LMFDB - Embedded newform 539.4.a.e.1.2

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

Display 100 coefficients 10 coefficients

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 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \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	$-0.339561\pi$
$2$	2.73205	0.965926	0.482963	+	0.875641i	$0.339561\pi$
$3$	7.92820	1.52578	0.762892	−	0.646526i	$-0.223779\pi$
$3$	7.92820	1.52578	0.762892	+	0.646526i	$0.223779\pi$
$4$	−0.535898	−0.0669873
$5$	−14.8564	−1.32880	−0.664399	−	0.747378i	$-0.731312\pi$
$5$	−14.8564	−1.32880	−0.664399	+	0.747378i	$0.731312\pi$
$6$	21.6603	1.47379
$7$	0	0
$7$	0	0
$8$	−23.3205	−1.03063
$9$	35.8564	1.32802
$10$	−40.5885	−1.28352
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	−4.24871	−0.102208
$13$	−5.35898	−0.114332	−0.0571659	−	0.998365i	$-0.518206\pi$
$13$	−5.35898	−0.114332	−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	$-0.405011\pi$
$17$	41.2154	0.588012	0.294006	+	0.955804i	$0.405011\pi$
$18$	97.9615	1.28276
$19$	−139.923	−1.68950	−0.844751	−	0.535159i	$-0.820252\pi$
$19$	−139.923	−1.68950	−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	$-0.668418\pi$
$23$	−111.354	−1.00952	−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	$-0.525500\pi$
$29$	−24.9948	−0.160049	−0.0800246	+	0.996793i	$0.525500\pi$
$30$	−321.794	−1.95837
$31$	−31.4974	−0.182487	−0.0912436	−	0.995829i	$-0.529084\pi$
$31$	−31.4974	−0.182487	−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	$-0.490704\pi$
$37$	13.1436	0.0583998	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	$-0.665648\pi$
$41$	−261.072	−0.994453	−0.497226	+	0.867621i	$0.665648\pi$
$42$	0	0
$43$	−57.7128	−0.204677	−0.102339	−	0.994750i	$-0.532633\pi$
$43$	−57.7128	−0.204677	−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	$-0.320852\pi$
$47$	343.846	1.06713	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	$-0.646608\pi$
$53$	−342.995	−0.888943	−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	$-0.531084\pi$
$59$	−88.3693	−0.194995	−0.0974975	+	0.995236i	$0.531084\pi$
$60$	63.1206	0.135814
$61$	−738.697	−1.55050	−0.775250	−	0.631654i	$-0.782376\pi$
$61$	−738.697	−1.55050	−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	$-0.398957\pi$
$67$	342.359	0.624266	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	$-0.555445\pi$
$71$	−207.364	−0.346614	−0.173307	+	0.984868i	$0.555445\pi$
$72$	−836.190	−1.36869
$73$	1010.60	1.62030	0.810149	−	0.586224i	$-0.199386\pi$
$73$	1010.60	1.62030	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	$-0.126888\pi$
$79$	1294.23	1.84319	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	$-0.594375\pi$
$83$	−441.846	−0.584324	−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	$-0.152942\pi$
$89$	1489.11	1.77355	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	$-0.748909\pi$
$97$	−1346.42	−1.40936	−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	$-0.474657\pi$
$101$	161.461	0.159069	0.0795347	+	0.996832i	$0.474657\pi$
$102$	892.736	0.866608
$103$	34.7592	0.0332517	0.0166259	−	0.999862i	$-0.494708\pi$
$103$	34.7592	0.0332517	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	$-0.377322\pi$
$107$	832.179	0.751867	0.375934	+	0.926647i	$0.377322\pi$
$108$	−37.6283	−0.0335258
$109$	1044.26	0.917629	0.458815	−	0.888532i	$-0.348274\pi$
$109$	1044.26	0.917629	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	$-0.460804\pi$
$113$	295.082	0.245654	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	$-0.652261\pi$
$127$	−1317.60	−0.920618	−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	$-0.320765\pi$
$131$	1600.71	1.06759	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	$-0.332400\pi$
$137$	1611.68	1.00507	0.502536	+	0.864556i	$0.332400\pi$
$138$	−2411.95	−1.48782
$139$	31.8619	0.0194424	0.00972120	−	0.999953i	$-0.496906\pi$
$139$	31.8619	0.0194424	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	$-0.732668\pi$
$149$	−2428.34	−1.33515	−0.667576	+	0.744542i	$0.732668\pi$
$150$	2073.16	1.12849
$151$	−2576.68	−1.38866	−0.694328	−	0.719659i	$-0.744298\pi$
$151$	−2576.68	−1.38866	−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	$-0.716660\pi$
$157$	−2475.94	−1.25861	−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	$-0.727224\pi$
$163$	−2725.11	−1.30949	−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	$-0.718665\pi$
$167$	−2737.30	−1.26837	−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	$-0.669254\pi$
$173$	−2307.42	−1.01404	−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	$-0.588331\pi$
$179$	−1312.15	−0.547905	−0.273953	+	0.961743i	$0.588331\pi$
$180$	285.472	0.118210
$181$	803.174	0.329831	0.164916	−	0.986308i	$-0.447265\pi$
$181$	803.174	0.329831	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	$-0.394479\pi$
$191$	1718.25	0.650932	0.325466	+	0.945554i	$0.394479\pi$
$192$	4293.51	1.61384
$193$	1340.18	0.499837	0.249919	−	0.968267i	$-0.419596\pi$
$193$	1340.18	0.499837	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	$-0.719503\pi$
$197$	−3518.33	−1.27244	−0.636220	+	0.771508i	$0.719503\pi$
$198$	−1077.58	−0.386768
$199$	−823.692	−0.293417	−0.146709	−	0.989180i	$-0.546868\pi$
$199$	−823.692	−0.293417	−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	$-0.505574\pi$
$211$	−107.343	−0.0350228	−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	$-0.298881\pi$
$223$	3933.68	1.18125	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	$-0.416593\pi$
$227$	1771.90	0.518085	0.259042	+	0.965866i	$0.416593\pi$
$228$	594.493	0.172681
$229$	−1915.37	−0.552713	−0.276356	−	0.961055i	$-0.589127\pi$
$229$	−1915.37	−0.552713	−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	$-0.287922\pi$
$233$	4396.32	1.23610	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	$-0.686413\pi$
$239$	−4084.49	−1.10546	−0.552728	+	0.833362i	$0.686413\pi$
$240$	6999.42	1.88255
$241$	−3908.58	−1.04471	−0.522353	−	0.852730i	$-0.674946\pi$
$241$	−3908.58	−1.04471	−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	$-0.543960\pi$
$251$	−1094.89	−0.275335	−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	$-0.530300\pi$
$257$	−783.179	−0.190091	−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	$-0.242078\pi$
$263$	6180.06	1.44897	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	$-0.535678\pi$
$269$	−986.965	−0.223704	−0.111852	+	0.993725i	$0.535678\pi$
$270$	−2849.93	−0.642376
$271$	−4576.99	−1.02595	−0.512975	−	0.858404i	$-0.671457\pi$
$271$	−4576.99	−1.02595	−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	$-0.480385\pi$
$277$	567.836	0.123169	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	$-0.309357\pi$
$281$	5311.01	1.12750	0.563752	+	0.825944i	$0.309357\pi$
$282$	7447.79	1.57273
$283$	4728.44	0.993204	0.496602	−	0.867978i	$-0.334581\pi$
$283$	4728.44	0.993204	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	$-0.574585\pi$
$293$	−2328.92	−0.464358	−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	$-0.450146\pi$
$307$	1678.07	0.311962	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	$-0.605602\pi$
$311$	−3572.71	−0.651413	−0.325707	+	0.945471i	$0.605602\pi$
$312$	990.821	0.179789
$313$	−7184.36	−1.29739	−0.648697	−	0.761047i	$-0.724686\pi$
$313$	−7184.36	−1.29739	−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	$-0.500443\pi$
$317$	−15.7077	−0.00278306	−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	$-0.534928\pi$
$331$	−1318.95	−0.219022	−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	$-0.506154\pi$
$337$	−239.183	−0.0386621	−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	$-0.649826\pi$
$347$	−5862.79	−0.907006	−0.453503	+	0.891255i	$0.649826\pi$
$348$	106.196	0.0163583
$349$	−3491.73	−0.535553	−0.267776	−	0.963481i	$-0.586289\pi$
$349$	−3491.73	−0.535553	−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	$-0.192299\pi$
$353$	10916.7	1.64600	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	$-0.820628\pi$
$359$	−11500.7	−1.69077	−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	$-0.659816\pi$
$367$	−6767.01	−0.962493	−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	$-0.620152\pi$
$373$	−5310.22	−0.737139	−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	$-0.518092\pi$
$379$	−838.267	−0.113612	−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	$-0.439500\pi$
$383$	2832.16	0.377851	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	$-0.435009\pi$
$389$	3111.25	0.405519	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	$-0.855075\pi$
$397$	−14208.7	−1.79626	−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	$-0.627488\pi$
$401$	−6261.68	−0.779784	−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	$-0.418441\pi$
$409$	4192.50	0.506860	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	$-0.317888\pi$
$419$	9287.15	1.08283	0.541416	+	0.840755i	$0.317888\pi$
$420$	0	0
$421$	13146.0	1.52185	0.760923	−	0.648842i	$-0.224746\pi$
$421$	13146.0	1.52185	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	$-0.411537\pi$
$431$	4909.67	0.548701	0.274351	+	0.961630i	$0.411537\pi$
$432$	−4172.59	−0.464708
$433$	11743.3	1.30334	0.651671	−	0.758502i	$-0.274068\pi$
$433$	11743.3	1.30334	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	$-0.277792\pi$
$439$	11824.2	1.28551	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	$-0.317772\pi$
$443$	10102.1	1.08344	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	$-0.505776\pi$
$449$	−345.254	−0.0362885	−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	$-0.681886\pi$
$457$	−10567.1	−1.08164	−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	$-0.576864\pi$
$461$	−4733.96	−0.478270	−0.239135	+	0.970986i	$0.576864\pi$
$462$	0	0
$463$	3431.20	0.344409	0.172204	−	0.985061i	$-0.444911\pi$
$463$	3431.20	0.344409	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	$-0.581587\pi$
$467$	−5116.96	−0.507034	−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	$-0.686011\pi$
$479$	−11566.9	−1.10335	−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	$-0.824990\pi$
$487$	−18326.5	−1.70525	−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	$-0.613845\pi$
$491$	−7617.58	−0.700156	−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	$-0.303424\pi$
$499$	12909.1	1.15810	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	$-0.648776\pi$
$503$	−10165.7	−0.901121	−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	$-0.590611\pi$
$509$	−6449.93	−0.561666	−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	$-0.198041\pi$
$521$	19327.4	1.62524	0.812620	+	0.582794i	$0.198041\pi$
$522$	−2448.53	−0.205305
$523$	−6259.09	−0.523310	−0.261655	−	0.965161i	$-0.584268\pi$
$523$	−6259.09	−0.523310	−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	$-0.687900\pi$
$541$	−14008.2	−1.11323	−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	$-0.561964\pi$
$547$	−4949.45	−0.386879	−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	$-0.546188\pi$
$557$	−3801.58	−0.289188	−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	$-0.379169\pi$
$563$	9900.11	0.741101	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	$-0.437102\pi$
$569$	5329.16	0.392636	0.196318	+	0.980540i	$0.437102\pi$
$570$	45026.3	3.30868
$571$	−16962.6	−1.24319	−0.621597	−	0.783337i	$-0.713516\pi$
$571$	−16962.6	−1.24319	−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	$-0.311303\pi$
$577$	15487.0	1.11738	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	$-0.627417\pi$
$587$	−11084.2	−0.779373	−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	$-0.548123\pi$
$593$	−4349.68	−0.301214	−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	$-0.351549\pi$
$599$	13183.9	0.899299	0.449650	+	0.893205i	$0.351549\pi$
$600$	−17696.3	−1.20408
$601$	18765.0	1.27361	0.636806	−	0.771024i	$-0.280255\pi$
$601$	18765.0	1.27361	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	$-0.761060\pi$
$607$	−21871.4	−1.46249	−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	$-0.537079\pi$
$613$	−3527.85	−0.232445	−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	$-0.765881\pi$
$617$	−22728.1	−1.48298	−0.741490	+	0.670963i	$0.765881\pi$
$618$	752.893	0.0490062
$619$	21443.3	1.39237	0.696187	−	0.717861i	$-0.254879\pi$
$619$	21443.3	1.39237	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	$-0.262319\pi$
$631$	21532.0	1.35844	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	$-0.286826\pi$
$641$	20148.3	1.24151	0.620756	+	0.784004i	$0.286826\pi$
$642$	18025.2	1.10810
$643$	−28869.7	−1.77062	−0.885310	−	0.465000i	$-0.846054\pi$
$643$	−28869.7	−1.77062	−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	$-0.484617\pi$
$647$	1590.02	0.0966155	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	$-0.295119\pi$
$653$	20028.1	1.20024	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	$-0.600647\pi$
$659$	−10520.7	−0.621897	−0.310948	+	0.950427i	$0.600647\pi$
$660$	−694.326	−0.0409494
$661$	−3295.83	−0.193938	−0.0969690	−	0.995287i	$-0.530915\pi$
$661$	−3295.83	−0.193938	−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	$-0.510828\pi$
$673$	−1187.64	−0.0680239	−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	$-0.622456\pi$
$677$	−13221.4	−0.750574	−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	$-0.626641\pi$
$683$	−13831.4	−0.774882	−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	$-0.412900\pi$
$691$	9817.07	0.540462	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	$-0.201177\pi$
$701$	29949.8	1.61368	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	$-0.403187\pi$
$709$	11307.5	0.598959	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	$-0.178962\pi$
$719$	32623.4	1.69214	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	$-0.495920\pi$
$727$	502.545	0.0256373	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	$-0.569780\pi$
$733$	−8631.37	−0.434935	−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	$-0.651039\pi$
$739$	−18357.5	−0.913792	−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	$-0.410965\pi$
$743$	11182.6	0.552155	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	$-0.366738\pi$
$751$	16733.4	0.813063	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	$-0.699225\pi$
$757$	−24402.4	−1.17163	−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	$-0.564652\pi$
$761$	−8469.33	−0.403434	−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	$-0.779676\pi$
$769$	−32834.7	−1.53973	−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	$-0.189726\pi$
$773$	35571.4	1.65513	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	$-0.615937\pi$
$787$	−15729.6	−0.712452	−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	$-0.556086\pi$
$797$	−7888.07	−0.350577	−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	$-0.459100\pi$
$809$	5896.97	0.256275	0.128138	+	0.991756i	$0.459100\pi$
$810$	16699.8	0.724407
$811$	−14197.9	−0.614744	−0.307372	−	0.951589i	$-0.599450\pi$
$811$	−14197.9	−0.614744	−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	$-0.638577\pi$
$821$	−19841.7	−0.843459	−0.421729	+	0.906722i	$0.638577\pi$
$822$	34909.3	1.48127
$823$	−28202.2	−1.19449	−0.597246	−	0.802058i	$-0.703738\pi$
$823$	−28202.2	−1.19449	−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	$-0.246216\pi$
$827$	34031.0	1.43092	0.715462	+	0.698651i	$0.246216\pi$
$828$	2139.71	0.0898067
$829$	−4931.55	−0.206610	−0.103305	−	0.994650i	$-0.532942\pi$
$829$	−4931.55	−0.206610	−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	$-0.212287\pi$
$839$	38189.8	1.57146	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	$-0.831003\pi$
$853$	−42966.8	−1.72469	−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	$-0.388078\pi$
$857$	17281.5	0.688828	0.344414	+	0.938818i	$0.388078\pi$
$858$	1276.85	0.0508052
$859$	−9316.75	−0.370062	−0.185031	−	0.982733i	$-0.559239\pi$
$859$	−9316.75	−0.370062	−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	$-0.560937\pi$
$863$	−9647.65	−0.380544	−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	$-0.375989\pi$
$877$	19728.7	0.759624	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	$-0.621450\pi$
$881$	−19473.9	−0.744712	−0.372356	+	0.928090i	$0.621450\pi$
$882$	0	0
$883$	49092.4	1.87100	0.935499	−	0.353329i	$-0.114950\pi$
$883$	49092.4	1.87100	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	$-0.556279\pi$
$887$	−9292.86	−0.351774	−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	$-0.257665\pi$
$907$	37688.7	1.37975	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	$-0.294778\pi$
$911$	33049.6	1.20196	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	$-0.636373\pi$
$919$	−23148.0	−0.830883	−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	$-0.365780\pi$
$929$	23177.9	0.818561	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	$-0.294082\pi$
$937$	34574.7	1.20545	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	$-0.757966\pi$
$941$	−41831.2	−1.44916	−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	$-0.345259\pi$
$947$	27231.2	0.934419	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	$-0.255058\pi$
$953$	40939.4	1.39156	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	$-0.778624\pi$
$967$	−46173.1	−1.53550	−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	$-0.472857\pi$
$971$	5153.91	0.170337	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	$-0.449273\pi$
$977$	9692.13	0.317378	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	$-0.679313\pi$
$983$	−32915.7	−1.06800	−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	$-0.343370\pi$
$991$	29477.9	0.944901	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	$-0.330619\pi$
$997$	31944.4	1.01473	0.507366	+	0.861731i	$0.330619\pi$
$998$	35268.4	1.11864
$999$	922.883	0.0292279

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 539 = 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	539.a (trivial)

\(f(q)\)	\(=\)	\(q+2.73205 q^{2} +7.92820 q^{3} -0.535898 q^{4} -14.8564 q^{5} +21.6603 q^{6} -23.3205 q^{8} +35.8564 q^{9} +O(q^{10})\)	\(q+2.73205 q^{2} +7.92820 q^{3} -0.535898 q^{4} -14.8564 q^{5} +21.6603 q^{6} -23.3205 q^{8} +35.8564 q^{9} -40.5885 q^{10} -11.0000 q^{11} -4.24871 q^{12} -5.35898 q^{13} -117.785 q^{15} -59.4256 q^{16} +41.2154 q^{17} +97.9615 q^{18} -139.923 q^{19} +7.96152 q^{20} -30.0526 q^{22} -111.354 q^{23} -184.890 q^{24} +95.7128 q^{25} -14.6410 q^{26} +70.2154 q^{27} -24.9948 q^{29} -321.794 q^{30} -31.4974 q^{31} +24.2102 q^{32} -87.2102 q^{33} +112.603 q^{34} -19.2154 q^{36} +13.1436 q^{37} -382.277 q^{38} -42.4871 q^{39} +346.459 q^{40} -261.072 q^{41} -57.7128 q^{43} +5.89488 q^{44} -532.697 q^{45} -304.224 q^{46} +343.846 q^{47} -471.138 q^{48} +261.492 q^{50} +326.764 q^{51} +2.87187 q^{52} -342.995 q^{53} +191.832 q^{54} +163.420 q^{55} -1109.34 q^{57} -68.2872 q^{58} -88.3693 q^{59} +63.1206 q^{60} -738.697 q^{61} -86.0526 q^{62} +541.549 q^{64} +79.6152 q^{65} -238.263 q^{66} +342.359 q^{67} -22.0873 q^{68} -882.836 q^{69} -207.364 q^{71} -836.190 q^{72} +1010.60 q^{73} +35.9090 q^{74} +758.831 q^{75} +74.9845 q^{76} -116.077 q^{78} +1294.23 q^{79} +882.851 q^{80} -411.441 q^{81} -713.261 q^{82} -441.846 q^{83} -612.313 q^{85} -157.674 q^{86} -198.164 q^{87} +256.526 q^{88} +1489.11 q^{89} -1455.36 q^{90} +59.6743 q^{92} -249.718 q^{93} +939.405 q^{94} +2078.75 q^{95} +191.944 q^{96} -1346.42 q^{97} -394.420 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{3} - 8 q^{4} - 2 q^{5} + 26 q^{6} - 12 q^{8} + 44 q^{9}+O(q^{10}) \) 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

Introduction

L-functions

Modular forms

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