$ −45.6359 −0.220396
$$36$$ −19.2154 −0.0889601
$$37$$ 13.1436 0.0583998 0.0291999 0.999574i $$-0.490704\pi$$
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 $$0.665648\pi$$
$$42$$ −66.5359 −0.244446
$$43$$ 57.7128 0.204677 0.102339 0.994750i $$-0.467367\pi$$
0.102339 + 0.994750i $$0.467367\pi$$
$$44$$ 0 0
$$45$$ 532.697 1.76466
$$46$$ 304.224 0.975118
$$47$$ −343.846 −1.06713 −0.533565 0.845759i $$-0.679148\pi$$
−0.533565 + 0.845759i $$0.679148\pi$$
$$48$$ 471.138 1.41673
$$49$$ −333.564 −0.972490
$$50$$ −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
$$55$$ 0 0
$$56$$ −71.6359 −0.170942
$$57$$ 1109.34 2.57782
$$58$$ −68.2872 −0.154596
$$59$$ 88.3693 0.194995 0.0974975 0.995236i $$-0.468916\pi$$
0.0974975 + 0.995236i $$0.468916\pi$$
$$60$$ 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
$$63$$ −110.144 −0.220266
$$64$$ 541.549 1.05771
$$65$$ −79.6152 −0.151924
$$66$$ 0 0
$$67$$ 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
$$70$$ 124.679 0.212886
$$71$$ −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 $$-0.873112\pi$$
−0.921593 + 0.388157i $$0.873112\pi$$
$$80$$ −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
$$88$$ 0 0
$$89$$ −1489.11 −1.77355 −0.886773 0.462205i $$-0.847058\pi$$
−0.886773 + 0.462205i $$0.847058\pi$$
$$90$$ −1455.36 −1.70453
$$91$$ 16.4617 0.0189633
$$92$$ 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 $$-0.251091\pi$$
0.704679 + 0.709526i $$0.251091\pi$$
$$98$$ 911.314 0.939353
$$99$$ 0 0
$$100$$ −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 $$-0.505292\pi$$
−0.0166259 + 0.999862i $$0.505292\pi$$
$$104$$ −124.974 −0.117834
$$105$$ 361.810 0.336277
$$106$$ 937.079 0.858653
$$107$$ −832.179 −0.751867 −0.375934 0.926647i $$-0.622678\pi$$
−0.375934 + 0.926647i $$0.622678\pi$$
$$108$$ 37.6283 0.0335258
$$109$$ −1044.26 −0.917629 −0.458815 0.888532i $$-0.651726\pi$$
−0.458815 + 0.888532i $$0.651726\pi$$
$$110$$ 0 0
$$111$$ −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
$$117$$ −192.154 −0.151834
$$118$$ −241.429 −0.188351
$$119$$ −126.605 −0.0975285
$$120$$ −2746.80 −2.08956
$$121$$ 0 0
$$122$$ 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 $$-0.347739\pi$$
0.460309 + 0.887759i $$0.347739\pi$$
$$128$$ −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 $$0.320765\pi$$
$$132$$ 0 0
$$133$$ 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 $$-0.332400\pi$$
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 $$0.496906\pi$$
$$140$$ 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
$$154$$ 0 0
$$155$$ 467.939 0.242489
$$156$$ −22.7688 −0.0116856
$$157$$ 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 $$-0.727224\pi$$
−0.654745 + 0.755850i $$0.727224\pi$$
$$164$$ 139.908 0.0666157
$$165$$ 0 0
$$166$$ 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
$$169$$ −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
$$175$$ −294.010 −0.127001
$$176$$ 0 0
$$177$$ −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 $$0.552735\pi$$
$$182$$ −44.9742 −0.0183171
$$183$$ 5856.54 2.36573
$$184$$ −2596.83 −1.04044
$$185$$ 195.267 0.0776015
$$186$$ 682.242 0.268949
$$187$$ 0 0
$$188$$ 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
$$195$$ 631.206 0.231803
$$196$$ 178.756 0.0651445
$$197$$ 3518.33 1.27244 0.636220 0.771508i $$-0.280497\pi$$
0.636220 + 0.771508i $$0.280497\pi$$
$$198$$ 0 0
$$199$$ 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
$$208$$ 318.461 0.106160
$$209$$ 0 0
$$210$$ −988.484 −0.324819
$$211$$ 107.343 0.0350228 0.0175114 0.999847i $$-0.494426\pi$$
0.0175114 + 0.999847i $$0.494426\pi$$
$$212$$ 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
$$220$$ 0 0
$$221$$ −220.873 −0.0672285
$$222$$ 284.694 0.0860693
$$223$$ −3933.68 −1.18125 −0.590625 0.806946i $$-0.701119\pi$$
−0.590625 + 0.806946i $$0.701119\pi$$
$$224$$ 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 $$-0.410873\pi$$
0.276356 + 0.961055i $$0.410873\pi$$
$$230$$ 4519.68 1.29573
$$231$$ 0 0
$$232$$ 582.892 0.164952
$$233$$ −4396.32 −1.23610 −0.618052 0.786137i $$-0.712078\pi$$
−0.618052 + 0.786137i $$0.712078\pi$$
$$234$$ 524.974 0.146661
$$235$$ −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 $$0.674946\pi$$
$$242$$ 0 0
$$243$$ 5157.80 1.36162
$$244$$ 395.867 0.103864
$$245$$ −4955.56 −1.29224
$$246$$ −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 $$-0.456040\pi$$
0.137667 + 0.990479i $$0.456040\pi$$
$$252$$ 59.0258 0.0147551
$$253$$ 0 0
$$254$$ −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
$$260$$ 42.6657 0.0101770
$$261$$ 896.225 0.212548
$$262$$ −4373.23 −1.03122
$$263$$ −6180.06 −1.44897 −0.724484 0.689292i $$-0.757922\pi$$
−0.724484 + 0.689292i $$0.757922\pi$$
$$264$$ 0 0
$$265$$ −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
$$273$$ −130.512 −0.0289338
$$274$$ −4403.18 −0.970825
$$275$$ 0 0
$$276$$ −473.110 −0.103181
$$277$$ −567.836 −0.123169 −0.0615847 0.998102i $$-0.519615\pi$$
−0.0615847 + 0.998102i $$0.519615\pi$$
$$278$$ −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 $$-0.334581\pi$$
0.496602 + 0.867978i $$0.334581\pi$$
$$284$$ 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
$$297$$ 0 0
$$298$$ −6634.36 −1.28966
$$299$$ 596.743 0.115420
$$300$$ 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 $$0.450146\pi$$
$$308$$ 0 0
$$309$$ 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 $$0.394398\pi$$
$$312$$ 990.821 0.179789
$$313$$ 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 $$-0.500443\pi$$
−0.00139153 + 0.999999i $$0.500443\pi$$
$$318$$ −7429.36 −1.31012
$$319$$ 0 0
$$320$$ 8045.47 1.40549
$$321$$ 6597.69 1.14719
$$322$$ −934.515 −0.161734
$$323$$ −5766.98 −0.993447
$$324$$ 220.491 0.0378070
$$325$$ −512.923 −0.0875442
$$326$$ 7445.13 1.26487
$$327$$ 8279.08 1.40010
$$328$$ −6088.33 −1.02491
$$329$$ 1056.23 0.176996
$$330$$ 0 0
$$331$$ −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
$$336$$ −1447.24 −0.234981
$$337$$ 239.183 0.0386621 0.0193310 0.999813i $$-0.493846\pi$$
0.0193310 + 0.999813i $$0.493846\pi$$
$$338$$ 5923.85 0.953299
$$339$$ −2339.47 −0.374816
$$340$$ −328.137 −0.0523404
$$341$$ 0 0
$$342$$ 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 $$-0.350174\pi$$
0.453503 + 0.891255i $$0.350174\pi$$
$$348$$ 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
$$351$$ 376.283 0.0572208
$$352$$ 0 0
$$353$$ −10916.7 −1.64600 −0.822999 0.568043i $$-0.807701\pi$$
−0.822999 + 0.568043i $$0.807701\pi$$
$$354$$ 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
$$363$$ 0 0
$$364$$ −8.82180 −0.00127030
$$365$$ 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 $$-0.379848\pi$$
0.368569 + 0.929600i $$0.379848\pi$$
$$374$$ 0 0
$$375$$ 3449.58 0.475028
$$376$$ −8018.67 −1.09982
$$377$$ −133.947 −0.0182987
$$378$$ −589.269 −0.0801818
$$379$$ −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 $$-0.560500\pi$$
−0.188925 + 0.981991i $$0.560500\pi$$
$$384$$ 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
$$392$$ −7778.88 −1.00228
$$393$$ −12690.8 −1.62892
$$394$$ −9612.25 −1.22908
$$395$$ −19227.5 −2.44922
$$396$$ 0 0
$$397$$ 14208.7 1.79626 0.898131 0.439728i $$-0.144925\pi$$
0.898131 + 0.439728i $$0.144925\pi$$
$$398$$ −2250.37 −0.283419
$$399$$ −3407.66 −0.427560
$$400$$ −5687.79 −0.710974
$$401$$ −6261.68 −0.779784 −0.389892 0.920861i $$-0.627488\pi$$
−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
$$406$$ 209.764 0.0256415
$$407$$ 0 0
$$408$$ −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
$$416$$ 129.742 0.0152912
$$417$$ −252.608 −0.0296649
$$418$$ 0 0
$$419$$ −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 $$-0.224746\pi$$
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
$$427$$ 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 $$-0.277792\pi$$
0.642754 + 0.766073i $$0.277792\pi$$
$$440$$ 0 0
$$441$$ −11960.4 −1.29148
$$442$$ 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
$$448$$ −1663.53 −0.175434
$$449$$ −345.254 −0.0362885 −0.0181443 0.999835i $$-0.505776\pi$$
−0.0181443 + 0.999835i $$0.505776\pi$$
$$450$$ −9376.17 −0.982216
$$451$$ 0 0
$$452$$ −158.134 −0.0164557
$$453$$ −20428.4 −2.11879
$$454$$ −4840.93 −0.500431
$$455$$ 244.562 0.0251983
$$456$$ 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 $$0.576864\pi$$
$$462$$ 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 $$-0.418413\pi$$
0.253517 + 0.967331i $$0.418413\pi$$
$$468$$ 102.975 0.0101710
$$469$$ −1051.66 −0.103542
$$470$$ 13956.2 1.36968
$$471$$ −19629.8 −1.92037
$$472$$ 2060.82 0.200968
$$473$$ 0 0
$$474$$ −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
$$481$$ −70.4363 −0.00667696
$$482$$ 10678.4 1.00911
$$483$$ −2711.89 −0.255477
$$484$$ 0 0
$$485$$ 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
$$490$$ 13538.9 1.24821
$$491$$ 7617.58 0.700156 0.350078 0.936721i $$-0.386155\pi$$
0.350078 + 0.936721i $$0.386155\pi$$
$$492$$ −1109.22 −0.101641
$$493$$ 1030.17 0.0941108
$$494$$ −2048.62 −0.186582
$$495$$ 0 0
$$496$$ −1871.75 −0.169444
$$497$$ 636.980 0.0574899
$$498$$ −9570.50 −0.861173
$$499$$ 12909.1 1.15810 0.579050 0.815292i $$-0.303424\pi$$
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 $$0.648776\pi$$
$$504$$ −2568.60 −0.227013
$$505$$ 2398.74 0.211371
$$506$$ 0 0
$$507$$ 17190.6 1.50584
$$508$$ −706.102 −0.0616697
$$509$$ 6449.93 0.561666 0.280833 0.959757i $$-0.409389\pi$$
0.280833 + 0.959757i $$0.409389\pi$$
$$510$$ 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
$$517$$ 0 0
$$518$$ 110.305 0.00935623
$$519$$ 18293.7 1.54721
$$520$$ −1856.67 −0.156577
$$521$$ −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 $$-0.584268\pi$$
−0.261655 + 0.965161i $$0.584268\pi$$
$$524$$ −857.819 −0.0715153
$$525$$ 2330.97 0.193775
$$526$$ 16884.2 1.39960
$$527$$ 1298.18 0.107305
$$528$$ 0 0
$$529$$ 232.675 0.0191235
$$530$$ 13921.6 1.14098
$$531$$ 3168.61 0.258956
$$532$$ −230.337 −0.0187714
$$533$$ 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 $$-0.312100\pi$$
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
$$546$$ 356.565 0.0279479
$$547$$ 4949.45 0.386879 0.193440 0.981112i $$-0.438036\pi$$
0.193440 + 0.981112i $$0.438036\pi$$
$$548$$ −863.695 −0.0673270
$$549$$ −26487.0 −2.05909
$$550$$ 0 0
$$551$$ −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
$$559$$ −309.282 −0.0234011
$$560$$ 2711.94 0.204644
$$561$$ 0 0
$$562$$ 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
$$567$$ 1263.86 0.0936107
$$568$$ −4835.84 −0.357231
$$569$$ −5329.16 −0.392636 −0.196318 0.980540i $$-0.562898\pi$$
−0.196318 + 0.980540i $$0.562898\pi$$
$$570$$ −45026.3 −3.30868
$$571$$ 16962.6 1.24319 0.621597 0.783337i $$-0.286484\pi$$
0.621597 + 0.783337i $$0.286484\pi$$
$$572$$ 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 $$-0.688697\pi$$
−0.558692 + 0.829375i $$0.688697\pi$$
$$578$$ 8781.61 0.631949
$$579$$ 10625.3 0.762643
$$580$$ −198.997 −0.0142464
$$581$$ 1357.26 0.0969169
$$582$$ 29163.7 2.07710
$$583$$ 0 0
$$584$$ 23567.7 1.66993
$$585$$ −2854.72 −0.201757
$$586$$ 6362.72 0.448535
$$587$$ 11084.2 0.779373 0.389686 0.920948i $$-0.372583\pi$$
0.389686 + 0.920948i $$0.372583\pi$$
$$588$$ −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 $$0.548123\pi$$
$$594$$ 0 0
$$595$$ −1880.90 −0.129596
$$596$$ −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 $$0.280255\pi$$
$$602$$ 484.344 0.0327913
$$603$$ 12275.8 0.829034
$$604$$ −1380.84 −0.0930223
$$605$$ 0 0
$$606$$ 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
$$609$$ 608.720 0.0405034
$$610$$ 29982.6 1.99010
$$611$$ 1842.67 0.122007
$$612$$ −791.970 −0.0523096
$$613$$ 3527.85 0.232445 0.116222 0.993223i $$-0.462921\pi$$
0.116222 + 0.993223i $$0.462921\pi$$
$$614$$ −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 $$-0.745121\pi$$
−0.696187 + 0.717861i $$0.745121\pi$$
$$620$$ −250.767 −0.0162437
$$621$$ 7818.75 0.505243
$$622$$ −9760.81 −0.629217
$$623$$ 4574.25 0.294163
$$624$$ −2524.82 −0.161977
$$625$$ −18428.2 −1.17940
$$626$$ −19628.0 −1.25319
$$627$$ 0 0
$$628$$ −1326.85 −0.0843109
$$629$$ 541.718 0.0343398
$$630$$ 4470.56 0.282716
$$631$$ 21532.0 1.35844 0.679219 0.733936i $$-0.262319\pi$$
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
$$637$$ 1787.56 0.111187
$$638$$ 0 0
$$639$$ −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 $$-0.153946\pi$$
0.885310 + 0.465000i $$0.153946\pi$$
$$644$$ −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
$$649$$ 0 0
$$650$$ 1401.33 0.0845612
$$651$$ 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 $$-0.399353\pi$$
0.310948 + 0.950427i $$0.399353\pi$$
$$660$$ 0 0
$$661$$ 3295.83 0.193938 0.0969690 0.995287i $$-0.469085\pi$$
0.0969690 + 0.995287i $$0.469085\pi$$
$$662$$ 3603.45 0.211559
$$663$$ 1751.12 0.102576
$$664$$ −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
$$671$$ 0 0
$$672$$ −589.612 −0.0338464
$$673$$ 1187.64 0.0680239 0.0340119 0.999421i $$-0.489172\pi$$
0.0340119 + 0.999421i $$0.489172\pi$$
$$674$$ −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
$$679$$ −4135.91 −0.233758
$$680$$ 14279.4 0.805282
$$681$$ −14048.0 −0.790485
$$682$$ 0 0
$$683$$ −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
$$686$$ −5677.93 −0.316012
$$687$$ −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 $$-0.587100\pi$$
−0.270231 + 0.962796i $$0.587100\pi$$
$$692$$ 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
$$700$$ 157.560 0.00850742
$$701$$ −29949.8 −1.61368 −0.806838 0.590773i $$-0.798823\pi$$
−0.806838 + 0.590773i $$0.798823\pi$$
$$702$$ −1028.02 −0.0552711
$$703$$ −1839.09 −0.0986667
$$704$$ 0 0
$$705$$ 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 $$-0.403187\pi$$
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
$$714$$ −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
$$726$$ 0 0
$$727$$ −502.545 −0.0256373 −0.0128187 0.999918i $$-0.504080\pi$$
−0.0128187 + 0.999918i $$0.504080\pi$$
$$728$$ 383.895 0.0195441
$$729$$ −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
$$737$$ 0 0
$$738$$ 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
$$741$$ −5944.93 −0.294726
$$742$$ −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
$$748$$ 0 0
$$749$$ 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
$$754$$ 365.950 0.0176752
$$755$$ 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
$$759$$ 0 0
$$760$$ −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
$$767$$ −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 $$-0.810274\pi$$
−0.827564 + 0.561371i $$0.810274\pi$$
$$774$$ −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
$$780$$ −338.262 −0.0155279
$$781$$ 0 0
$$782$$ 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
$$792$$ 0 0
$$793$$ 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 $$-0.443914\pi$$
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
$$803$$ 0 0
$$804$$ 1454.58 0.0638050
$$805$$ 5081.73 0.222494
$$806$$ 461.154 0.0201532
$$807$$ −7824.86 −0.341323
$$808$$ 3765.36 0.163942
$$809$$ −5896.97 −0.256275 −0.128138 0.991756i $$-0.540900\pi$$
−0.128138 + 0.991756i $$0.540900\pi$$
$$810$$ 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
$$814$$ 0 0
$$815$$ −40485.3 −1.74005
$$816$$ 19418.2 0.833053
$$817$$ −8075.35 −0.345803
$$818$$ −11454.1 −0.489589
$$819$$ 590.258 0.0251835
$$820$$ 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
$$825$$ 0 0
$$826$$ 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
$$832$$ −2902.15 −0.120930
$$833$$ −13748.0 −0.571836
$$834$$ 690.138 0.0286541
$$835$$ −40666.4 −1.68541
$$836$$ 0 0
$$837$$ −2211.60 −0.0913312
$$838$$ 25373.0 1.04594
$$839$$ −38189.8 −1.57146 −0.785731 0.618568i $$-0.787713\pi$$
−0.785731 + 0.618568i $$0.787713\pi$$
$$840$$ 8437.60 0.346577
$$841$$ −23764.3 −0.974384
$$842$$ −35915.6 −1.46999
$$843$$ 42106.8 1.72033
$$844$$ −57.5250 −0.00234608
$$845$$ −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 $$0.831003\pi$$
$$854$$ −6199.37 −0.248405
$$855$$ −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 $$0.388078\pi$$
$$858$$ 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 $$-0.560937\pi$$
−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
$$868$$ 51.8501 0.00202754
$$869$$ 0 0
$$870$$ 8043.18 0.313436
$$871$$ −1834.70 −0.0713735
$$872$$ −24352.6 −0.945737
$$873$$ 48277.6 1.87165
$$874$$ −42568.0 −1.64746
$$875$$ 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
$$880$$ 0 0
$$881$$ 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 $$0.114950\pi$$
$$884$$ 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
$$889$$ −4047.41 −0.152695
$$890$$ 60440.8 2.27638
$$891$$ 0 0
$$892$$ 2108.05 0.0791288
$$893$$ 48112.0 1.80292
$$894$$ 52598.5 1.96774
$$895$$ −19493.9 −0.728055
$$896$$ 3949.89 0.147273
$$897$$ −4731.10 −0.176106
$$898$$ 943.252 0.0350520
$$899$$ 787.273 0.0292069
$$900$$ −1839.16 −0.0681170
$$901$$ −14136.7 −0.522709
$$902$$ 0 0
$$903$$ 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
$$910$$ −668.155 −0.0243397
$$911$$ 33049.6 1.20196 0.600979 0.799265i $$-0.294778\pi$$
0.600979 + 0.799265i $$0.294778\pi$$
$$912$$ −65923.1 −2.39357
$$913$$ 0 0
$$914$$ −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 $$-0.363627\pi$$
0.415442 + 0.909620i $$0.363627\pi$$
$$920$$ −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 $$-0.634220\pi$$
−0.409280 + 0.912409i $$0.634220\pi$$
$$930$$ 10135.7 0.357378
$$931$$ 46673.3 1.64302
$$932$$ 2355.98 0.0828033
$$933$$ −28325.1 −0.993916
$$934$$ −13979.8 −0.489757
$$935$$ 0 0
$$936$$ −4481.13 −0.156485
$$937$$ 34574.7 1.20545 0.602724 0.797950i $$-0.294082\pi$$
0.602724 + 0.797950i $$0.294082\pi$$
$$938$$ 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
$$946$$ 0 0
$$947$$ 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
$$952$$ −2952.50 −0.100516
$$953$$ −40939.4 −1.39156 −0.695781 0.718254i $$-0.744942\pi$$
−0.695781 + 0.718254i $$0.744942\pi$$
$$954$$ 33600.3 1.14030
$$955$$ 25527.0 0.864956
$$956$$ −2188.87 −0.0740515
$$957$$ 0 0
$$958$$ 31601.3 1.06575
$$959$$ −4950.74 −0.166703
$$960$$ −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
$$966$$ 7409.03 0.246772
$$967$$ 46173.1 1.53550 0.767750 0.640750i $$-0.221376\pi$$
0.767750 + 0.640750i $$0.221376\pi$$
$$968$$ 0 0
$$969$$ 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
$$975$$ 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
$$979$$ 0 0
$$980$$ 2655.68 0.0865638
$$981$$ −37443.3 −1.21863
$$982$$ −20811.6 −0.676299
$$983$$ 32915.7 1.06800 0.534002 0.845483i $$-0.320687\pi$$
0.534002 + 0.845483i $$0.320687\pi$$
$$984$$ 48269.5 1.56380
$$985$$ 52269.7 1.69081
$$986$$ −2814.48 −0.0909041
$$987$$ −8373.97 −0.270057
$$988$$ −401.841 −0.0129395
$$989$$ −6426.54 −0.206625
$$990$$ 0 0
$$991$$ 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 $$-0.330619\pi$$
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 121.4.a.c.1.1 2
3.2 odd 2 1089.4.a.v.1.2 2
4.3 odd 2 1936.4.a.w.1.2 2
11.2 odd 10 121.4.c.c.81.1 8
11.3 even 5 121.4.c.f.9.1 8
11.4 even 5 121.4.c.f.27.1 8
11.5 even 5 121.4.c.f.3.2 8
11.6 odd 10 121.4.c.c.3.1 8
11.7 odd 10 121.4.c.c.27.2 8
11.8 odd 10 121.4.c.c.9.2 8
11.9 even 5 121.4.c.f.81.2 8
11.10 odd 2 11.4.a.a.1.2 2
33.32 even 2 99.4.a.c.1.1 2
44.43 even 2 176.4.a.i.1.2 2
55.32 even 4 275.4.b.c.199.4 4
55.43 even 4 275.4.b.c.199.1 4
55.54 odd 2 275.4.a.b.1.1 2
77.76 even 2 539.4.a.e.1.2 2
88.21 odd 2 704.4.a.p.1.2 2
88.43 even 2 704.4.a.n.1.1 2
132.131 odd 2 1584.4.a.bc.1.1 2
143.142 odd 2 1859.4.a.a.1.1 2
165.164 even 2 2475.4.a.q.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
11.4.a.a.1.2 2 11.10 odd 2
99.4.a.c.1.1 2 33.32 even 2
121.4.a.c.1.1 2 1.1 even 1 trivial
121.4.c.c.3.1 8 11.6 odd 10
121.4.c.c.9.2 8 11.8 odd 10
121.4.c.c.27.2 8 11.7 odd 10
121.4.c.c.81.1 8 11.2 odd 10
121.4.c.f.3.2 8 11.5 even 5
121.4.c.f.9.1 8 11.3 even 5
121.4.c.f.27.1 8 11.4 even 5
121.4.c.f.81.2 8 11.9 even 5
176.4.a.i.1.2 2 44.43 even 2
275.4.a.b.1.1 2 55.54 odd 2
275.4.b.c.199.1 4 55.43 even 4
275.4.b.c.199.4 4 55.32 even 4
539.4.a.e.1.2 2 77.76 even 2
704.4.a.n.1.1 2 88.43 even 2
704.4.a.p.1.2 2 88.21 odd 2
1089.4.a.v.1.2 2 3.2 odd 2
1584.4.a.bc.1.1 2 132.131 odd 2
1859.4.a.a.1.1 2 143.142 odd 2
1936.4.a.w.1.2 2 4.3 odd 2
2475.4.a.q.1.2 2 165.164 even 2