LMFDB - Embedded newform 1045.2.a.d.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1045,2,Mod(1,1045)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1045.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1045, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1045 = 5 \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1045.a (trivial)

Newform invariants

comment:select newform

sage:traces = [5,-3,-7,5,5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$8.34436701122$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	$\Q(\zeta_{22})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 4x^{3} + 3x^{2} + 3x - 1 $ x^5 - x^4 - 4x^3 + 3x^2 + 3*x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.68251$ of defining polynomial
Character	$\chi$	$=$	1045.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.37279 q^{2} -1.23648 q^{3} -0.115460 q^{4} +1.00000 q^{5} +1.69742 q^{6} +2.43232 q^{7} +2.90407 q^{8} -1.47112 q^{9} -1.37279 q^{10} -1.00000 q^{11} +0.142763 q^{12} -4.84815 q^{13} -3.33906 q^{14} -1.23648 q^{15} -3.75575 q^{16} -1.04001 q^{17} +2.01953 q^{18} +1.00000 q^{19} -0.115460 q^{20} -3.00752 q^{21} +1.37279 q^{22} +0.377030 q^{23} -3.59082 q^{24} +1.00000 q^{25} +6.65547 q^{26} +5.52845 q^{27} -0.280835 q^{28} +4.50688 q^{29} +1.69742 q^{30} +4.76341 q^{31} -0.652306 q^{32} +1.23648 q^{33} +1.42770 q^{34} +2.43232 q^{35} +0.169855 q^{36} +3.01428 q^{37} -1.37279 q^{38} +5.99464 q^{39} +2.90407 q^{40} +0.790451 q^{41} +4.12867 q^{42} -7.46777 q^{43} +0.115460 q^{44} -1.47112 q^{45} -0.517582 q^{46} -9.67305 q^{47} +4.64391 q^{48} -1.08380 q^{49} -1.37279 q^{50} +1.28594 q^{51} +0.559766 q^{52} -12.7465 q^{53} -7.58937 q^{54} -1.00000 q^{55} +7.06364 q^{56} -1.23648 q^{57} -6.18698 q^{58} +1.32111 q^{59} +0.142763 q^{60} -5.58779 q^{61} -6.53915 q^{62} -3.57824 q^{63} +8.40698 q^{64} -4.84815 q^{65} -1.69742 q^{66} -1.20728 q^{67} +0.120079 q^{68} -0.466190 q^{69} -3.33906 q^{70} -0.259538 q^{71} -4.27224 q^{72} -6.27686 q^{73} -4.13797 q^{74} -1.23648 q^{75} -0.115460 q^{76} -2.43232 q^{77} -8.22935 q^{78} -9.16926 q^{79} -3.75575 q^{80} -2.42245 q^{81} -1.08512 q^{82} -14.2430 q^{83} +0.347247 q^{84} -1.04001 q^{85} +10.2517 q^{86} -5.57266 q^{87} -2.90407 q^{88} -1.23983 q^{89} +2.01953 q^{90} -11.7923 q^{91} -0.0435318 q^{92} -5.88986 q^{93} +13.2790 q^{94} +1.00000 q^{95} +0.806563 q^{96} +2.95633 q^{97} +1.48783 q^{98} +1.47112 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 3 q^{2} - 7 q^{3} + 5 q^{4} + 5 q^{5} + 2 q^{6} - 11 q^{7} + 3 q^{8} + 8 q^{9} - 3 q^{10} - 5 q^{11} - 7 q^{12} + q^{13} - 7 q^{15} - 3 q^{16} - 3 q^{17} - 7 q^{18} + 5 q^{19} + 5 q^{20} + 11 q^{21}+ \cdots - 8 q^{99}+O(q^{100}) $ 5 * q - 3 * q^2 - 7 * q^3 + 5 * q^4 + 5 * q^5 + 2 * q^6 - 11 * q^7 + 3 * q^8 + 8 * q^9 - 3 * q^10 - 5 * q^11 - 7 * q^12 + q^13 - 7 * q^15 - 3 * q^16 - 3 * q^17 - 7 * q^18 + 5 * q^19 + 5 * q^20 + 11 * q^21 + 3 * q^22 - 8 * q^23 + 9 * q^24 + 5 * q^25 - 16 * q^26 - 10 * q^27 - 22 * q^28 + 11 * q^29 + 2 * q^30 - 5 * q^31 - 2 * q^32 + 7 * q^33 + 4 * q^34 - 11 * q^35 - 3 * q^36 - 9 * q^37 - 3 * q^38 - 8 * q^39 + 3 * q^40 + 15 * q^41 + 11 * q^42 - 13 * q^43 - 5 * q^44 + 8 * q^45 + 18 * q^46 - 20 * q^47 - 20 * q^48 + 20 * q^49 - 3 * q^50 + 24 * q^51 + q^52 - 5 * q^53 + 17 * q^54 - 5 * q^55 - 7 * q^57 - 33 * q^58 - 17 * q^59 - 7 * q^60 + 3 * q^61 + 14 * q^62 - 22 * q^63 - 17 * q^64 + q^65 - 2 * q^66 - 28 * q^67 - 25 * q^68 - 2 * q^69 - 6 * q^71 - 26 * q^72 - 16 * q^73 - 21 * q^74 - 7 * q^75 + 5 * q^76 + 11 * q^77 + 29 * q^78 + 3 * q^79 - 3 * q^80 + q^81 + 2 * q^82 - 33 * q^83 + 33 * q^84 - 3 * q^85 + 10 * q^86 - 3 * q^88 - 16 * q^89 - 7 * q^90 - 22 * q^91 - 19 * q^92 - 26 * q^93 - 10 * q^94 + 5 * q^95 + 5 * q^96 - 14 * q^97 + 10 * q^98 - 8 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (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 $

$2$	−1.37279	−0.970706	−0.485353	−	0.874318i	$-0.661309\pi$
$2$	−1.37279	−0.970706	−0.485353	+	0.874318i	$0.661309\pi$
$3$	−1.23648	−0.713881	−0.356941	−	0.934127i	$-0.616180\pi$
$3$	−1.23648	−0.713881	−0.356941	+	0.934127i	$0.616180\pi$
$4$	−0.115460	−0.0577299
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	1.69742	0.692969
$7$	2.43232	0.919332	0.459666	−	0.888092i	$-0.347969\pi$
$7$	2.43232	0.919332	0.459666	+	0.888092i	$0.347969\pi$
$8$	2.90407	1.02674
$9$	−1.47112	−0.490373
$10$	−1.37279	−0.434113
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0.142763	0.0412123
$13$	−4.84815	−1.34463	−0.672317	−	0.740263i	$-0.734701\pi$
$13$	−4.84815	−1.34463	−0.672317	+	0.740263i	$0.734701\pi$
$14$	−3.33906	−0.892401
$15$	−1.23648	−0.319257
$16$	−3.75575	−0.938937
$17$	−1.04001	−0.252238	−0.126119	−	0.992015i	$-0.540252\pi$
$17$	−1.04001	−0.252238	−0.126119	+	0.992015i	$0.540252\pi$
$18$	2.01953	0.476008
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	−0.115460	−0.0258176
$21$	−3.00752	−0.656294
$22$	1.37279	0.292679
$23$	0.377030	0.0786163	0.0393081	−	0.999227i	$-0.487485\pi$
$23$	0.377030	0.0786163	0.0393081	+	0.999227i	$0.487485\pi$
$24$	−3.59082	−0.732974
$25$	1.00000	0.200000
$26$	6.65547	1.30525
$27$	5.52845	1.06395
$28$	−0.280835	−0.0530729
$29$	4.50688	0.836907	0.418453	−	0.908238i	$-0.362572\pi$
$29$	4.50688	0.836907	0.418453	+	0.908238i	$0.362572\pi$
$30$	1.69742	0.309905
$31$	4.76341	0.855535	0.427767	−	0.903889i	$-0.359300\pi$
$31$	4.76341	0.855535	0.427767	+	0.903889i	$0.359300\pi$
$32$	−0.652306	−0.115313
$33$	1.23648	0.215243
$34$	1.42770	0.244849
$35$	2.43232	0.411138
$36$	0.169855	0.0283092
$37$	3.01428	0.495545	0.247773	−	0.968818i	$-0.420301\pi$
$37$	3.01428	0.495545	0.247773	+	0.968818i	$0.420301\pi$
$38$	−1.37279	−0.222695
$39$	5.99464	0.959910
$40$	2.90407	0.459174
$41$	0.790451	0.123448	0.0617238	−	0.998093i	$-0.480340\pi$
$41$	0.790451	0.123448	0.0617238	+	0.998093i	$0.480340\pi$
$42$	4.12867	0.637068
$43$	−7.46777	−1.13882	−0.569412	−	0.822052i	$-0.692829\pi$
$43$	−7.46777	−1.13882	−0.569412	+	0.822052i	$0.692829\pi$
$44$	0.115460	0.0174062
$45$	−1.47112	−0.219302
$46$	−0.517582	−0.0763133
$47$	−9.67305	−1.41096	−0.705479	−	0.708730i	$-0.749268\pi$
$47$	−9.67305	−1.41096	−0.705479	+	0.708730i	$0.749268\pi$
$48$	4.64391	0.670290
$49$	−1.08380	−0.154829
$50$	−1.37279	−0.194141
$51$	1.28594	0.180068
$52$	0.559766	0.0776256
$53$	−12.7465	−1.75087	−0.875435	−	0.483336i	$-0.839425\pi$
$53$	−12.7465	−1.75087	−0.875435	+	0.483336i	$0.839425\pi$
$54$	−7.58937	−1.03278
$55$	−1.00000	−0.134840
$56$	7.06364	0.943919
$57$	−1.23648	−0.163776
$58$	−6.18698	−0.812390
$59$	1.32111	0.171994	0.0859968	−	0.996295i	$-0.472593\pi$
$59$	1.32111	0.171994	0.0859968	+	0.996295i	$0.472593\pi$
$60$	0.142763	0.0184307
$61$	−5.58779	−0.715443	−0.357721	−	0.933828i	$-0.616446\pi$
$61$	−5.58779	−0.715443	−0.357721	+	0.933828i	$0.616446\pi$
$62$	−6.53915	−0.830472
$63$	−3.57824	−0.450816
$64$	8.40698	1.05087
$65$	−4.84815	−0.601339
$66$	−1.69742	−0.208938
$67$	−1.20728	−0.147493	−0.0737465	−	0.997277i	$-0.523496\pi$
$67$	−1.20728	−0.147493	−0.0737465	+	0.997277i	$0.523496\pi$
$68$	0.120079	0.0145617
$69$	−0.466190	−0.0561227
$70$	−3.33906	−0.399094
$71$	−0.259538	−0.0308015	−0.0154007	−	0.999881i	$-0.504902\pi$
$71$	−0.259538	−0.0308015	−0.0154007	+	0.999881i	$0.504902\pi$
$72$	−4.27224	−0.503488
$73$	−6.27686	−0.734651	−0.367325	−	0.930093i	$-0.619726\pi$
$73$	−6.27686	−0.734651	−0.367325	+	0.930093i	$0.619726\pi$
$74$	−4.13797	−0.481029
$75$	−1.23648	−0.142776
$76$	−0.115460	−0.0132441
$77$	−2.43232	−0.277189
$78$	−8.22935	−0.931790
$79$	−9.16926	−1.03162	−0.515811	−	0.856702i	$-0.672509\pi$
$79$	−9.16926	−1.03162	−0.515811	+	0.856702i	$0.672509\pi$
$80$	−3.75575	−0.419906
$81$	−2.42245	−0.269161
$82$	−1.08512	−0.119831
$83$	−14.2430	−1.56337	−0.781686	−	0.623673i	$-0.785640\pi$
$83$	−14.2430	−1.56337	−0.781686	+	0.623673i	$0.785640\pi$
$84$	0.347247	0.0378877
$85$	−1.04001	−0.112804
$86$	10.2517	1.10546
$87$	−5.57266	−0.597452
$88$	−2.90407	−0.309575
$89$	−1.23983	−0.131421	−0.0657107	−	0.997839i	$-0.520931\pi$
$89$	−1.23983	−0.131421	−0.0657107	+	0.997839i	$0.520931\pi$
$90$	2.01953	0.212877
$91$	−11.7923	−1.23617
$92$	−0.0435318	−0.00453851
$93$	−5.88986	−0.610750
$94$	13.2790	1.36963
$95$	1.00000	0.102598
$96$	0.806563	0.0823195
$97$	2.95633	0.300170	0.150085	−	0.988673i	$-0.452045\pi$
$97$	2.95633	0.300170	0.150085	+	0.988673i	$0.452045\pi$
$98$	1.48783	0.150294
$99$	1.47112	0.147853
$100$	−0.115460	−0.0115460
$101$	−1.55010	−0.154241	−0.0771204	−	0.997022i	$-0.524573\pi$
$101$	−1.55010	−0.154241	−0.0771204	+	0.997022i	$0.524573\pi$
$102$	−1.76533	−0.174793
$103$	−0.339313	−0.0334335	−0.0167168	−	0.999860i	$-0.505321\pi$
$103$	−0.339313	−0.0334335	−0.0167168	+	0.999860i	$0.505321\pi$
$104$	−14.0794	−1.38060
$105$	−3.00752	−0.293504
$106$	17.4982	1.69958
$107$	−9.05124	−0.875017	−0.437508	−	0.899214i	$-0.644139\pi$
$107$	−9.05124	−0.875017	−0.437508	+	0.899214i	$0.644139\pi$
$108$	−0.638313	−0.0614217
$109$	14.7515	1.41294	0.706471	−	0.707742i	$-0.250286\pi$
$109$	14.7515	1.41294	0.706471	+	0.707742i	$0.250286\pi$
$110$	1.37279	0.130890
$111$	−3.72710	−0.353761
$112$	−9.13520	−0.863195
$113$	−4.71030	−0.443108	−0.221554	−	0.975148i	$-0.571113\pi$
$113$	−4.71030	−0.443108	−0.221554	+	0.975148i	$0.571113\pi$
$114$	1.69742	0.158978
$115$	0.377030	0.0351583
$116$	−0.520363	−0.0483145
$117$	7.13221	0.659373
$118$	−1.81360	−0.166955
$119$	−2.52963	−0.231891
$120$	−3.59082	−0.327796
$121$	1.00000	0.0909091
$122$	7.67083	0.694485
$123$	−0.977376	−0.0881270
$124$	−0.549982	−0.0493899
$125$	1.00000	0.0894427
$126$	4.91215	0.437610
$127$	14.7860	1.31205	0.656024	−	0.754740i	$-0.272237\pi$
$127$	14.7860	1.31205	0.656024	+	0.754740i	$0.272237\pi$
$128$	−10.2364	−0.904775
$129$	9.23374	0.812986
$130$	6.65547	0.583723
$131$	−7.82065	−0.683294	−0.341647	−	0.939828i	$-0.610985\pi$
$131$	−7.82065	−0.683294	−0.341647	+	0.939828i	$0.610985\pi$
$132$	−0.142763	−0.0124260
$133$	2.43232	0.210909
$134$	1.65734	0.143172
$135$	5.52845	0.475813
$136$	−3.02025	−0.258984
$137$	−7.70625	−0.658389	−0.329195	−	0.944262i	$-0.606777\pi$
$137$	−7.70625	−0.658389	−0.329195	+	0.944262i	$0.606777\pi$
$138$	0.639979	0.0544786
$139$	14.3953	1.22099	0.610495	−	0.792020i	$-0.290970\pi$
$139$	14.3953	1.22099	0.610495	+	0.792020i	$0.290970\pi$
$140$	−0.280835	−0.0237349
$141$	11.9605	1.00726
$142$	0.356290	0.0298992
$143$	4.84815	0.405423
$144$	5.52516	0.460430
$145$	4.50688	0.374276
$146$	8.61678	0.713130
$147$	1.34010	0.110530
$148$	−0.348028	−0.0286078
$149$	−13.8399	−1.13381	−0.566906	−	0.823783i	$-0.691860\pi$
$149$	−13.8399	−1.13381	−0.566906	+	0.823783i	$0.691860\pi$
$150$	1.69742	0.138594
$151$	−8.10281	−0.659398	−0.329699	−	0.944086i	$-0.606947\pi$
$151$	−8.10281	−0.659398	−0.329699	+	0.944086i	$0.606947\pi$
$152$	2.90407	0.235551
$153$	1.52997	0.123691
$154$	3.33906	0.269069
$155$	4.76341	0.382607
$156$	−0.692139	−0.0554155
$157$	−7.13378	−0.569338	−0.284669	−	0.958626i	$-0.591884\pi$
$157$	−7.13378	−0.569338	−0.284669	+	0.958626i	$0.591884\pi$
$158$	12.5874	1.00140
$159$	15.7608	1.24991
$160$	−0.652306	−0.0515693
$161$	0.917060	0.0722744
$162$	3.32550	0.261276
$163$	−23.8705	−1.86968	−0.934840	−	0.355068i	$-0.884458\pi$
$163$	−23.8705	−1.86968	−0.934840	+	0.355068i	$0.884458\pi$
$164$	−0.0912652	−0.00712661
$165$	1.23648	0.0962598
$166$	19.5526	1.51757
$167$	−8.49981	−0.657735	−0.328868	−	0.944376i	$-0.606667\pi$
$167$	−8.49981	−0.657735	−0.328868	+	0.944376i	$0.606667\pi$
$168$	−8.73405	−0.673846
$169$	10.5046	0.808043
$170$	1.42770	0.109500
$171$	−1.47112	−0.112499
$172$	0.862227	0.0657442
$173$	−3.76732	−0.286424	−0.143212	−	0.989692i	$-0.545743\pi$
$173$	−3.76732	−0.286424	−0.143212	+	0.989692i	$0.545743\pi$
$174$	7.65007	0.579950
$175$	2.43232	0.183866
$176$	3.75575	0.283100
$177$	−1.63352	−0.122783
$178$	1.70202	0.127571
$179$	−12.0321	−0.899323	−0.449661	−	0.893199i	$-0.648455\pi$
$179$	−12.0321	−0.899323	−0.449661	+	0.893199i	$0.648455\pi$
$180$	0.169855	0.0126602
$181$	−9.01903	−0.670380	−0.335190	−	0.942151i	$-0.608800\pi$
$181$	−9.01903	−0.670380	−0.335190	+	0.942151i	$0.608800\pi$
$182$	16.1883	1.19995
$183$	6.90918	0.510741
$184$	1.09492	0.0807188
$185$	3.01428	0.221615
$186$	8.08552	0.592859
$187$	1.04001	0.0760527
$188$	1.11685	0.0814544
$189$	13.4470	0.978123
$190$	−1.37279	−0.0995923
$191$	−15.2598	−1.10416	−0.552079	−	0.833792i	$-0.686165\pi$
$191$	−15.2598	−1.10416	−0.552079	+	0.833792i	$0.686165\pi$
$192$	−10.3950	−0.750198
$193$	3.17246	0.228359	0.114179	−	0.993460i	$-0.463576\pi$
$193$	3.17246	0.228359	0.114179	+	0.993460i	$0.463576\pi$
$194$	−4.05841	−0.291377
$195$	5.99464	0.429285
$196$	0.125136	0.00893827
$197$	−6.09397	−0.434178	−0.217089	−	0.976152i	$-0.569656\pi$
$197$	−6.09397	−0.434178	−0.217089	+	0.976152i	$0.569656\pi$
$198$	−2.01953	−0.143522
$199$	−15.8148	−1.12108	−0.560541	−	0.828127i	$-0.689407\pi$
$199$	−15.8148	−1.12108	−0.560541	+	0.828127i	$0.689407\pi$
$200$	2.90407	0.205349
$201$	1.49278	0.105292
$202$	2.12796	0.149723
$203$	10.9622	0.769395
$204$	−0.148475	−0.0103953
$205$	0.790451	0.0552075
$206$	0.465805	0.0324541
$207$	−0.554657	−0.0385513
$208$	18.2084	1.26253
$209$	−1.00000	−0.0691714
$210$	4.12867	0.284906
$211$	10.4371	0.718522	0.359261	−	0.933237i	$-0.383029\pi$
$211$	10.4371	0.718522	0.359261	+	0.933237i	$0.383029\pi$
$212$	1.47171	0.101077
$213$	0.320913	0.0219886
$214$	12.4254	0.849384
$215$	−7.46777	−0.509298
$216$	16.0550	1.09240
$217$	11.5862	0.786520
$218$	−20.2507	−1.37155
$219$	7.76120	0.524453
$220$	0.115460	0.00778429
$221$	5.04210	0.339168
$222$	5.11651	0.343397
$223$	−15.2161	−1.01895	−0.509474	−	0.860486i	$-0.670160\pi$
$223$	−15.2161	−1.01895	−0.509474	+	0.860486i	$0.670160\pi$
$224$	−1.58662	−0.106010
$225$	−1.47112	−0.0980747
$226$	6.46623	0.430127
$227$	−15.4170	−1.02326	−0.511630	−	0.859206i	$-0.670958\pi$
$227$	−15.4170	−1.02326	−0.511630	+	0.859206i	$0.670958\pi$
$228$	0.142763	0.00945474
$229$	18.2529	1.20618	0.603091	−	0.797672i	$-0.293935\pi$
$229$	18.2529	1.20618	0.603091	+	0.797672i	$0.293935\pi$
$230$	−0.517582	−0.0341283
$231$	3.00752	0.197880
$232$	13.0883	0.859289
$233$	19.1041	1.25155	0.625776	−	0.780003i	$-0.284782\pi$
$233$	19.1041	1.25155	0.625776	+	0.780003i	$0.284782\pi$
$234$	−9.79100	−0.640057
$235$	−9.67305	−0.631000
$236$	−0.152535	−0.00992917
$237$	11.3376	0.736456
$238$	3.47264	0.225098
$239$	20.6551	1.33607	0.668034	−	0.744131i	$-0.267136\pi$
$239$	20.6551	1.33607	0.668034	+	0.744131i	$0.267136\pi$
$240$	4.64391	0.299763
$241$	15.9963	1.03042	0.515208	−	0.857065i	$-0.327715\pi$
$241$	15.9963	1.03042	0.515208	+	0.857065i	$0.327715\pi$
$242$	−1.37279	−0.0882460
$243$	−13.5900	−0.871801
$244$	0.645164	0.0413024
$245$	−1.08380	−0.0692417
$246$	1.34173	0.0855454
$247$	−4.84815	−0.308480
$248$	13.8333	0.878416
$249$	17.6112	1.11606
$250$	−1.37279	−0.0868226
$251$	−11.3379	−0.715639	−0.357820	−	0.933791i	$-0.616480\pi$
$251$	−11.3379	−0.715639	−0.357820	+	0.933791i	$0.616480\pi$
$252$	0.413142	0.0260255
$253$	−0.377030	−0.0237037
$254$	−20.2981	−1.27361
$255$	1.28594	0.0805290
$256$	−2.76162	−0.172601
$257$	9.80023	0.611322	0.305661	−	0.952140i	$-0.401123\pi$
$257$	9.80023	0.611322	0.305661	+	0.952140i	$0.401123\pi$
$258$	−12.6759	−0.789170
$259$	7.33171	0.455570
$260$	0.559766	0.0347152
$261$	−6.63016	−0.410397
$262$	10.7361	0.663277
$263$	−21.2899	−1.31279	−0.656395	−	0.754418i	$-0.727919\pi$
$263$	−21.2899	−1.31279	−0.656395	+	0.754418i	$0.727919\pi$
$264$	3.59082	0.221000
$265$	−12.7465	−0.783013
$266$	−3.33906	−0.204731
$267$	1.53302	0.0938192
$268$	0.139392	0.00851475
$269$	−18.2788	−1.11448	−0.557240	−	0.830352i	$-0.688140\pi$
$269$	−18.2788	−1.11448	−0.557240	+	0.830352i	$0.688140\pi$
$270$	−7.58937	−0.461874
$271$	19.7872	1.20199	0.600994	−	0.799253i	$-0.294772\pi$
$271$	19.7872	1.20199	0.600994	+	0.799253i	$0.294772\pi$
$272$	3.90600	0.236836
$273$	14.5809	0.882476
$274$	10.5790	0.639103
$275$	−1.00000	−0.0603023
$276$	0.0538262	0.00323995
$277$	24.5658	1.47602	0.738008	−	0.674792i	$-0.235767\pi$
$277$	24.5658	1.47602	0.738008	+	0.674792i	$0.235767\pi$
$278$	−19.7616	−1.18522
$279$	−7.00755	−0.419531
$280$	7.06364	0.422133
$281$	14.0898	0.840524	0.420262	−	0.907403i	$-0.361938\pi$
$281$	14.0898	0.840524	0.420262	+	0.907403i	$0.361938\pi$
$282$	−16.4192	−0.977751
$283$	−18.8367	−1.11973	−0.559863	−	0.828585i	$-0.689146\pi$
$283$	−18.8367	−1.11973	−0.559863	+	0.828585i	$0.689146\pi$
$284$	0.0299662	0.00177816
$285$	−1.23648	−0.0732427
$286$	−6.65547	−0.393546
$287$	1.92263	0.113489
$288$	0.959621	0.0565462
$289$	−15.9184	−0.936376
$290$	−6.18698	−0.363312
$291$	−3.65544	−0.214286
$292$	0.724724	0.0424113
$293$	2.08270	0.121673	0.0608364	−	0.998148i	$-0.480623\pi$
$293$	2.08270	0.121673	0.0608364	+	0.998148i	$0.480623\pi$
$294$	−1.83967	−0.107292
$295$	1.32111	0.0769179
$296$	8.75370	0.508798
$297$	−5.52845	−0.320793
$298$	18.9993	1.10060
$299$	−1.82790	−0.105710
$300$	0.142763	0.00824245
$301$	−18.1640	−1.04696
$302$	11.1234	0.640081
$303$	1.91667	0.110110
$304$	−3.75575	−0.215407
$305$	−5.58779	−0.319956
$306$	−2.10032	−0.120068
$307$	19.8442	1.13257	0.566284	−	0.824210i	$-0.308381\pi$
$307$	19.8442	1.13257	0.566284	+	0.824210i	$0.308381\pi$
$308$	0.280835	0.0160021
$309$	0.419554	0.0238676
$310$	−6.53915	−0.371399
$311$	3.71814	0.210836	0.105418	−	0.994428i	$-0.466382\pi$
$311$	3.71814	0.210836	0.105418	+	0.994428i	$0.466382\pi$
$312$	17.4089	0.985582
$313$	27.0866	1.53102	0.765511	−	0.643423i	$-0.222486\pi$
$313$	27.0866	1.53102	0.765511	+	0.643423i	$0.222486\pi$
$314$	9.79316	0.552660
$315$	−3.57824	−0.201611
$316$	1.05868	0.0595554
$317$	9.50445	0.533823	0.266911	−	0.963721i	$-0.413997\pi$
$317$	9.50445	0.533823	0.266911	+	0.963721i	$0.413997\pi$
$318$	−21.6362	−1.21330
$319$	−4.50688	−0.252337
$320$	8.40698	0.469964
$321$	11.1917	0.624658
$322$	−1.25893	−0.0701572
$323$	−1.04001	−0.0578674
$324$	0.279695	0.0155386
$325$	−4.84815	−0.268927
$326$	32.7691	1.81491
$327$	−18.2400	−1.00867
$328$	2.29553	0.126749
$329$	−23.5280	−1.29714
$330$	−1.69742	−0.0934399
$331$	7.27869	0.400073	0.200036	−	0.979788i	$-0.435894\pi$
$331$	7.27869	0.400073	0.200036	+	0.979788i	$0.435894\pi$
$332$	1.64449	0.0902532
$333$	−4.43437	−0.243002
$334$	11.6684	0.638467
$335$	−1.20728	−0.0659609
$336$	11.2955	0.616219
$337$	29.3790	1.60037	0.800187	−	0.599750i	$-0.204733\pi$
$337$	29.3790	1.60037	0.800187	+	0.599750i	$0.204733\pi$
$338$	−14.4205	−0.784372
$339$	5.82419	0.316326
$340$	0.120079	0.00651218
$341$	−4.76341	−0.257953
$342$	2.01953	0.109204
$343$	−19.6624	−1.06167
$344$	−21.6870	−1.16928
$345$	−0.466190	−0.0250988
$346$	5.17173	0.278034
$347$	28.1028	1.50864	0.754318	−	0.656509i	$-0.227968\pi$
$347$	28.1028	1.50864	0.754318	+	0.656509i	$0.227968\pi$
$348$	0.643418	0.0344908
$349$	20.9948	1.12383	0.561913	−	0.827196i	$-0.310065\pi$
$349$	20.9948	1.12383	0.561913	+	0.827196i	$0.310065\pi$
$350$	−3.33906	−0.178480
$351$	−26.8027	−1.43062
$352$	0.652306	0.0347680
$353$	15.1874	0.808345	0.404172	−	0.914683i	$-0.367560\pi$
$353$	15.1874	0.808345	0.404172	+	0.914683i	$0.367560\pi$
$354$	2.24248	0.119186
$355$	−0.259538	−0.0137748
$356$	0.143150	0.00758693
$357$	3.12783	0.165542
$358$	16.5175	0.872978
$359$	11.5028	0.607093	0.303546	−	0.952817i	$-0.401829\pi$
$359$	11.5028	0.607093	0.303546	+	0.952817i	$0.401829\pi$
$360$	−4.27224	−0.225167
$361$	1.00000	0.0526316
$362$	12.3812	0.650741
$363$	−1.23648	−0.0648983
$364$	1.36153	0.0713637
$365$	−6.27686	−0.328546
$366$	−9.48483	−0.495780
$367$	−1.52355	−0.0795284	−0.0397642	−	0.999209i	$-0.512661\pi$
$367$	−1.52355	−0.0795284	−0.0397642	+	0.999209i	$0.512661\pi$
$368$	−1.41603	−0.0738158
$369$	−1.16285	−0.0605354
$370$	−4.13797	−0.215123
$371$	−31.0037	−1.60963
$372$	0.680042	0.0352585
$373$	16.5646	0.857682	0.428841	−	0.903380i	$-0.358922\pi$
$373$	16.5646	0.857682	0.428841	+	0.903380i	$0.358922\pi$
$374$	−1.42770	−0.0738248
$375$	−1.23648	−0.0638515
$376$	−28.0912	−1.44869
$377$	−21.8500	−1.12533
$378$	−18.4598	−0.949470
$379$	−0.0190602	−0.000979058 0	−0.000489529	−	1.00000i	$-0.500156\pi$
$379$	−0.0190602	−0.000979058 0	−0.000489529		1.00000i	$0.500156\pi$
$380$	−0.115460	−0.00592296
$381$	−18.2826	−0.936647
$382$	20.9484	1.07181
$383$	18.2085	0.930412	0.465206	−	0.885202i	$-0.345980\pi$
$383$	18.2085	0.930412	0.465206	+	0.885202i	$0.345980\pi$
$384$	12.6570	0.645902
$385$	−2.43232	−0.123963
$386$	−4.35511	−0.221669
$387$	10.9860	0.558449
$388$	−0.341337	−0.0173288
$389$	−31.4031	−1.59220	−0.796099	−	0.605166i	$-0.793107\pi$
$389$	−31.4031	−1.59220	−0.796099	+	0.605166i	$0.793107\pi$
$390$	−8.22935	−0.416709
$391$	−0.392114	−0.0198300
$392$	−3.14745	−0.158970
$393$	9.67007	0.487791
$394$	8.36572	0.421459
$395$	−9.16926	−0.461355
$396$	−0.169855	−0.00853554
$397$	−25.8619	−1.29797	−0.648985	−	0.760801i	$-0.724806\pi$
$397$	−25.8619	−1.29797	−0.648985	+	0.760801i	$0.724806\pi$
$398$	21.7103	1.08824
$399$	−3.00752	−0.150564
$400$	−3.75575	−0.187787
$401$	−15.5141	−0.774737	−0.387369	−	0.921925i	$-0.626616\pi$
$401$	−15.5141	−0.774737	−0.387369	+	0.921925i	$0.626616\pi$
$402$	−2.04926	−0.102208
$403$	−23.0938	−1.15038
$404$	0.178974	0.00890430
$405$	−2.42245	−0.120372
$406$	−15.0487	−0.746856
$407$	−3.01428	−0.149412
$408$	3.73448	0.184884
$409$	20.7451	1.02578	0.512889	−	0.858455i	$-0.328575\pi$
$409$	20.7451	1.02578	0.512889	+	0.858455i	$0.328575\pi$
$410$	−1.08512	−0.0535902
$411$	9.52862	0.470012
$412$	0.0391770	0.00193011
$413$	3.21336	0.158119
$414$	0.761425	0.0374220
$415$	−14.2430	−0.699161
$416$	3.16248	0.155053
$417$	−17.7994	−0.871643
$418$	1.37279	0.0671451
$419$	6.94955	0.339508	0.169754	−	0.985487i	$-0.445703\pi$
$419$	6.94955	0.339508	0.169754	+	0.985487i	$0.445703\pi$
$420$	0.347247	0.0169439
$421$	17.0486	0.830897	0.415448	−	0.909617i	$-0.363625\pi$
$421$	17.0486	0.830897	0.415448	+	0.909617i	$0.363625\pi$
$422$	−14.3280	−0.697474
$423$	14.2302	0.691897
$424$	−37.0168	−1.79770
$425$	−1.04001	−0.0504477
$426$	−0.440545	−0.0213445
$427$	−13.5913	−0.657729
$428$	1.04505	0.0505146
$429$	−5.99464	−0.289424
$430$	10.2517	0.494378
$431$	10.6174	0.511421	0.255711	−	0.966753i	$-0.417691\pi$
$431$	10.6174	0.511421	0.255711	+	0.966753i	$0.417691\pi$
$432$	−20.7635	−0.998982
$433$	−5.21952	−0.250834	−0.125417	−	0.992104i	$-0.540027\pi$
$433$	−5.21952	−0.250834	−0.125417	+	0.992104i	$0.540027\pi$
$434$	−15.9053	−0.763480
$435$	−5.57266	−0.267189
$436$	−1.70321	−0.0815689
$437$	0.377030	0.0180358
$438$	−10.6545	−0.509090
$439$	−25.8096	−1.23182	−0.615912	−	0.787815i	$-0.711212\pi$
$439$	−25.8096	−1.23182	−0.615912	+	0.787815i	$0.711212\pi$
$440$	−2.90407	−0.138446
$441$	1.59441	0.0759241
$442$	−6.92172	−0.329233
$443$	−30.3917	−1.44395	−0.721977	−	0.691917i	$-0.756766\pi$
$443$	−30.3917	−1.44395	−0.721977	+	0.691917i	$0.756766\pi$
$444$	0.430330	0.0204225
$445$	−1.23983	−0.0587734
$446$	20.8885	0.989099
$447$	17.1128	0.809407
$448$	20.4485	0.966100
$449$	10.8574	0.512394	0.256197	−	0.966625i	$-0.417530\pi$
$449$	10.8574	0.512394	0.256197	+	0.966625i	$0.417530\pi$
$450$	2.01953	0.0952017
$451$	−0.790451	−0.0372209
$452$	0.543850	0.0255805
$453$	10.0190	0.470732
$454$	21.1642	0.993285
$455$	−11.7923	−0.552830
$456$	−3.59082	−0.168156
$457$	18.3849	0.860008	0.430004	−	0.902827i	$-0.358512\pi$
$457$	18.3849	0.860008	0.430004	+	0.902827i	$0.358512\pi$
$458$	−25.0573	−1.17085
$459$	−5.74961	−0.268369
$460$	−0.0435318	−0.00202968
$461$	−28.2662	−1.31649	−0.658245	−	0.752804i	$-0.728701\pi$
$461$	−28.2662	−1.31649	−0.658245	+	0.752804i	$0.728701\pi$
$462$	−4.12867	−0.192083
$463$	33.4266	1.55346	0.776732	−	0.629831i	$-0.216876\pi$
$463$	33.4266	1.55346	0.776732	+	0.629831i	$0.216876\pi$
$464$	−16.9267	−0.785803
$465$	−5.88986	−0.273136
$466$	−26.2258	−1.21489
$467$	−1.21297	−0.0561297	−0.0280648	−	0.999606i	$-0.508934\pi$
$467$	−1.21297	−0.0561297	−0.0280648	+	0.999606i	$0.508934\pi$
$468$	−0.823483	−0.0380655
$469$	−2.93650	−0.135595
$470$	13.2790	0.612515
$471$	8.82077	0.406440
$472$	3.83659	0.176594
$473$	7.46777	0.343369
$474$	−15.5641	−0.714882
$475$	1.00000	0.0458831
$476$	0.292070	0.0133870
$477$	18.7517	0.858580
$478$	−28.3550	−1.29693
$479$	−25.9098	−1.18385	−0.591923	−	0.805994i	$-0.701631\pi$
$479$	−25.9098	−1.18385	−0.591923	+	0.805994i	$0.701631\pi$
$480$	0.806563	0.0368144
$481$	−14.6137	−0.666327
$482$	−21.9596	−1.00023
$483$	−1.13392	−0.0515954
$484$	−0.115460	−0.00524817
$485$	2.95633	0.134240
$486$	18.6562	0.846262
$487$	−3.09538	−0.140265	−0.0701326	−	0.997538i	$-0.522342\pi$
$487$	−3.09538	−0.140265	−0.0701326	+	0.997538i	$0.522342\pi$
$488$	−16.2273	−0.734577
$489$	29.5154	1.33473
$490$	1.48783	0.0672134
$491$	17.4687	0.788352	0.394176	−	0.919035i	$-0.371030\pi$
$491$	17.4687	0.788352	0.394176	+	0.919035i	$0.371030\pi$
$492$	0.112847	0.00508756
$493$	−4.68718	−0.211100
$494$	6.65547	0.299444
$495$	1.47112	0.0661219
$496$	−17.8902	−0.803293
$497$	−0.631280	−0.0283168
$498$	−24.1763	−1.08337
$499$	−6.47276	−0.289760	−0.144880	−	0.989449i	$-0.546280\pi$
$499$	−6.47276	−0.289760	−0.144880	+	0.989449i	$0.546280\pi$
$500$	−0.115460	−0.00516351
$501$	10.5098	0.469545
$502$	15.5644	0.694675
$503$	−15.5931	−0.695262	−0.347631	−	0.937631i	$-0.613014\pi$
$503$	−15.5931	−0.695262	−0.347631	+	0.937631i	$0.613014\pi$
$504$	−10.3915	−0.462873
$505$	−1.55010	−0.0689786
$506$	0.517582	0.0230093
$507$	−12.9887	−0.576847
$508$	−1.70719	−0.0757444
$509$	6.14404	0.272330	0.136165	−	0.990686i	$-0.456522\pi$
$509$	6.14404	0.272330	0.136165	+	0.990686i	$0.456522\pi$
$510$	−1.76533	−0.0781699
$511$	−15.2673	−0.675388
$512$	24.2638	1.07232
$513$	5.52845	0.244087
$514$	−13.4536	−0.593414
$515$	−0.339313	−0.0149519
$516$	−1.06613	−0.0469335
$517$	9.67305	0.425420
$518$	−10.0649	−0.442225
$519$	4.65822	0.204473
$520$	−14.0794	−0.617422
$521$	41.4684	1.81676	0.908382	−	0.418141i	$-0.137318\pi$
$521$	41.4684	1.81676	0.908382	+	0.418141i	$0.137318\pi$
$522$	9.10179	0.398375
$523$	2.76287	0.120812	0.0604059	−	0.998174i	$-0.480760\pi$
$523$	2.76287	0.120812	0.0604059	+	0.998174i	$0.480760\pi$
$524$	0.902970	0.0394464
$525$	−3.00752	−0.131259
$526$	29.2264	1.27433
$527$	−4.95398	−0.215799
$528$	−4.64391	−0.202100
$529$	−22.8578	−0.993819
$530$	17.4982	0.760075
$531$	−1.94351	−0.0843411
$532$	−0.280835	−0.0121758
$533$	−3.83222	−0.165992
$534$	−2.10451	−0.0910709
$535$	−9.05124	−0.391319
$536$	−3.50603	−0.151438
$537$	14.8775	0.642010
$538$	25.0929	1.08183
$539$	1.08380	0.0466828
$540$	−0.638313	−0.0274686
$541$	29.6566	1.27504	0.637519	−	0.770435i	$-0.279961\pi$
$541$	29.6566	1.27504	0.637519	+	0.770435i	$0.279961\pi$
$542$	−27.1636	−1.16678
$543$	11.1518	0.478572
$544$	0.678402	0.0290862
$545$	14.7515	0.631886
$546$	−20.0164	−0.856624
$547$	−21.7318	−0.929184	−0.464592	−	0.885525i	$-0.653799\pi$
$547$	−21.7318	−0.929184	−0.464592	+	0.885525i	$0.653799\pi$
$548$	0.889761	0.0380087
$549$	8.22031	0.350834
$550$	1.37279	0.0585358
$551$	4.50688	0.192000
$552$	−1.35385	−0.0576237
$553$	−22.3026	−0.948403
$554$	−33.7236	−1.43278
$555$	−3.72710	−0.158207
$556$	−1.66207	−0.0704876
$557$	8.08244	0.342464	0.171232	−	0.985231i	$-0.445225\pi$
$557$	8.08244	0.342464	0.171232	+	0.985231i	$0.445225\pi$
$558$	9.61987	0.407242
$559$	36.2049	1.53130
$560$	−9.13520	−0.386033
$561$	−1.28594	−0.0542926
$562$	−19.3422	−0.815902
$563$	−25.7685	−1.08601	−0.543006	−	0.839729i	$-0.682714\pi$
$563$	−25.7685	−1.08601	−0.543006	+	0.839729i	$0.682714\pi$
$564$	−1.38096	−0.0581488
$565$	−4.71030	−0.198164
$566$	25.8588	1.08693
$567$	−5.89217	−0.247448
$568$	−0.753717	−0.0316253
$569$	−19.2047	−0.805102	−0.402551	−	0.915398i	$-0.631876\pi$
$569$	−19.2047	−0.805102	−0.402551	+	0.915398i	$0.631876\pi$
$570$	1.69742	0.0710971
$571$	−28.0284	−1.17295	−0.586477	−	0.809966i	$-0.699486\pi$
$571$	−28.0284	−1.17295	−0.586477	+	0.809966i	$0.699486\pi$
$572$	−0.559766	−0.0234050
$573$	18.8684	0.788237
$574$	−2.63936	−0.110165
$575$	0.377030	0.0157233
$576$	−12.3677	−0.515320
$577$	−34.8547	−1.45102	−0.725509	−	0.688212i	$-0.758396\pi$
$577$	−34.8547	−1.45102	−0.725509	+	0.688212i	$0.758396\pi$
$578$	21.8525	0.908946
$579$	−3.92268	−0.163021
$580$	−0.520363	−0.0216069
$581$	−34.6436	−1.43726
$582$	5.01813	0.208008
$583$	12.7465	0.527907
$584$	−18.2285	−0.754299
$585$	7.13221	0.294881
$586$	−2.85910	−0.118109
$587$	7.98152	0.329433	0.164716	−	0.986341i	$-0.447329\pi$
$587$	7.98152	0.329433	0.164716	+	0.986341i	$0.447329\pi$
$588$	−0.154728	−0.00638086
$589$	4.76341	0.196273
$590$	−1.81360	−0.0746647
$591$	7.53507	0.309951
$592$	−11.3209	−0.465286
$593$	11.6876	0.479951	0.239976	−	0.970779i	$-0.422861\pi$
$593$	11.6876	0.479951	0.239976	+	0.970779i	$0.422861\pi$
$594$	7.58937	0.311396
$595$	−2.52963	−0.103705
$596$	1.59796	0.0654548
$597$	19.5547	0.800320
$598$	2.50931	0.102614
$599$	−20.3132	−0.829973	−0.414986	−	0.909828i	$-0.636214\pi$
$599$	−20.3132	−0.829973	−0.414986	+	0.909828i	$0.636214\pi$
$600$	−3.59082	−0.146595
$601$	32.4805	1.32491	0.662453	−	0.749104i	$-0.269516\pi$
$601$	32.4805	1.32491	0.662453	+	0.749104i	$0.269516\pi$
$602$	24.9353	1.01629
$603$	1.77606	0.0723266
$604$	0.935548	0.0380669
$605$	1.00000	0.0406558
$606$	−2.63117	−0.106884
$607$	−45.4701	−1.84557	−0.922787	−	0.385311i	$-0.874094\pi$
$607$	−45.4701	−1.84557	−0.922787	+	0.385311i	$0.874094\pi$
$608$	−0.652306	−0.0264545
$609$	−13.5545	−0.549257
$610$	7.67083	0.310583
$611$	46.8964	1.89722
$612$	−0.176650	−0.00714066
$613$	4.06460	0.164168	0.0820839	−	0.996625i	$-0.473842\pi$
$613$	4.06460	0.164168	0.0820839	+	0.996625i	$0.473842\pi$
$614$	−27.2418	−1.09939
$615$	−0.977376	−0.0394116
$616$	−7.06364	−0.284602
$617$	−27.7980	−1.11911	−0.559554	−	0.828794i	$-0.689027\pi$
$617$	−27.7980	−1.11911	−0.559554	+	0.828794i	$0.689027\pi$
$618$	−0.575958	−0.0231684
$619$	31.8288	1.27931	0.639654	−	0.768663i	$-0.279077\pi$
$619$	31.8288	1.27931	0.639654	+	0.768663i	$0.279077\pi$
$620$	−0.549982	−0.0220878
$621$	2.08439	0.0836438
$622$	−5.10420	−0.204660
$623$	−3.01566	−0.120820
$624$	−22.5144	−0.901295
$625$	1.00000	0.0400000
$626$	−37.1840	−1.48617
$627$	1.23648	0.0493802
$628$	0.823665	0.0328678
$629$	−3.13487	−0.124995
$630$	4.91215	0.195705
$631$	32.8526	1.30784	0.653920	−	0.756564i	$-0.273123\pi$
$631$	32.8526	1.30784	0.653920	+	0.756564i	$0.273123\pi$
$632$	−26.6282	−1.05921
$633$	−12.9053	−0.512940
$634$	−13.0476	−0.518185
$635$	14.7860	0.586766
$636$	−1.81974	−0.0721573
$637$	5.25445	0.208189
$638$	6.18698	0.244945
$639$	0.381811	0.0151042
$640$	−10.2364	−0.404628
$641$	31.2727	1.23520	0.617599	−	0.786493i	$-0.288106\pi$
$641$	31.2727	1.23520	0.617599	+	0.786493i	$0.288106\pi$
$642$	−15.3638	−0.606359
$643$	−13.7579	−0.542557	−0.271279	−	0.962501i	$-0.587446\pi$
$643$	−13.7579	−0.542557	−0.271279	+	0.962501i	$0.587446\pi$
$644$	−0.105883	−0.00417239
$645$	9.23374	0.363578
$646$	1.42770	0.0561723
$647$	20.3525	0.800140	0.400070	−	0.916485i	$-0.368986\pi$
$647$	20.3525	0.800140	0.400070	+	0.916485i	$0.368986\pi$
$648$	−7.03496	−0.276359
$649$	−1.32111	−0.0518580
$650$	6.65547	0.261049
$651$	−14.3260	−0.561482
$652$	2.75608	0.107936
$653$	−4.04869	−0.158437	−0.0792187	−	0.996857i	$-0.525243\pi$
$653$	−4.04869	−0.158437	−0.0792187	+	0.996857i	$0.525243\pi$
$654$	25.0396	0.979124
$655$	−7.82065	−0.305578
$656$	−2.96873	−0.115910
$657$	9.23401	0.360253
$658$	32.2989	1.25914
$659$	−13.0892	−0.509882	−0.254941	−	0.966957i	$-0.582056\pi$
$659$	−13.0892	−0.509882	−0.254941	+	0.966957i	$0.582056\pi$
$660$	−0.142763	−0.00555706
$661$	9.55227	0.371540	0.185770	−	0.982593i	$-0.440522\pi$
$661$	9.55227	0.371540	0.185770	+	0.982593i	$0.440522\pi$
$662$	−9.99208	−0.388353
$663$	−6.23445	−0.242126
$664$	−41.3627	−1.60518
$665$	2.43232	0.0943214
$666$	6.08744	0.235884
$667$	1.69923	0.0657945
$668$	0.981386	0.0379710
$669$	18.8144	0.727408
$670$	1.65734	0.0640286
$671$	5.58779	0.215714
$672$	1.96182	0.0756789
$673$	31.4730	1.21320	0.606598	−	0.795009i	$-0.292534\pi$
$673$	31.4730	1.21320	0.606598	+	0.795009i	$0.292534\pi$
$674$	−40.3310	−1.55349
$675$	5.52845	0.212790
$676$	−1.21285	−0.0466482
$677$	−35.4776	−1.36352	−0.681758	−	0.731578i	$-0.738784\pi$
$677$	−35.4776	−1.36352	−0.681758	+	0.731578i	$0.738784\pi$
$678$	−7.99536	−0.307060
$679$	7.19075	0.275956
$680$	−3.02025	−0.115821
$681$	19.0628	0.730487
$682$	6.53915	0.250397
$683$	−40.6216	−1.55434	−0.777171	−	0.629289i	$-0.783346\pi$
$683$	−40.6216	−1.55434	−0.777171	+	0.629289i	$0.783346\pi$
$684$	0.169855	0.00649457
$685$	−7.70625	−0.294441
$686$	26.9923	1.03057
$687$	−22.5693	−0.861072
$688$	28.0471	1.06929
$689$	61.7971	2.35428
$690$	0.639979	0.0243636
$691$	−12.5025	−0.475616	−0.237808	−	0.971312i	$-0.576429\pi$
$691$	−12.5025	−0.475616	−0.237808	+	0.971312i	$0.576429\pi$
$692$	0.434974	0.0165352
$693$	3.57824	0.135926
$694$	−38.5791	−1.46444
$695$	14.3953	0.546044
$696$	−16.1834	−0.613431
$697$	−0.822073	−0.0311382
$698$	−28.8214	−1.09091
$699$	−23.6218	−0.893460
$700$	−0.280835	−0.0106146
$701$	−22.1578	−0.836887	−0.418443	−	0.908243i	$-0.637424\pi$
$701$	−22.1578	−0.836887	−0.418443	+	0.908243i	$0.637424\pi$
$702$	36.7944	1.38872
$703$	3.01428	0.113686
$704$	−8.40698	−0.316850
$705$	11.9605	0.450459
$706$	−20.8491	−0.784665
$707$	−3.77035	−0.141798
$708$	0.188606	0.00708825
$709$	−32.1717	−1.20824	−0.604118	−	0.796895i	$-0.706474\pi$
$709$	−32.1717	−1.20824	−0.604118	+	0.796895i	$0.706474\pi$
$710$	0.356290	0.0133713
$711$	13.4891	0.505880
$712$	−3.60055	−0.134936
$713$	1.79595	0.0672589
$714$	−4.29384	−0.160693
$715$	4.84815	0.181311
$716$	1.38922	0.0519178
$717$	−25.5396	−0.953794
$718$	−15.7908	−0.589308
$719$	4.63511	0.172860	0.0864302	−	0.996258i	$-0.472454\pi$
$719$	4.63511	0.172860	0.0864302	+	0.996258i	$0.472454\pi$
$720$	5.52516	0.205910
$721$	−0.825320	−0.0307365
$722$	−1.37279	−0.0510898
$723$	−19.7791	−0.735594
$724$	1.04134	0.0387009
$725$	4.50688	0.167381
$726$	1.69742	0.0629972
$727$	−29.6755	−1.10060	−0.550302	−	0.834966i	$-0.685487\pi$
$727$	−29.6755	−1.10060	−0.550302	+	0.834966i	$0.685487\pi$
$728$	−34.2456	−1.26923
$729$	24.0711	0.891523
$730$	8.61678	0.318921
$731$	7.76652	0.287255
$732$	−0.797732	−0.0294850
$733$	36.0642	1.33206	0.666031	−	0.745924i	$-0.267992\pi$
$733$	36.0642	1.33206	0.666031	+	0.745924i	$0.267992\pi$
$734$	2.09150	0.0771987
$735$	1.34010	0.0494304
$736$	−0.245939	−0.00906544
$737$	1.20728	0.0444708
$738$	1.59634	0.0587621
$739$	13.1048	0.482066	0.241033	−	0.970517i	$-0.422514\pi$
$739$	13.1048	0.482066	0.241033	+	0.970517i	$0.422514\pi$
$740$	−0.348028	−0.0127938
$741$	5.99464	0.220218
$742$	42.5614	1.56248
$743$	49.4215	1.81310	0.906550	−	0.422098i	$-0.138706\pi$
$743$	49.4215	1.81310	0.906550	+	0.422098i	$0.138706\pi$
$744$	−17.1046	−0.627085
$745$	−13.8399	−0.507056
$746$	−22.7396	−0.832557
$747$	20.9531	0.766636
$748$	−0.120079	−0.00439051
$749$	−22.0155	−0.804431
$750$	1.69742	0.0619810
$751$	5.10832	0.186405	0.0932026	−	0.995647i	$-0.470290\pi$
$751$	5.10832	0.186405	0.0932026	+	0.995647i	$0.470290\pi$
$752$	36.3295	1.32480
$753$	14.0190	0.510882
$754$	29.9954	1.09237
$755$	−8.10281	−0.294892
$756$	−1.55258	−0.0564669
$757$	−38.3696	−1.39457	−0.697283	−	0.716796i	$-0.745608\pi$
$757$	−38.3696	−1.39457	−0.697283	+	0.716796i	$0.745608\pi$
$758$	0.0261656	0.000950378 0
$759$	0.466190	0.0169216
$760$	2.90407	0.105342
$761$	−34.9845	−1.26819	−0.634093	−	0.773257i	$-0.718626\pi$
$761$	−34.9845	−1.26819	−0.634093	+	0.773257i	$0.718626\pi$
$762$	25.0981	0.909209
$763$	35.8805	1.29896
$764$	1.76189	0.0637428
$765$	1.52997	0.0553163
$766$	−24.9964	−0.903157
$767$	−6.40493	−0.231269
$768$	3.41468	0.123217
$769$	4.72143	0.170259	0.0851296	−	0.996370i	$-0.472870\pi$
$769$	4.72143	0.170259	0.0851296	+	0.996370i	$0.472870\pi$
$770$	3.33906	0.120331
$771$	−12.1178	−0.436411
$772$	−0.366292	−0.0131831
$773$	47.5946	1.71186	0.855930	−	0.517092i	$-0.172986\pi$
$773$	47.5946	1.71186	0.855930	+	0.517092i	$0.172986\pi$
$774$	−15.0814	−0.542090
$775$	4.76341	0.171107
$776$	8.58539	0.308198
$777$	−9.06551	−0.325223
$778$	43.1097	1.54556
$779$	0.790451	0.0283208
$780$	−0.692139	−0.0247825
$781$	0.259538	0.00928700
$782$	0.538288	0.0192491
$783$	24.9160	0.890427
$784$	4.07050	0.145375
$785$	−7.13378	−0.254616
$786$	−13.2749	−0.473501
$787$	1.02158	0.0364154	0.0182077	−	0.999834i	$-0.494204\pi$
$787$	1.02158	0.0364154	0.0182077	+	0.999834i	$0.494204\pi$
$788$	0.703608	0.0250650
$789$	26.3245	0.937176
$790$	12.5874	0.447841
$791$	−11.4570	−0.407363
$792$	4.27224	0.151807
$793$	27.0904	0.962010
$794$	35.5028	1.25995
$795$	15.7608	0.558978
$796$	1.82597	0.0647199
$797$	−37.8469	−1.34061	−0.670304	−	0.742087i	$-0.733836\pi$
$797$	−37.8469	−1.34061	−0.670304	+	0.742087i	$0.733836\pi$
$798$	4.12867	0.146154
$799$	10.0600	0.355898
$800$	−0.652306	−0.0230625
$801$	1.82393	0.0644455
$802$	21.2975	0.752042
$803$	6.27686	0.221506
$804$	−0.172356	−0.00607852
$805$	0.917060	0.0323221
$806$	31.7028	1.11668
$807$	22.6014	0.795606
$808$	−4.50161	−0.158366
$809$	25.6686	0.902461	0.451231	−	0.892407i	$-0.350985\pi$
$809$	25.6686	0.902461	0.451231	+	0.892407i	$0.350985\pi$
$810$	3.32550	0.116846
$811$	−6.16852	−0.216606	−0.108303	−	0.994118i	$-0.534542\pi$
$811$	−6.16852	−0.216606	−0.108303	+	0.994118i	$0.534542\pi$
$812$	−1.26569	−0.0444170
$813$	−24.4665	−0.858077
$814$	4.13797	0.145036
$815$	−23.8705	−0.836147
$816$	−4.82969	−0.169073
$817$	−7.46777	−0.261264
$818$	−28.4785	−0.995729
$819$	17.3478	0.606183
$820$	−0.0912652	−0.00318712
$821$	−30.9910	−1.08159	−0.540796	−	0.841154i	$-0.681877\pi$
$821$	−30.9910	−1.08159	−0.540796	+	0.841154i	$0.681877\pi$
$822$	−13.0807	−0.456243
$823$	−40.7706	−1.42117	−0.710587	−	0.703609i	$-0.751571\pi$
$823$	−40.7706	−1.42117	−0.710587	+	0.703609i	$0.751571\pi$
$824$	−0.985391	−0.0343277
$825$	1.23648	0.0430487
$826$	−4.41126	−0.153487
$827$	46.2940	1.60980	0.804900	−	0.593410i	$-0.202219\pi$
$827$	46.2940	1.60980	0.804900	+	0.593410i	$0.202219\pi$
$828$	0.0640405	0.00222556
$829$	−14.8471	−0.515660	−0.257830	−	0.966190i	$-0.583007\pi$
$829$	−14.8471	−0.515660	−0.257830	+	0.966190i	$0.583007\pi$
$830$	19.5526	0.678680
$831$	−30.3751	−1.05370
$832$	−40.7583	−1.41304
$833$	1.12716	0.0390539
$834$	24.4348	0.846109
$835$	−8.49981	−0.294148
$836$	0.115460	0.00399326
$837$	26.3343	0.910246
$838$	−9.54024	−0.329562
$839$	36.8854	1.27343	0.636713	−	0.771101i	$-0.280294\pi$
$839$	36.8854	1.27343	0.636713	+	0.771101i	$0.280294\pi$
$840$	−8.73405	−0.301353
$841$	−8.68803	−0.299587
$842$	−23.4040	−0.806556
$843$	−17.4217	−0.600035
$844$	−1.20507	−0.0414802
$845$	10.5046	0.361368
$846$	−19.5350	−0.671628
$847$	2.43232	0.0835756
$848$	47.8728	1.64396
$849$	23.2912	0.799352
$850$	1.42770	0.0489698
$851$	1.13648	0.0389579
$852$	−0.0370525	−0.00126940
$853$	9.84716	0.337160	0.168580	−	0.985688i	$-0.446082\pi$
$853$	9.84716	0.337160	0.168580	+	0.985688i	$0.446082\pi$
$854$	18.6579	0.638462
$855$	−1.47112	−0.0503112
$856$	−26.2855	−0.898419
$857$	−25.6874	−0.877464	−0.438732	−	0.898618i	$-0.644572\pi$
$857$	−25.6874	−0.877464	−0.438732	+	0.898618i	$0.644572\pi$
$858$	8.22935	0.280945
$859$	48.4118	1.65179	0.825894	−	0.563825i	$-0.190671\pi$
$859$	48.4118	1.65179	0.825894	+	0.563825i	$0.190671\pi$
$860$	0.862227	0.0294017
$861$	−2.37729	−0.0810179
$862$	−14.5754	−0.496439
$863$	15.3028	0.520913	0.260457	−	0.965486i	$-0.416127\pi$
$863$	15.3028	0.520913	0.260457	+	0.965486i	$0.416127\pi$
$864$	−3.60624	−0.122687
$865$	−3.76732	−0.128093
$866$	7.16528	0.243486
$867$	19.6828	0.668461
$868$	−1.33773	−0.0454057
$869$	9.16926	0.311046
$870$	7.65007	0.259362
$871$	5.85308	0.198324
$872$	42.8395	1.45073
$873$	−4.34911	−0.147195
$874$	−0.517582	−0.0175075
$875$	2.43232	0.0822275
$876$	−0.896106	−0.0302766
$877$	−34.7213	−1.17245	−0.586227	−	0.810147i	$-0.699387\pi$
$877$	−34.7213	−1.17245	−0.586227	+	0.810147i	$0.699387\pi$
$878$	35.4310	1.19574
$879$	−2.57522	−0.0868600
$880$	3.75575	0.126606
$881$	33.7901	1.13842	0.569208	−	0.822194i	$-0.307250\pi$
$881$	33.7901	1.13842	0.569208	+	0.822194i	$0.307250\pi$
$882$	−2.18878	−0.0737000
$883$	54.4286	1.83167	0.915835	−	0.401556i	$-0.131530\pi$
$883$	54.4286	1.83167	0.915835	+	0.401556i	$0.131530\pi$
$884$	−0.582160	−0.0195801
$885$	−1.63352	−0.0549103
$886$	41.7213	1.40166
$887$	14.5748	0.489373	0.244686	−	0.969602i	$-0.421315\pi$
$887$	14.5748	0.489373	0.244686	+	0.969602i	$0.421315\pi$
$888$	−10.8238	−0.363222
$889$	35.9644	1.20621
$890$	1.70202	0.0570517
$891$	2.42245	0.0811550
$892$	1.75685	0.0588237
$893$	−9.67305	−0.323696
$894$	−23.4922	−0.785696
$895$	−12.0321	−0.402189
$896$	−24.8981	−0.831789
$897$	2.26016	0.0754645
$898$	−14.9049	−0.497384
$899$	21.4681	0.716003
$900$	0.169855	0.00566184
$901$	13.2565	0.441636
$902$	1.08512	0.0361305
$903$	22.4594	0.747404
$904$	−13.6791	−0.454959
$905$	−9.01903	−0.299803
$906$	−13.7539	−0.456942
$907$	−36.7115	−1.21898	−0.609492	−	0.792792i	$-0.708627\pi$
$907$	−36.7115	−1.21898	−0.609492	+	0.792792i	$0.708627\pi$
$908$	1.78004	0.0590727
$909$	2.28038	0.0756356
$910$	16.1883	0.536635
$911$	18.5321	0.613997	0.306999	−	0.951710i	$-0.400675\pi$
$911$	18.5321	0.613997	0.306999	+	0.951710i	$0.400675\pi$
$912$	4.64391	0.153775
$913$	14.2430	0.471374
$914$	−25.2385	−0.834815
$915$	6.90918	0.228411
$916$	−2.10747	−0.0696328
$917$	−19.0224	−0.628173
$918$	7.89298	0.260507
$919$	−30.1650	−0.995052	−0.497526	−	0.867449i	$-0.665758\pi$
$919$	−30.1650	−0.995052	−0.497526	+	0.867449i	$0.665758\pi$
$920$	1.09492	0.0360986
$921$	−24.5369	−0.808519
$922$	38.8035	1.27792
$923$	1.25828	0.0414167
$924$	−0.347247	−0.0114236
$925$	3.01428	0.0991090
$926$	−45.8875	−1.50796
$927$	0.499171	0.0163949
$928$	−2.93987	−0.0965058
$929$	−53.1253	−1.74298	−0.871492	−	0.490409i	$-0.836847\pi$
$929$	−53.1253	−1.74298	−0.871492	+	0.490409i	$0.836847\pi$
$930$	8.08552	0.265135
$931$	−1.08380	−0.0355203
$932$	−2.20575	−0.0722519
$933$	−4.59740	−0.150512
$934$	1.66515	0.0544854
$935$	1.04001	0.0340118
$936$	20.7125	0.677008
$937$	46.9097	1.53247	0.766237	−	0.642558i	$-0.222127\pi$
$937$	46.9097	1.53247	0.766237	+	0.642558i	$0.222127\pi$
$938$	4.03118	0.131623
$939$	−33.4919	−1.09297
$940$	1.11685	0.0364275
$941$	36.0038	1.17369	0.586845	−	0.809699i	$-0.300370\pi$
$941$	36.0038	1.17369	0.586845	+	0.809699i	$0.300370\pi$
$942$	−12.1090	−0.394534
$943$	0.298024	0.00970499
$944$	−4.96175	−0.161491
$945$	13.4470	0.437430
$946$	−10.2517	−0.333310
$947$	−21.3336	−0.693249	−0.346625	−	0.938004i	$-0.612672\pi$
$947$	−21.3336	−0.693249	−0.346625	+	0.938004i	$0.612672\pi$
$948$	−1.30904	−0.0425155
$949$	30.4312	0.987837
$950$	−1.37279	−0.0445390
$951$	−11.7520	−0.381086
$952$	−7.34622	−0.238093
$953$	0.777926	0.0251995	0.0125997	−	0.999921i	$-0.495989\pi$
$953$	0.777926	0.0251995	0.0125997	+	0.999921i	$0.495989\pi$
$954$	−25.7420	−0.833429
$955$	−15.2598	−0.493794
$956$	−2.38483	−0.0771310
$957$	5.57266	0.180139
$958$	35.5685	1.14917
$959$	−18.7441	−0.605278
$960$	−10.3950	−0.335499
$961$	−8.30988	−0.268061
$962$	20.0615	0.646808
$963$	13.3155	0.429085
$964$	−1.84693	−0.0594857
$965$	3.17246	0.102125
$966$	1.55664	0.0500839
$967$	41.5502	1.33616	0.668082	−	0.744088i	$-0.267116\pi$
$967$	41.5502	1.33616	0.668082	+	0.744088i	$0.267116\pi$
$968$	2.90407	0.0933404
$969$	1.28594	0.0413105
$970$	−4.05841	−0.130308
$971$	41.5722	1.33412	0.667058	−	0.745006i	$-0.267553\pi$
$971$	41.5722	1.33412	0.667058	+	0.745006i	$0.267553\pi$
$972$	1.56910	0.0503289
$973$	35.0139	1.12250
$974$	4.24930	0.136156
$975$	5.99464	0.191982
$976$	20.9863	0.671756
$977$	−4.43754	−0.141969	−0.0709847	−	0.997477i	$-0.522614\pi$
$977$	−4.43754	−0.141969	−0.0709847	+	0.997477i	$0.522614\pi$
$978$	−40.5183	−1.29563
$979$	1.23983	0.0396250
$980$	0.125136	0.00399732
$981$	−21.7013	−0.692869
$982$	−23.9808	−0.765258
$983$	4.98576	0.159021	0.0795106	−	0.996834i	$-0.474664\pi$
$983$	4.98576	0.159021	0.0795106	+	0.996834i	$0.474664\pi$
$984$	−2.83837	−0.0904839
$985$	−6.09397	−0.194170
$986$	6.43449	0.204916
$987$	29.0918	0.926004
$988$	0.559766	0.0178085
$989$	−2.81558	−0.0895301
$990$	−2.01953	−0.0641850
$991$	−39.9837	−1.27012	−0.635062	−	0.772461i	$-0.719025\pi$
$991$	−39.9837	−1.27012	−0.635062	+	0.772461i	$0.719025\pi$
$992$	−3.10720	−0.0986539
$993$	−8.99994	−0.285605
$994$	0.866612	0.0274873
$995$	−15.8148	−0.501363
$996$	−2.03338	−0.0644301
$997$	−31.3940	−0.994258	−0.497129	−	0.867677i	$-0.665612\pi$
$997$	−31.3940	−0.994258	−0.497129	+	0.867677i	$0.665612\pi$
$998$	8.88571	0.281272
$999$	16.6643	0.527235

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1045.2.a.d.1.3	✓	5
3.2	odd	2		9405.2.a.v.1.3		5
5.4	even	2		5225.2.a.j.1.3		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1045.2.a.d.1.3	✓	5	1.1	even	1	trivial
5225.2.a.j.1.3		5	5.4	even	2
9405.2.a.v.1.3		5	3.2	odd	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1045 = 5 \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1045.a (trivial)

\(f(q)\)	\(=\)	\(q-1.37279 q^{2} -1.23648 q^{3} -0.115460 q^{4} +1.00000 q^{5} +1.69742 q^{6} +2.43232 q^{7} +2.90407 q^{8} -1.47112 q^{9} -1.37279 q^{10} -1.00000 q^{11} +0.142763 q^{12} -4.84815 q^{13} -3.33906 q^{14} -1.23648 q^{15} -3.75575 q^{16} -1.04001 q^{17} +2.01953 q^{18} +1.00000 q^{19} -0.115460 q^{20} -3.00752 q^{21} +1.37279 q^{22} +0.377030 q^{23} -3.59082 q^{24} +1.00000 q^{25} +6.65547 q^{26} +5.52845 q^{27} -0.280835 q^{28} +4.50688 q^{29} +1.69742 q^{30} +4.76341 q^{31} -0.652306 q^{32} +1.23648 q^{33} +1.42770 q^{34} +2.43232 q^{35} +0.169855 q^{36} +3.01428 q^{37} -1.37279 q^{38} +5.99464 q^{39} +2.90407 q^{40} +0.790451 q^{41} +4.12867 q^{42} -7.46777 q^{43} +0.115460 q^{44} -1.47112 q^{45} -0.517582 q^{46} -9.67305 q^{47} +4.64391 q^{48} -1.08380 q^{49} -1.37279 q^{50} +1.28594 q^{51} +0.559766 q^{52} -12.7465 q^{53} -7.58937 q^{54} -1.00000 q^{55} +7.06364 q^{56} -1.23648 q^{57} -6.18698 q^{58} +1.32111 q^{59} +0.142763 q^{60} -5.58779 q^{61} -6.53915 q^{62} -3.57824 q^{63} +8.40698 q^{64} -4.84815 q^{65} -1.69742 q^{66} -1.20728 q^{67} +0.120079 q^{68} -0.466190 q^{69} -3.33906 q^{70} -0.259538 q^{71} -4.27224 q^{72} -6.27686 q^{73} -4.13797 q^{74} -1.23648 q^{75} -0.115460 q^{76} -2.43232 q^{77} -8.22935 q^{78} -9.16926 q^{79} -3.75575 q^{80} -2.42245 q^{81} -1.08512 q^{82} -14.2430 q^{83} +0.347247 q^{84} -1.04001 q^{85} +10.2517 q^{86} -5.57266 q^{87} -2.90407 q^{88} -1.23983 q^{89} +2.01953 q^{90} -11.7923 q^{91} -0.0435318 q^{92} -5.88986 q^{93} +13.2790 q^{94} +1.00000 q^{95} +0.806563 q^{96} +2.95633 q^{97} +1.48783 q^{98} +1.47112 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 3 q^{2} - 7 q^{3} + 5 q^{4} + 5 q^{5} + 2 q^{6} - 11 q^{7} + 3 q^{8} + 8 q^{9} - 3 q^{10} - 5 q^{11} - 7 q^{12} + q^{13} - 7 q^{15} - 3 q^{16} - 3 q^{17} - 7 q^{18} + 5 q^{19} + 5 q^{20} + 11 q^{21}+ \cdots - 8 q^{99}+O(q^{100}) \) 5 * q - 3 * q^2 - 7 * q^3 + 5 * q^4 + 5 * q^5 + 2 * q^6 - 11 * q^7 + 3 * q^8 + 8 * q^9 - 3 * q^10 - 5 * q^11 - 7 * q^12 + q^13 - 7 * q^15 - 3 * q^16 - 3 * q^17 - 7 * q^18 + 5 * q^19 + 5 * q^20 + 11 * q^21 + 3 * q^22 - 8 * q^23 + 9 * q^24 + 5 * q^25 - 16 * q^26 - 10 * q^27 - 22 * q^28 + 11 * q^29 + 2 * q^30 - 5 * q^31 - 2 * q^32 + 7 * q^33 + 4 * q^34 - 11 * q^35 - 3 * q^36 - 9 * q^37 - 3 * q^38 - 8 * q^39 + 3 * q^40 + 15 * q^41 + 11 * q^42 - 13 * q^43 - 5 * q^44 + 8 * q^45 + 18 * q^46 - 20 * q^47 - 20 * q^48 + 20 * q^49 - 3 * q^50 + 24 * q^51 + q^52 - 5 * q^53 + 17 * q^54 - 5 * q^55 - 7 * q^57 - 33 * q^58 - 17 * q^59 - 7 * q^60 + 3 * q^61 + 14 * q^62 - 22 * q^63 - 17 * q^64 + q^65 - 2 * q^66 - 28 * q^67 - 25 * q^68 - 2 * q^69 - 6 * q^71 - 26 * q^72 - 16 * q^73 - 21 * q^74 - 7 * q^75 + 5 * q^76 + 11 * q^77 + 29 * q^78 + 3 * q^79 - 3 * q^80 + q^81 + 2 * q^82 - 33 * q^83 + 33 * q^84 - 3 * q^85 + 10 * q^86 - 3 * q^88 - 16 * q^89 - 7 * q^90 - 22 * q^91 - 19 * q^92 - 26 * q^93 - 10 * q^94 + 5 * q^95 + 5 * q^96 - 14 * q^97 + 10 * q^98 - 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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