LMFDB - Embedded newform 8670.2.a.ce.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8670,2,Mod(1,8670)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8670, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("8670.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8670 = 2 \cdot 3 \cdot 5 \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8670.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:	$69.2302985525$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.204493248.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 24x^{4} + 189x^{2} - 487 $ x^6 - 24x^4 + 189x^2 - 487
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-3.08741$ of defining polynomial
Character	$\chi$	$=$	8670.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} -3.08741 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} -3.08741 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} +1.80714 q^{11} +1.00000 q^{12} -5.95687 q^{13} +3.08741 q^{14} +1.00000 q^{15} +1.00000 q^{16} -1.00000 q^{18} -8.33452 q^{19} +1.00000 q^{20} -3.08741 q^{21} -1.80714 q^{22} +4.61950 q^{23} -1.00000 q^{24} +1.00000 q^{25} +5.95687 q^{26} +1.00000 q^{27} -3.08741 q^{28} -9.88177 q^{29} -1.00000 q^{30} -8.72556 q^{31} -1.00000 q^{32} +1.80714 q^{33} -3.08741 q^{35} +1.00000 q^{36} -8.97729 q^{37} +8.33452 q^{38} -5.95687 q^{39} -1.00000 q^{40} +5.78913 q^{41} +3.08741 q^{42} +10.8211 q^{43} +1.80714 q^{44} +1.00000 q^{45} -4.61950 q^{46} +0.765251 q^{47} +1.00000 q^{48} +2.53209 q^{49} -1.00000 q^{50} -5.95687 q^{52} +1.70789 q^{53} -1.00000 q^{54} +1.80714 q^{55} +3.08741 q^{56} -8.33452 q^{57} +9.88177 q^{58} +8.71741 q^{59} +1.00000 q^{60} +5.38634 q^{61} +8.72556 q^{62} -3.08741 q^{63} +1.00000 q^{64} -5.95687 q^{65} -1.80714 q^{66} +11.1900 q^{67} +4.61950 q^{69} +3.08741 q^{70} +15.9282 q^{71} -1.00000 q^{72} -9.05308 q^{73} +8.97729 q^{74} +1.00000 q^{75} -8.33452 q^{76} -5.57938 q^{77} +5.95687 q^{78} -0.639279 q^{79} +1.00000 q^{80} +1.00000 q^{81} -5.78913 q^{82} +13.0462 q^{83} -3.08741 q^{84} -10.8211 q^{86} -9.88177 q^{87} -1.80714 q^{88} -6.44277 q^{89} -1.00000 q^{90} +18.3913 q^{91} +4.61950 q^{92} -8.72556 q^{93} -0.765251 q^{94} -8.33452 q^{95} -1.00000 q^{96} +14.8647 q^{97} -2.53209 q^{98} +1.80714 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{2} + 6 q^{3} + 6 q^{4} + 6 q^{5} - 6 q^{6} - 6 q^{8} + 6 q^{9}+O(q^{10}) $ 6 * q - 6 * q^2 + 6 * q^3 + 6 * q^4 + 6 * q^5 - 6 * q^6 - 6 * q^8 + 6 * q^9	$ 6 q - 6 q^{2} + 6 q^{3} + 6 q^{4} + 6 q^{5} - 6 q^{6} - 6 q^{8} + 6 q^{9} - 6 q^{10} + 6 q^{11} + 6 q^{12} + 6 q^{13} + 6 q^{15} + 6 q^{16} - 6 q^{18} - 6 q^{19} + 6 q^{20} - 6 q^{22} - 6 q^{24} + 6 q^{25} - 6 q^{26} + 6 q^{27} + 6 q^{29} - 6 q^{30} - 6 q^{32} + 6 q^{33} + 6 q^{36} + 6 q^{38} + 6 q^{39} - 6 q^{40} + 12 q^{41} + 24 q^{43} + 6 q^{44} + 6 q^{45} + 6 q^{47} + 6 q^{48} + 6 q^{49} - 6 q^{50} + 6 q^{52} - 6 q^{53} - 6 q^{54} + 6 q^{55} - 6 q^{57} - 6 q^{58} - 18 q^{59} + 6 q^{60} - 6 q^{61} + 6 q^{64} + 6 q^{65} - 6 q^{66} + 36 q^{67} + 30 q^{71} - 6 q^{72} + 12 q^{73} + 6 q^{75} - 6 q^{76} - 6 q^{77} - 6 q^{78} + 12 q^{79} + 6 q^{80} + 6 q^{81} - 12 q^{82} + 36 q^{83} - 24 q^{86} + 6 q^{87} - 6 q^{88} - 30 q^{89} - 6 q^{90} + 12 q^{91} - 6 q^{94} - 6 q^{95} - 6 q^{96} + 18 q^{97} - 6 q^{98} + 6 q^{99}+O(q^{100}) $ 6 * q - 6 * q^2 + 6 * q^3 + 6 * q^4 + 6 * q^5 - 6 * q^6 - 6 * q^8 + 6 * q^9 - 6 * q^10 + 6 * q^11 + 6 * q^12 + 6 * q^13 + 6 * q^15 + 6 * q^16 - 6 * q^18 - 6 * q^19 + 6 * q^20 - 6 * q^22 - 6 * q^24 + 6 * q^25 - 6 * q^26 + 6 * q^27 + 6 * q^29 - 6 * q^30 - 6 * q^32 + 6 * q^33 + 6 * q^36 + 6 * q^38 + 6 * q^39 - 6 * q^40 + 12 * q^41 + 24 * q^43 + 6 * q^44 + 6 * q^45 + 6 * q^47 + 6 * q^48 + 6 * q^49 - 6 * q^50 + 6 * q^52 - 6 * q^53 - 6 * q^54 + 6 * q^55 - 6 * q^57 - 6 * q^58 - 18 * q^59 + 6 * q^60 - 6 * q^61 + 6 * q^64 + 6 * q^65 - 6 * q^66 + 36 * q^67 + 30 * q^71 - 6 * q^72 + 12 * q^73 + 6 * q^75 - 6 * q^76 - 6 * q^77 - 6 * q^78 + 12 * q^79 + 6 * q^80 + 6 * q^81 - 12 * q^82 + 36 * q^83 - 24 * q^86 + 6 * q^87 - 6 * q^88 - 30 * q^89 - 6 * q^90 + 12 * q^91 - 6 * q^94 - 6 * q^95 - 6 * q^96 + 18 * q^97 - 6 * q^98 + 6 * 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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	−1.00000	−0.408248
$7$	−3.08741	−1.16693	−0.583465	−	0.812138i	$-0.698304\pi$
$7$	−3.08741	−1.16693	−0.583465	+	0.812138i	$0.698304\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	−1.00000	−0.316228
$11$	1.80714	0.544873	0.272437	−	0.962174i	$-0.412171\pi$
$11$	1.80714	0.544873	0.272437	+	0.962174i	$0.412171\pi$
$12$	1.00000	0.288675
$13$	−5.95687	−1.65214	−0.826069	−	0.563570i	$-0.809428\pi$
$13$	−5.95687	−1.65214	−0.826069	+	0.563570i	$0.809428\pi$
$14$	3.08741	0.825145
$15$	1.00000	0.258199
$16$	1.00000	0.250000
$17$	0	0
$17$	0	0
$18$	−1.00000	−0.235702
$19$	−8.33452	−1.91207	−0.956035	−	0.293253i	$-0.905262\pi$
$19$	−8.33452	−1.91207	−0.956035	+	0.293253i	$0.905262\pi$
$20$	1.00000	0.223607
$21$	−3.08741	−0.673728
$22$	−1.80714	−0.385283
$23$	4.61950	0.963232	0.481616	−	0.876382i	$-0.340050\pi$
$23$	4.61950	0.963232	0.481616	+	0.876382i	$0.340050\pi$
$24$	−1.00000	−0.204124
$25$	1.00000	0.200000
$26$	5.95687	1.16824
$27$	1.00000	0.192450
$28$	−3.08741	−0.583465
$29$	−9.88177	−1.83500	−0.917499	−	0.397737i	$-0.869795\pi$
$29$	−9.88177	−1.83500	−0.917499	+	0.397737i	$0.869795\pi$
$30$	−1.00000	−0.182574
$31$	−8.72556	−1.56716	−0.783579	−	0.621292i	$-0.786608\pi$
$31$	−8.72556	−1.56716	−0.783579	+	0.621292i	$0.786608\pi$
$32$	−1.00000	−0.176777
$33$	1.80714	0.314583
$34$	0	0
$35$	−3.08741	−0.521867
$36$	1.00000	0.166667
$37$	−8.97729	−1.47586	−0.737929	−	0.674878i	$-0.764196\pi$
$37$	−8.97729	−1.47586	−0.737929	+	0.674878i	$0.764196\pi$
$38$	8.33452	1.35204
$39$	−5.95687	−0.953862
$40$	−1.00000	−0.158114
$41$	5.78913	0.904110	0.452055	−	0.891990i	$-0.350691\pi$
$41$	5.78913	0.904110	0.452055	+	0.891990i	$0.350691\pi$
$42$	3.08741	0.476397
$43$	10.8211	1.65020	0.825100	−	0.564986i	$-0.191119\pi$
$43$	10.8211	1.65020	0.825100	+	0.564986i	$0.191119\pi$
$44$	1.80714	0.272437
$45$	1.00000	0.149071
$46$	−4.61950	−0.681108
$47$	0.765251	0.111623	0.0558116	−	0.998441i	$-0.482225\pi$
$47$	0.765251	0.111623	0.0558116	+	0.998441i	$0.482225\pi$
$48$	1.00000	0.144338
$49$	2.53209	0.361727
$50$	−1.00000	−0.141421
$51$	0	0
$52$	−5.95687	−0.826069
$53$	1.70789	0.234597	0.117299	−	0.993097i	$-0.462577\pi$
$53$	1.70789	0.234597	0.117299	+	0.993097i	$0.462577\pi$
$54$	−1.00000	−0.136083
$55$	1.80714	0.243675
$56$	3.08741	0.412572
$57$	−8.33452	−1.10393
$58$	9.88177	1.29754
$59$	8.71741	1.13491	0.567455	−	0.823405i	$-0.307928\pi$
$59$	8.71741	1.13491	0.567455	+	0.823405i	$0.307928\pi$
$60$	1.00000	0.129099
$61$	5.38634	0.689650	0.344825	−	0.938667i	$-0.387938\pi$
$61$	5.38634	0.689650	0.344825	+	0.938667i	$0.387938\pi$
$62$	8.72556	1.10815
$63$	−3.08741	−0.388977
$64$	1.00000	0.125000
$65$	−5.95687	−0.738858
$66$	−1.80714	−0.222444
$67$	11.1900	1.36708	0.683540	−	0.729913i	$-0.260440\pi$
$67$	11.1900	1.36708	0.683540	+	0.729913i	$0.260440\pi$
$68$	0	0
$69$	4.61950	0.556122
$70$	3.08741	0.369016
$71$	15.9282	1.89033	0.945167	−	0.326589i	$-0.105899\pi$
$71$	15.9282	1.89033	0.945167	+	0.326589i	$0.105899\pi$
$72$	−1.00000	−0.117851
$73$	−9.05308	−1.05958	−0.529791	−	0.848128i	$-0.677730\pi$
$73$	−9.05308	−1.05958	−0.529791	+	0.848128i	$0.677730\pi$
$74$	8.97729	1.04359
$75$	1.00000	0.115470
$76$	−8.33452	−0.956035
$77$	−5.57938	−0.635829
$78$	5.95687	0.674482
$79$	−0.639279	−0.0719245	−0.0359623	−	0.999353i	$-0.511450\pi$
$79$	−0.639279	−0.0719245	−0.0359623	+	0.999353i	$0.511450\pi$
$80$	1.00000	0.111803
$81$	1.00000	0.111111
$82$	−5.78913	−0.639302
$83$	13.0462	1.43200	0.716002	−	0.698098i	$-0.245970\pi$
$83$	13.0462	1.43200	0.716002	+	0.698098i	$0.245970\pi$
$84$	−3.08741	−0.336864
$85$	0	0
$86$	−10.8211	−1.16687
$87$	−9.88177	−1.05944
$88$	−1.80714	−0.192642
$89$	−6.44277	−0.682932	−0.341466	−	0.939894i	$-0.610924\pi$
$89$	−6.44277	−0.682932	−0.341466	+	0.939894i	$0.610924\pi$
$90$	−1.00000	−0.105409
$91$	18.3913	1.92793
$92$	4.61950	0.481616
$93$	−8.72556	−0.904799
$94$	−0.765251	−0.0789296
$95$	−8.33452	−0.855104
$96$	−1.00000	−0.102062
$97$	14.8647	1.50928	0.754641	−	0.656138i	$-0.227811\pi$
$97$	14.8647	1.50928	0.754641	+	0.656138i	$0.227811\pi$
$98$	−2.53209	−0.255780
$99$	1.80714	0.181624
$100$	1.00000	0.100000
$101$	15.3888	1.53125	0.765623	−	0.643289i	$-0.222431\pi$
$101$	15.3888	1.53125	0.765623	+	0.643289i	$0.222431\pi$
$102$	0	0
$103$	−1.83745	−0.181049	−0.0905246	−	0.995894i	$-0.528854\pi$
$103$	−1.83745	−0.181049	−0.0905246	+	0.995894i	$0.528854\pi$
$104$	5.95687	0.584119
$105$	−3.08741	−0.301300
$106$	−1.70789	−0.165885
$107$	14.9068	1.44109	0.720546	−	0.693407i	$-0.243891\pi$
$107$	14.9068	1.44109	0.720546	+	0.693407i	$0.243891\pi$
$108$	1.00000	0.0962250
$109$	1.39808	0.133912	0.0669560	−	0.997756i	$-0.478671\pi$
$109$	1.39808	0.133912	0.0669560	+	0.997756i	$0.478671\pi$
$110$	−1.80714	−0.172304
$111$	−8.97729	−0.852087
$112$	−3.08741	−0.291733
$113$	15.2757	1.43701	0.718507	−	0.695520i	$-0.244826\pi$
$113$	15.2757	1.43701	0.718507	+	0.695520i	$0.244826\pi$
$114$	8.33452	0.780599
$115$	4.61950	0.430770
$116$	−9.88177	−0.917499
$117$	−5.95687	−0.550712
$118$	−8.71741	−0.802502
$119$	0	0
$120$	−1.00000	−0.0912871
$121$	−7.73425	−0.703113
$122$	−5.38634	−0.487656
$123$	5.78913	0.521988
$124$	−8.72556	−0.783579
$125$	1.00000	0.0894427
$126$	3.08741	0.275048
$127$	−0.290811	−0.0258053	−0.0129027	−	0.999917i	$-0.504107\pi$
$127$	−0.290811	−0.0258053	−0.0129027	+	0.999917i	$0.504107\pi$
$128$	−1.00000	−0.0883883
$129$	10.8211	0.952744
$130$	5.95687	0.522452
$131$	1.73113	0.151249	0.0756246	−	0.997136i	$-0.475905\pi$
$131$	1.73113	0.151249	0.0756246	+	0.997136i	$0.475905\pi$
$132$	1.80714	0.157291
$133$	25.7321	2.23125
$134$	−11.1900	−0.966671
$135$	1.00000	0.0860663
$136$	0	0
$137$	−6.01833	−0.514181	−0.257090	−	0.966387i	$-0.582764\pi$
$137$	−6.01833	−0.514181	−0.257090	+	0.966387i	$0.582764\pi$
$138$	−4.61950	−0.393238
$139$	−7.07929	−0.600458	−0.300229	−	0.953867i	$-0.597063\pi$
$139$	−7.07929	−0.600458	−0.300229	+	0.953867i	$0.597063\pi$
$140$	−3.08741	−0.260934
$141$	0.765251	0.0644457
$142$	−15.9282	−1.33667
$143$	−10.7649	−0.900205
$144$	1.00000	0.0833333
$145$	−9.88177	−0.820636
$146$	9.05308	0.749238
$147$	2.53209	0.208843
$148$	−8.97729	−0.737929
$149$	−3.53675	−0.289742	−0.144871	−	0.989451i	$-0.546277\pi$
$149$	−3.53675	−0.289742	−0.144871	+	0.989451i	$0.546277\pi$
$150$	−1.00000	−0.0816497
$151$	−8.82481	−0.718153	−0.359076	−	0.933308i	$-0.616908\pi$
$151$	−8.82481	−0.718153	−0.359076	+	0.933308i	$0.616908\pi$
$152$	8.33452	0.676019
$153$	0	0
$154$	5.57938	0.449599
$155$	−8.72556	−0.700854
$156$	−5.95687	−0.476931
$157$	17.9931	1.43601	0.718005	−	0.696038i	$-0.245056\pi$
$157$	17.9931	1.43601	0.718005	+	0.696038i	$0.245056\pi$
$158$	0.639279	0.0508583
$159$	1.70789	0.135445
$160$	−1.00000	−0.0790569
$161$	−14.2623	−1.12402
$162$	−1.00000	−0.0785674
$163$	−4.85740	−0.380461	−0.190230	−	0.981739i	$-0.560924\pi$
$163$	−4.85740	−0.380461	−0.190230	+	0.981739i	$0.560924\pi$
$164$	5.78913	0.452055
$165$	1.80714	0.140686
$166$	−13.0462	−1.01258
$167$	14.3949	1.11391	0.556956	−	0.830542i	$-0.311969\pi$
$167$	14.3949	1.11391	0.556956	+	0.830542i	$0.311969\pi$
$168$	3.08741	0.238199
$169$	22.4842	1.72956
$170$	0	0
$171$	−8.33452	−0.637357
$172$	10.8211	0.825100
$173$	−4.94538	−0.375990	−0.187995	−	0.982170i	$-0.560199\pi$
$173$	−4.94538	−0.375990	−0.187995	+	0.982170i	$0.560199\pi$
$174$	9.88177	0.749135
$175$	−3.08741	−0.233386
$176$	1.80714	0.136218
$177$	8.71741	0.655240
$178$	6.44277	0.482906
$179$	−3.51711	−0.262881	−0.131441	−	0.991324i	$-0.541960\pi$
$179$	−3.51711	−0.262881	−0.131441	+	0.991324i	$0.541960\pi$
$180$	1.00000	0.0745356
$181$	22.0727	1.64065	0.820326	−	0.571896i	$-0.193792\pi$
$181$	22.0727	1.64065	0.820326	+	0.571896i	$0.193792\pi$
$182$	−18.3913	−1.36325
$183$	5.38634	0.398169
$184$	−4.61950	−0.340554
$185$	−8.97729	−0.660024
$186$	8.72556	0.639789
$187$	0	0
$188$	0.765251	0.0558116
$189$	−3.08741	−0.224576
$190$	8.33452	0.604649
$191$	−17.5621	−1.27075	−0.635376	−	0.772203i	$-0.719155\pi$
$191$	−17.5621	−1.27075	−0.635376	+	0.772203i	$0.719155\pi$
$192$	1.00000	0.0721688
$193$	−12.2609	−0.882560	−0.441280	−	0.897369i	$-0.645475\pi$
$193$	−12.2609	−0.882560	−0.441280	+	0.897369i	$0.645475\pi$
$194$	−14.8647	−1.06722
$195$	−5.95687	−0.426580
$196$	2.53209	0.180863
$197$	6.46320	0.460484	0.230242	−	0.973133i	$-0.426048\pi$
$197$	6.46320	0.460484	0.230242	+	0.973133i	$0.426048\pi$
$198$	−1.80714	−0.128428
$199$	−12.6004	−0.893216	−0.446608	−	0.894730i	$-0.647368\pi$
$199$	−12.6004	−0.893216	−0.446608	+	0.894730i	$0.647368\pi$
$200$	−1.00000	−0.0707107
$201$	11.1900	0.789284
$202$	−15.3888	−1.08276
$203$	30.5091	2.14132
$204$	0	0
$205$	5.78913	0.404330
$206$	1.83745	0.128021
$207$	4.61950	0.321077
$208$	−5.95687	−0.413034
$209$	−15.0616	−1.04184
$210$	3.08741	0.213051
$211$	3.35385	0.230889	0.115444	−	0.993314i	$-0.463171\pi$
$211$	3.35385	0.230889	0.115444	+	0.993314i	$0.463171\pi$
$212$	1.70789	0.117299
$213$	15.9282	1.09138
$214$	−14.9068	−1.01901
$215$	10.8211	0.737992
$216$	−1.00000	−0.0680414
$217$	26.9394	1.82876
$218$	−1.39808	−0.0946900
$219$	−9.05308	−0.611750
$220$	1.80714	0.121837
$221$	0	0
$222$	8.97729	0.602517
$223$	19.6916	1.31865	0.659324	−	0.751859i	$-0.270843\pi$
$223$	19.6916	1.31865	0.659324	+	0.751859i	$0.270843\pi$
$224$	3.08741	0.206286
$225$	1.00000	0.0666667
$226$	−15.2757	−1.01612
$227$	16.9010	1.12176	0.560880	−	0.827897i	$-0.310463\pi$
$227$	16.9010	1.12176	0.560880	+	0.827897i	$0.310463\pi$
$228$	−8.33452	−0.551967
$229$	7.06507	0.466873	0.233437	−	0.972372i	$-0.425003\pi$
$229$	7.06507	0.466873	0.233437	+	0.972372i	$0.425003\pi$
$230$	−4.61950	−0.304601
$231$	−5.57938	−0.367096
$232$	9.88177	0.648770
$233$	1.92951	0.126406	0.0632031	−	0.998001i	$-0.479868\pi$
$233$	1.92951	0.126406	0.0632031	+	0.998001i	$0.479868\pi$
$234$	5.95687	0.389412
$235$	0.765251	0.0499194
$236$	8.71741	0.567455
$237$	−0.639279	−0.0415257
$238$	0	0
$239$	−11.9374	−0.772166	−0.386083	−	0.922464i	$-0.626172\pi$
$239$	−11.9374	−0.772166	−0.386083	+	0.922464i	$0.626172\pi$
$240$	1.00000	0.0645497
$241$	4.77789	0.307771	0.153885	−	0.988089i	$-0.450821\pi$
$241$	4.77789	0.307771	0.153885	+	0.988089i	$0.450821\pi$
$242$	7.73425	0.497176
$243$	1.00000	0.0641500
$244$	5.38634	0.344825
$245$	2.53209	0.161769
$246$	−5.78913	−0.369101
$247$	49.6476	3.15900
$248$	8.72556	0.554074
$249$	13.0462	0.826768
$250$	−1.00000	−0.0632456
$251$	−19.9225	−1.25750	−0.628750	−	0.777608i	$-0.716433\pi$
$251$	−19.9225	−1.25750	−0.628750	+	0.777608i	$0.716433\pi$
$252$	−3.08741	−0.194488
$253$	8.34808	0.524839
$254$	0.290811	0.0182471
$255$	0	0
$256$	1.00000	0.0625000
$257$	−2.37593	−0.148207	−0.0741033	−	0.997251i	$-0.523609\pi$
$257$	−2.37593	−0.148207	−0.0741033	+	0.997251i	$0.523609\pi$
$258$	−10.8211	−0.673692
$259$	27.7166	1.72222
$260$	−5.95687	−0.369429
$261$	−9.88177	−0.611666
$262$	−1.73113	−0.106949
$263$	−27.7220	−1.70941	−0.854707	−	0.519111i	$-0.826263\pi$
$263$	−27.7220	−1.70941	−0.854707	+	0.519111i	$0.826263\pi$
$264$	−1.80714	−0.111222
$265$	1.70789	0.104915
$266$	−25.7321	−1.57773
$267$	−6.44277	−0.394291
$268$	11.1900	0.683540
$269$	8.15049	0.496944	0.248472	−	0.968639i	$-0.420072\pi$
$269$	8.15049	0.496944	0.248472	+	0.968639i	$0.420072\pi$
$270$	−1.00000	−0.0608581
$271$	−14.3589	−0.872240	−0.436120	−	0.899888i	$-0.643648\pi$
$271$	−14.3589	−0.872240	−0.436120	+	0.899888i	$0.643648\pi$
$272$	0	0
$273$	18.3913	1.11309
$274$	6.01833	0.363581
$275$	1.80714	0.108975
$276$	4.61950	0.278061
$277$	−5.17477	−0.310922	−0.155461	−	0.987842i	$-0.549686\pi$
$277$	−5.17477	−0.310922	−0.155461	+	0.987842i	$0.549686\pi$
$278$	7.07929	0.424588
$279$	−8.72556	−0.522386
$280$	3.08741	0.184508
$281$	27.9254	1.66589	0.832946	−	0.553354i	$-0.186652\pi$
$281$	27.9254	1.66589	0.832946	+	0.553354i	$0.186652\pi$
$282$	−0.765251	−0.0455700
$283$	−3.80624	−0.226258	−0.113129	−	0.993580i	$-0.536087\pi$
$283$	−3.80624	−0.226258	−0.113129	+	0.993580i	$0.536087\pi$
$284$	15.9282	0.945167
$285$	−8.33452	−0.493694
$286$	10.7649	0.636541
$287$	−17.8734	−1.05503
$288$	−1.00000	−0.0589256
$289$	0	0
$290$	9.88177	0.580277
$291$	14.8647	0.871384
$292$	−9.05308	−0.529791
$293$	−10.0310	−0.586018	−0.293009	−	0.956110i	$-0.594657\pi$
$293$	−10.0310	−0.586018	−0.293009	+	0.956110i	$0.594657\pi$
$294$	−2.53209	−0.147674
$295$	8.71741	0.507547
$296$	8.97729	0.521795
$297$	1.80714	0.104861
$298$	3.53675	0.204878
$299$	−27.5177	−1.59139
$300$	1.00000	0.0577350
$301$	−33.4091	−1.92567
$302$	8.82481	0.507811
$303$	15.3888	0.884066
$304$	−8.33452	−0.478017
$305$	5.38634	0.308421
$306$	0	0
$307$	−2.74331	−0.156569	−0.0782846	−	0.996931i	$-0.524944\pi$
$307$	−2.74331	−0.156569	−0.0782846	+	0.996931i	$0.524944\pi$
$308$	−5.57938	−0.317915
$309$	−1.83745	−0.104529
$310$	8.72556	0.495579
$311$	−11.2074	−0.635513	−0.317756	−	0.948172i	$-0.602929\pi$
$311$	−11.2074	−0.635513	−0.317756	+	0.948172i	$0.602929\pi$
$312$	5.95687	0.337241
$313$	23.4478	1.32535	0.662673	−	0.748909i	$-0.269422\pi$
$313$	23.4478	1.32535	0.662673	+	0.748909i	$0.269422\pi$
$314$	−17.9931	−1.01541
$315$	−3.08741	−0.173956
$316$	−0.639279	−0.0359623
$317$	−2.40030	−0.134814	−0.0674071	−	0.997726i	$-0.521473\pi$
$317$	−2.40030	−0.134814	−0.0674071	+	0.997726i	$0.521473\pi$
$318$	−1.70789	−0.0957740
$319$	−17.8577	−0.999841
$320$	1.00000	0.0559017
$321$	14.9068	0.832015
$322$	14.2623	0.794805
$323$	0	0
$324$	1.00000	0.0555556
$325$	−5.95687	−0.330427
$326$	4.85740	0.269027
$327$	1.39808	0.0773141
$328$	−5.78913	−0.319651
$329$	−2.36264	−0.130257
$330$	−1.80714	−0.0994798
$331$	−5.72448	−0.314646	−0.157323	−	0.987547i	$-0.550286\pi$
$331$	−5.72448	−0.314646	−0.157323	+	0.987547i	$0.550286\pi$
$332$	13.0462	0.716002
$333$	−8.97729	−0.491953
$334$	−14.3949	−0.787655
$335$	11.1900	0.611376
$336$	−3.08741	−0.168432
$337$	11.8506	0.645545	0.322773	−	0.946477i	$-0.395385\pi$
$337$	11.8506	0.645545	0.322773	+	0.946477i	$0.395385\pi$
$338$	−22.4842	−1.22298
$339$	15.2757	0.829660
$340$	0	0
$341$	−15.7683	−0.853902
$342$	8.33452	0.450679
$343$	13.7943	0.744820
$344$	−10.8211	−0.583434
$345$	4.61950	0.248705
$346$	4.94538	0.265865
$347$	−1.13688	−0.0610311	−0.0305156	−	0.999534i	$-0.509715\pi$
$347$	−1.13688	−0.0610311	−0.0305156	+	0.999534i	$0.509715\pi$
$348$	−9.88177	−0.529718
$349$	−20.9250	−1.12009	−0.560045	−	0.828462i	$-0.689216\pi$
$349$	−20.9250	−1.12009	−0.560045	+	0.828462i	$0.689216\pi$
$350$	3.08741	0.165029
$351$	−5.95687	−0.317954
$352$	−1.80714	−0.0963209
$353$	5.02419	0.267411	0.133705	−	0.991021i	$-0.457312\pi$
$353$	5.02419	0.267411	0.133705	+	0.991021i	$0.457312\pi$
$354$	−8.71741	−0.463325
$355$	15.9282	0.845383
$356$	−6.44277	−0.341466
$357$	0	0
$358$	3.51711	0.185885
$359$	−31.3545	−1.65483	−0.827413	−	0.561594i	$-0.810188\pi$
$359$	−31.3545	−1.65483	−0.827413	+	0.561594i	$0.810188\pi$
$360$	−1.00000	−0.0527046
$361$	50.4642	2.65601
$362$	−22.0727	−1.16012
$363$	−7.73425	−0.405943
$364$	18.3913	0.963965
$365$	−9.05308	−0.473860
$366$	−5.38634	−0.281548
$367$	−9.37526	−0.489385	−0.244692	−	0.969601i	$-0.578687\pi$
$367$	−9.37526	−0.489385	−0.244692	+	0.969601i	$0.578687\pi$
$368$	4.61950	0.240808
$369$	5.78913	0.301370
$370$	8.97729	0.466707
$371$	−5.27297	−0.273759
$372$	−8.72556	−0.452399
$373$	−8.20910	−0.425051	−0.212525	−	0.977156i	$-0.568169\pi$
$373$	−8.20910	−0.425051	−0.212525	+	0.977156i	$0.568169\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	−0.765251	−0.0394648
$377$	58.8644	3.03167
$378$	3.08741	0.158799
$379$	0.150095	0.00770986	0.00385493	−	0.999993i	$-0.498773\pi$
$379$	0.150095	0.00770986	0.00385493	+	0.999993i	$0.498773\pi$
$380$	−8.33452	−0.427552
$381$	−0.290811	−0.0148987
$382$	17.5621	0.898557
$383$	7.43165	0.379739	0.189870	−	0.981809i	$-0.439193\pi$
$383$	7.43165	0.379739	0.189870	+	0.981809i	$0.439193\pi$
$384$	−1.00000	−0.0510310
$385$	−5.57938	−0.284351
$386$	12.2609	0.624064
$387$	10.8211	0.550067
$388$	14.8647	0.754641
$389$	−6.50342	−0.329736	−0.164868	−	0.986316i	$-0.552720\pi$
$389$	−6.50342	−0.329736	−0.164868	+	0.986316i	$0.552720\pi$
$390$	5.95687	0.301638
$391$	0	0
$392$	−2.53209	−0.127890
$393$	1.73113	0.0873237
$394$	−6.46320	−0.325611
$395$	−0.639279	−0.0321656
$396$	1.80714	0.0908122
$397$	27.3742	1.37387	0.686936	−	0.726718i	$-0.258955\pi$
$397$	27.3742	1.37387	0.686936	+	0.726718i	$0.258955\pi$
$398$	12.6004	0.631599
$399$	25.7321	1.28821
$400$	1.00000	0.0500000
$401$	8.94064	0.446474	0.223237	−	0.974764i	$-0.428338\pi$
$401$	8.94064	0.446474	0.223237	+	0.974764i	$0.428338\pi$
$402$	−11.1900	−0.558108
$403$	51.9770	2.58916
$404$	15.3888	0.765623
$405$	1.00000	0.0496904
$406$	−30.5091	−1.51414
$407$	−16.2232	−0.804155
$408$	0	0
$409$	−29.7681	−1.47194	−0.735969	−	0.677015i	$-0.763273\pi$
$409$	−29.7681	−1.47194	−0.735969	+	0.677015i	$0.763273\pi$
$410$	−5.78913	−0.285905
$411$	−6.01833	−0.296862
$412$	−1.83745	−0.0905246
$413$	−26.9142	−1.32436
$414$	−4.61950	−0.227036
$415$	13.0462	0.640412
$416$	5.95687	0.292059
$417$	−7.07929	−0.346674
$418$	15.0616	0.736689
$419$	25.0180	1.22221	0.611104	−	0.791550i	$-0.290726\pi$
$419$	25.0180	1.22221	0.611104	+	0.791550i	$0.290726\pi$
$420$	−3.08741	−0.150650
$421$	−2.28209	−0.111222	−0.0556110	−	0.998453i	$-0.517711\pi$
$421$	−2.28209	−0.111222	−0.0556110	+	0.998453i	$0.517711\pi$
$422$	−3.35385	−0.163263
$423$	0.765251	0.0372078
$424$	−1.70789	−0.0829427
$425$	0	0
$426$	−15.9282	−0.771725
$427$	−16.6298	−0.804773
$428$	14.9068	0.720546
$429$	−10.7649	−0.519734
$430$	−10.8211	−0.521839
$431$	22.7448	1.09558	0.547788	−	0.836617i	$-0.315470\pi$
$431$	22.7448	1.09558	0.547788	+	0.836617i	$0.315470\pi$
$432$	1.00000	0.0481125
$433$	−18.1574	−0.872591	−0.436296	−	0.899803i	$-0.643710\pi$
$433$	−18.1574	−0.872591	−0.436296	+	0.899803i	$0.643710\pi$
$434$	−26.9394	−1.29313
$435$	−9.88177	−0.473795
$436$	1.39808	0.0669560
$437$	−38.5013	−1.84177
$438$	9.05308	0.432573
$439$	13.3730	0.638260	0.319130	−	0.947711i	$-0.396609\pi$
$439$	13.3730	0.638260	0.319130	+	0.947711i	$0.396609\pi$
$440$	−1.80714	−0.0861520
$441$	2.53209	0.120576
$442$	0	0
$443$	−17.4629	−0.829688	−0.414844	−	0.909893i	$-0.636164\pi$
$443$	−17.4629	−0.829688	−0.414844	+	0.909893i	$0.636164\pi$
$444$	−8.97729	−0.426044
$445$	−6.44277	−0.305417
$446$	−19.6916	−0.932424
$447$	−3.53675	−0.167283
$448$	−3.08741	−0.145866
$449$	−14.6209	−0.690004	−0.345002	−	0.938602i	$-0.612122\pi$
$449$	−14.6209	−0.690004	−0.345002	+	0.938602i	$0.612122\pi$
$450$	−1.00000	−0.0471405
$451$	10.4618	0.492625
$452$	15.2757	0.718507
$453$	−8.82481	−0.414626
$454$	−16.9010	−0.793204
$455$	18.3913	0.862196
$456$	8.33452	0.390300
$457$	28.3168	1.32461	0.662303	−	0.749236i	$-0.269579\pi$
$457$	28.3168	1.32461	0.662303	+	0.749236i	$0.269579\pi$
$458$	−7.06507	−0.330129
$459$	0	0
$460$	4.61950	0.215385
$461$	−18.2506	−0.850014	−0.425007	−	0.905190i	$-0.639728\pi$
$461$	−18.2506	−0.850014	−0.425007	+	0.905190i	$0.639728\pi$
$462$	5.57938	0.259576
$463$	7.07454	0.328782	0.164391	−	0.986395i	$-0.447434\pi$
$463$	7.07454	0.328782	0.164391	+	0.986395i	$0.447434\pi$
$464$	−9.88177	−0.458750
$465$	−8.72556	−0.404638
$466$	−1.92951	−0.0893827
$467$	20.9566	0.969754	0.484877	−	0.874582i	$-0.338864\pi$
$467$	20.9566	0.969754	0.484877	+	0.874582i	$0.338864\pi$
$468$	−5.95687	−0.275356
$469$	−34.5482	−1.59529
$470$	−0.765251	−0.0352984
$471$	17.9931	0.829080
$472$	−8.71741	−0.401251
$473$	19.5552	0.899150
$474$	0.639279	0.0293631
$475$	−8.33452	−0.382414
$476$	0	0
$477$	1.70789	0.0781991
$478$	11.9374	0.546004
$479$	−17.9922	−0.822085	−0.411043	−	0.911616i	$-0.634835\pi$
$479$	−17.9922	−0.822085	−0.411043	+	0.911616i	$0.634835\pi$
$480$	−1.00000	−0.0456435
$481$	53.4765	2.43832
$482$	−4.77789	−0.217627
$483$	−14.2623	−0.648956
$484$	−7.73425	−0.351557
$485$	14.8647	0.674971
$486$	−1.00000	−0.0453609
$487$	−21.5579	−0.976881	−0.488440	−	0.872597i	$-0.662434\pi$
$487$	−21.5579	−0.976881	−0.488440	+	0.872597i	$0.662434\pi$
$488$	−5.38634	−0.243828
$489$	−4.85740	−0.219659
$490$	−2.53209	−0.114388
$491$	−5.27264	−0.237951	−0.118976	−	0.992897i	$-0.537961\pi$
$491$	−5.27264	−0.237951	−0.118976	+	0.992897i	$0.537961\pi$
$492$	5.78913	0.260994
$493$	0	0
$494$	−49.6476	−2.23375
$495$	1.80714	0.0812249
$496$	−8.72556	−0.391789
$497$	−49.1769	−2.20589
$498$	−13.0462	−0.584613
$499$	−16.7839	−0.751350	−0.375675	−	0.926751i	$-0.622589\pi$
$499$	−16.7839	−0.751350	−0.375675	+	0.926751i	$0.622589\pi$
$500$	1.00000	0.0447214
$501$	14.3949	0.643118
$502$	19.9225	0.889186
$503$	0.629932	0.0280873	0.0140436	−	0.999901i	$-0.495530\pi$
$503$	0.629932	0.0280873	0.0140436	+	0.999901i	$0.495530\pi$
$504$	3.08741	0.137524
$505$	15.3888	0.684794
$506$	−8.34808	−0.371117
$507$	22.4842	0.998560
$508$	−0.290811	−0.0129027
$509$	1.19189	0.0528295	0.0264147	−	0.999651i	$-0.491591\pi$
$509$	1.19189	0.0528295	0.0264147	+	0.999651i	$0.491591\pi$
$510$	0	0
$511$	27.9505	1.23646
$512$	−1.00000	−0.0441942
$513$	−8.33452	−0.367978
$514$	2.37593	0.104798
$515$	−1.83745	−0.0809676
$516$	10.8211	0.476372
$517$	1.38291	0.0608205
$518$	−27.7166	−1.21780
$519$	−4.94538	−0.217078
$520$	5.95687	0.261226
$521$	21.1641	0.927214	0.463607	−	0.886041i	$-0.346555\pi$
$521$	21.1641	0.927214	0.463607	+	0.886041i	$0.346555\pi$
$522$	9.88177	0.432513
$523$	13.2666	0.580106	0.290053	−	0.957011i	$-0.406327\pi$
$523$	13.2666	0.580106	0.290053	+	0.957011i	$0.406327\pi$
$524$	1.73113	0.0756246
$525$	−3.08741	−0.134746
$526$	27.7220	1.20874
$527$	0	0
$528$	1.80714	0.0786457
$529$	−1.66025	−0.0721847
$530$	−1.70789	−0.0741862
$531$	8.71741	0.378303
$532$	25.7321	1.11563
$533$	−34.4850	−1.49371
$534$	6.44277	0.278806
$535$	14.9068	0.644476
$536$	−11.1900	−0.483336
$537$	−3.51711	−0.151774
$538$	−8.15049	−0.351393
$539$	4.57584	0.197095
$540$	1.00000	0.0430331
$541$	21.2515	0.913672	0.456836	−	0.889551i	$-0.348983\pi$
$541$	21.2515	0.913672	0.456836	+	0.889551i	$0.348983\pi$
$542$	14.3589	0.616767
$543$	22.0727	0.947231
$544$	0	0
$545$	1.39808	0.0598872
$546$	−18.3913	−0.787074
$547$	−29.7167	−1.27059	−0.635296	−	0.772268i	$-0.719122\pi$
$547$	−29.7167	−1.27059	−0.635296	+	0.772268i	$0.719122\pi$
$548$	−6.01833	−0.257090
$549$	5.38634	0.229883
$550$	−1.80714	−0.0770567
$551$	82.3598	3.50864
$552$	−4.61950	−0.196619
$553$	1.97372	0.0839310
$554$	5.17477	0.219855
$555$	−8.97729	−0.381065
$556$	−7.07929	−0.300229
$557$	−17.5412	−0.743243	−0.371621	−	0.928384i	$-0.621198\pi$
$557$	−17.5412	−0.743243	−0.371621	+	0.928384i	$0.621198\pi$
$558$	8.72556	0.369383
$559$	−64.4598	−2.72636
$560$	−3.08741	−0.130467
$561$	0	0
$562$	−27.9254	−1.17796
$563$	−1.30731	−0.0550966	−0.0275483	−	0.999620i	$-0.508770\pi$
$563$	−1.30731	−0.0550966	−0.0275483	+	0.999620i	$0.508770\pi$
$564$	0.765251	0.0322229
$565$	15.2757	0.642652
$566$	3.80624	0.159988
$567$	−3.08741	−0.129659
$568$	−15.9282	−0.668334
$569$	11.5562	0.484461	0.242231	−	0.970219i	$-0.422121\pi$
$569$	11.5562	0.484461	0.242231	+	0.970219i	$0.422121\pi$
$570$	8.33452	0.349095
$571$	−16.1686	−0.676635	−0.338318	−	0.941032i	$-0.609858\pi$
$571$	−16.1686	−0.676635	−0.338318	+	0.941032i	$0.609858\pi$
$572$	−10.7649	−0.450103
$573$	−17.5621	−0.733669
$574$	17.8734	0.746021
$575$	4.61950	0.192646
$576$	1.00000	0.0416667
$577$	−15.4178	−0.641850	−0.320925	−	0.947105i	$-0.603994\pi$
$577$	−15.4178	−0.641850	−0.320925	+	0.947105i	$0.603994\pi$
$578$	0	0
$579$	−12.2609	−0.509546
$580$	−9.88177	−0.410318
$581$	−40.2789	−1.67105
$582$	−14.8647	−0.616162
$583$	3.08640	0.127826
$584$	9.05308	0.374619
$585$	−5.95687	−0.246286
$586$	10.0310	0.414378
$587$	−20.5239	−0.847111	−0.423556	−	0.905870i	$-0.639218\pi$
$587$	−20.5239	−0.847111	−0.423556	+	0.905870i	$0.639218\pi$
$588$	2.53209	0.104422
$589$	72.7234	2.99651
$590$	−8.71741	−0.358890
$591$	6.46320	0.265861
$592$	−8.97729	−0.368964
$593$	43.7763	1.79768	0.898838	−	0.438281i	$-0.144413\pi$
$593$	43.7763	1.79768	0.898838	+	0.438281i	$0.144413\pi$
$594$	−1.80714	−0.0741478
$595$	0	0
$596$	−3.53675	−0.144871
$597$	−12.6004	−0.515698
$598$	27.5177	1.12528
$599$	−8.94338	−0.365417	−0.182708	−	0.983167i	$-0.558486\pi$
$599$	−8.94338	−0.365417	−0.182708	+	0.983167i	$0.558486\pi$
$600$	−1.00000	−0.0408248
$601$	41.2400	1.68222	0.841108	−	0.540867i	$-0.181904\pi$
$601$	41.2400	1.68222	0.841108	+	0.540867i	$0.181904\pi$
$602$	33.4091	1.36165
$603$	11.1900	0.455693
$604$	−8.82481	−0.359076
$605$	−7.73425	−0.314442
$606$	−15.3888	−0.625129
$607$	−24.8630	−1.00916	−0.504579	−	0.863366i	$-0.668352\pi$
$607$	−24.8630	−1.00916	−0.504579	+	0.863366i	$0.668352\pi$
$608$	8.33452	0.338009
$609$	30.5091	1.23629
$610$	−5.38634	−0.218086
$611$	−4.55849	−0.184417
$612$	0	0
$613$	12.4689	0.503612	0.251806	−	0.967778i	$-0.418975\pi$
$613$	12.4689	0.503612	0.251806	+	0.967778i	$0.418975\pi$
$614$	2.74331	0.110711
$615$	5.78913	0.233440
$616$	5.57938	0.224800
$617$	−21.1517	−0.851535	−0.425768	−	0.904833i	$-0.639996\pi$
$617$	−21.1517	−0.851535	−0.425768	+	0.904833i	$0.639996\pi$
$618$	1.83745	0.0739130
$619$	−0.442238	−0.0177750	−0.00888752	−	0.999961i	$-0.502829\pi$
$619$	−0.442238	−0.0177750	−0.00888752	+	0.999961i	$0.502829\pi$
$620$	−8.72556	−0.350427
$621$	4.61950	0.185374
$622$	11.2074	0.449375
$623$	19.8915	0.796935
$624$	−5.95687	−0.238465
$625$	1.00000	0.0400000
$626$	−23.4478	−0.937161
$627$	−15.0616	−0.601504
$628$	17.9931	0.718005
$629$	0	0
$630$	3.08741	0.123005
$631$	−38.7860	−1.54405	−0.772023	−	0.635595i	$-0.780755\pi$
$631$	−38.7860	−1.54405	−0.772023	+	0.635595i	$0.780755\pi$
$632$	0.639279	0.0254292
$633$	3.35385	0.133304
$634$	2.40030	0.0953281
$635$	−0.290811	−0.0115405
$636$	1.70789	0.0677224
$637$	−15.0833	−0.597623
$638$	17.8577	0.706995
$639$	15.9282	0.630111
$640$	−1.00000	−0.0395285
$641$	−5.12988	−0.202618	−0.101309	−	0.994855i	$-0.532303\pi$
$641$	−5.12988	−0.202618	−0.101309	+	0.994855i	$0.532303\pi$
$642$	−14.9068	−0.588323
$643$	8.24955	0.325330	0.162665	−	0.986681i	$-0.447991\pi$
$643$	8.24955	0.325330	0.162665	+	0.986681i	$0.447991\pi$
$644$	−14.2623	−0.562012
$645$	10.8211	0.426080
$646$	0	0
$647$	6.41676	0.252269	0.126134	−	0.992013i	$-0.459743\pi$
$647$	6.41676	0.252269	0.126134	+	0.992013i	$0.459743\pi$
$648$	−1.00000	−0.0392837
$649$	15.7536	0.618382
$650$	5.95687	0.233647
$651$	26.9394	1.05584
$652$	−4.85740	−0.190230
$653$	16.7201	0.654310	0.327155	−	0.944971i	$-0.393910\pi$
$653$	16.7201	0.654310	0.327155	+	0.944971i	$0.393910\pi$
$654$	−1.39808	−0.0546693
$655$	1.73113	0.0676407
$656$	5.78913	0.226027
$657$	−9.05308	−0.353194
$658$	2.36264	0.0921053
$659$	46.3917	1.80716	0.903581	−	0.428416i	$-0.140928\pi$
$659$	46.3917	1.80716	0.903581	+	0.428416i	$0.140928\pi$
$660$	1.80714	0.0703428
$661$	−15.2972	−0.594994	−0.297497	−	0.954723i	$-0.596152\pi$
$661$	−15.2972	−0.594994	−0.297497	+	0.954723i	$0.596152\pi$
$662$	5.72448	0.222488
$663$	0	0
$664$	−13.0462	−0.506290
$665$	25.7321	0.997846
$666$	8.97729	0.347863
$667$	−45.6488	−1.76753
$668$	14.3949	0.556956
$669$	19.6916	0.761321
$670$	−11.1900	−0.432308
$671$	9.73386	0.375772
$672$	3.08741	0.119099
$673$	−24.5104	−0.944805	−0.472402	−	0.881383i	$-0.656613\pi$
$673$	−24.5104	−0.944805	−0.472402	+	0.881383i	$0.656613\pi$
$674$	−11.8506	−0.456469
$675$	1.00000	0.0384900
$676$	22.4842	0.864779
$677$	−26.0682	−1.00188	−0.500941	−	0.865481i	$-0.667013\pi$
$677$	−26.0682	−1.00188	−0.500941	+	0.865481i	$0.667013\pi$
$678$	−15.2757	−0.586658
$679$	−45.8934	−1.76123
$680$	0	0
$681$	16.9010	0.647649
$682$	15.7683	0.603800
$683$	−32.5866	−1.24689	−0.623445	−	0.781867i	$-0.714267\pi$
$683$	−32.5866	−1.24689	−0.623445	+	0.781867i	$0.714267\pi$
$684$	−8.33452	−0.318678
$685$	−6.01833	−0.229949
$686$	−13.7943	−0.526667
$687$	7.06507	0.269549
$688$	10.8211	0.412550
$689$	−10.1737	−0.387587
$690$	−4.61950	−0.175861
$691$	−7.33726	−0.279123	−0.139561	−	0.990213i	$-0.544569\pi$
$691$	−7.33726	−0.279123	−0.139561	+	0.990213i	$0.544569\pi$
$692$	−4.94538	−0.187995
$693$	−5.57938	−0.211943
$694$	1.13688	0.0431555
$695$	−7.07929	−0.268533
$696$	9.88177	0.374568
$697$	0	0
$698$	20.9250	0.792023
$699$	1.92951	0.0729807
$700$	−3.08741	−0.116693
$701$	24.6215	0.929942	0.464971	−	0.885326i	$-0.346065\pi$
$701$	24.6215	0.929942	0.464971	+	0.885326i	$0.346065\pi$
$702$	5.95687	0.224827
$703$	74.8214	2.82194
$704$	1.80714	0.0681091
$705$	0.765251	0.0288210
$706$	−5.02419	−0.189088
$707$	−47.5116	−1.78686
$708$	8.71741	0.327620
$709$	27.2864	1.02476	0.512382	−	0.858758i	$-0.328763\pi$
$709$	27.2864	1.02476	0.512382	+	0.858758i	$0.328763\pi$
$710$	−15.9282	−0.597776
$711$	−0.639279	−0.0239748
$712$	6.44277	0.241453
$713$	−40.3077	−1.50954
$714$	0	0
$715$	−10.7649	−0.402584
$716$	−3.51711	−0.131441
$717$	−11.9374	−0.445811
$718$	31.3545	1.17014
$719$	22.5021	0.839186	0.419593	−	0.907712i	$-0.362173\pi$
$719$	22.5021	0.839186	0.419593	+	0.907712i	$0.362173\pi$
$720$	1.00000	0.0372678
$721$	5.67295	0.211272
$722$	−50.4642	−1.87808
$723$	4.77789	0.177692
$724$	22.0727	0.820326
$725$	−9.88177	−0.367000
$726$	7.73425	0.287045
$727$	10.6520	0.395059	0.197530	−	0.980297i	$-0.436708\pi$
$727$	10.6520	0.395059	0.197530	+	0.980297i	$0.436708\pi$
$728$	−18.3913	−0.681626
$729$	1.00000	0.0370370
$730$	9.05308	0.335069
$731$	0	0
$732$	5.38634	0.199085
$733$	30.0448	1.10973	0.554866	−	0.831940i	$-0.312770\pi$
$733$	30.0448	1.10973	0.554866	+	0.831940i	$0.312770\pi$
$734$	9.37526	0.346047
$735$	2.53209	0.0933975
$736$	−4.61950	−0.170277
$737$	20.2219	0.744885
$738$	−5.78913	−0.213101
$739$	−42.3019	−1.55610	−0.778050	−	0.628202i	$-0.783791\pi$
$739$	−42.3019	−1.55610	−0.778050	+	0.628202i	$0.783791\pi$
$740$	−8.97729	−0.330012
$741$	49.6476	1.82385
$742$	5.27297	0.193577
$743$	−32.6616	−1.19824	−0.599120	−	0.800659i	$-0.704483\pi$
$743$	−32.6616	−1.19824	−0.599120	+	0.800659i	$0.704483\pi$
$744$	8.72556	0.319895
$745$	−3.53675	−0.129577
$746$	8.20910	0.300556
$747$	13.0462	0.477335
$748$	0	0
$749$	−46.0233	−1.68165
$750$	−1.00000	−0.0365148
$751$	−10.1137	−0.369055	−0.184527	−	0.982827i	$-0.559075\pi$
$751$	−10.1137	−0.369055	−0.184527	+	0.982827i	$0.559075\pi$
$752$	0.765251	0.0279058
$753$	−19.9225	−0.726018
$754$	−58.8644	−2.14371
$755$	−8.82481	−0.321168
$756$	−3.08741	−0.112288
$757$	15.7508	0.572473	0.286236	−	0.958159i	$-0.407596\pi$
$757$	15.7508	0.572473	0.286236	+	0.958159i	$0.407596\pi$
$758$	−0.150095	−0.00545170
$759$	8.34808	0.303016
$760$	8.33452	0.302325
$761$	−17.2381	−0.624881	−0.312440	−	0.949937i	$-0.601146\pi$
$761$	−17.2381	−0.624881	−0.312440	+	0.949937i	$0.601146\pi$
$762$	0.290811	0.0105350
$763$	−4.31645	−0.156266
$764$	−17.5621	−0.635376
$765$	0	0
$766$	−7.43165	−0.268516
$767$	−51.9284	−1.87503
$768$	1.00000	0.0360844
$769$	−4.40614	−0.158890	−0.0794448	−	0.996839i	$-0.525315\pi$
$769$	−4.40614	−0.158890	−0.0794448	+	0.996839i	$0.525315\pi$
$770$	5.57938	0.201067
$771$	−2.37593	−0.0855671
$772$	−12.2609	−0.441280
$773$	−31.8516	−1.14562	−0.572811	−	0.819688i	$-0.694147\pi$
$773$	−31.8516	−1.14562	−0.572811	+	0.819688i	$0.694147\pi$
$774$	−10.8211	−0.388956
$775$	−8.72556	−0.313432
$776$	−14.8647	−0.533612
$777$	27.7166	0.994326
$778$	6.50342	0.233159
$779$	−48.2496	−1.72872
$780$	−5.95687	−0.213290
$781$	28.7845	1.02999
$782$	0	0
$783$	−9.88177	−0.353146
$784$	2.53209	0.0904317
$785$	17.9931	0.642203
$786$	−1.73113	−0.0617472
$787$	9.97897	0.355712	0.177856	−	0.984057i	$-0.443084\pi$
$787$	9.97897	0.355712	0.177856	+	0.984057i	$0.443084\pi$
$788$	6.46320	0.230242
$789$	−27.7220	−0.986931
$790$	0.639279	0.0227445
$791$	−47.1622	−1.67690
$792$	−1.80714	−0.0642139
$793$	−32.0857	−1.13940
$794$	−27.3742	−0.971475
$795$	1.70789	0.0605728
$796$	−12.6004	−0.446608
$797$	44.0640	1.56083	0.780413	−	0.625264i	$-0.215009\pi$
$797$	44.0640	1.56083	0.780413	+	0.625264i	$0.215009\pi$
$798$	−25.7321	−0.910905
$799$	0	0
$800$	−1.00000	−0.0353553
$801$	−6.44277	−0.227644
$802$	−8.94064	−0.315705
$803$	−16.3602	−0.577338
$804$	11.1900	0.394642
$805$	−14.2623	−0.502679
$806$	−51.9770	−1.83081
$807$	8.15049	0.286911
$808$	−15.3888	−0.541378
$809$	5.00555	0.175986	0.0879929	−	0.996121i	$-0.471955\pi$
$809$	5.00555	0.175986	0.0879929	+	0.996121i	$0.471955\pi$
$810$	−1.00000	−0.0351364
$811$	−7.49952	−0.263344	−0.131672	−	0.991293i	$-0.542035\pi$
$811$	−7.49952	−0.263344	−0.131672	+	0.991293i	$0.542035\pi$
$812$	30.5091	1.07066
$813$	−14.3589	−0.503588
$814$	16.2232	0.568624
$815$	−4.85740	−0.170147
$816$	0	0
$817$	−90.1886	−3.15530
$818$	29.7681	1.04082
$819$	18.3913	0.642643
$820$	5.78913	0.202165
$821$	−11.2150	−0.391405	−0.195703	−	0.980663i	$-0.562699\pi$
$821$	−11.2150	−0.391405	−0.195703	+	0.980663i	$0.562699\pi$
$822$	6.01833	0.209913
$823$	26.7246	0.931562	0.465781	−	0.884900i	$-0.345773\pi$
$823$	26.7246	0.931562	0.465781	+	0.884900i	$0.345773\pi$
$824$	1.83745	0.0640105
$825$	1.80714	0.0629165
$826$	26.9142	0.936464
$827$	−2.07743	−0.0722394	−0.0361197	−	0.999347i	$-0.511500\pi$
$827$	−2.07743	−0.0722394	−0.0361197	+	0.999347i	$0.511500\pi$
$828$	4.61950	0.160539
$829$	−15.1332	−0.525596	−0.262798	−	0.964851i	$-0.584645\pi$
$829$	−15.1332	−0.525596	−0.262798	+	0.964851i	$0.584645\pi$
$830$	−13.0462	−0.452840
$831$	−5.17477	−0.179511
$832$	−5.95687	−0.206517
$833$	0	0
$834$	7.07929	0.245136
$835$	14.3949	0.498157
$836$	−15.0616	−0.520918
$837$	−8.72556	−0.301600
$838$	−25.0180	−0.864231
$839$	28.6140	0.987866	0.493933	−	0.869500i	$-0.335559\pi$
$839$	28.6140	0.987866	0.493933	+	0.869500i	$0.335559\pi$
$840$	3.08741	0.106526
$841$	68.6494	2.36722
$842$	2.28209	0.0786459
$843$	27.9254	0.961803
$844$	3.35385	0.115444
$845$	22.4842	0.773481
$846$	−0.765251	−0.0263099
$847$	23.8788	0.820484
$848$	1.70789	0.0586494
$849$	−3.80624	−0.130630
$850$	0	0
$851$	−41.4706	−1.42159
$852$	15.9282	0.545692
$853$	15.0401	0.514963	0.257482	−	0.966283i	$-0.417107\pi$
$853$	15.0401	0.514963	0.257482	+	0.966283i	$0.417107\pi$
$854$	16.6298	0.569061
$855$	−8.33452	−0.285035
$856$	−14.9068	−0.509503
$857$	19.0785	0.651708	0.325854	−	0.945420i	$-0.394348\pi$
$857$	19.0785	0.651708	0.325854	+	0.945420i	$0.394348\pi$
$858$	10.7649	0.367507
$859$	−23.1551	−0.790043	−0.395021	−	0.918672i	$-0.629263\pi$
$859$	−23.1551	−0.790043	−0.395021	+	0.918672i	$0.629263\pi$
$860$	10.8211	0.368996
$861$	−17.8734	−0.609124
$862$	−22.7448	−0.774690
$863$	21.0343	0.716015	0.358008	−	0.933719i	$-0.383456\pi$
$863$	21.0343	0.716015	0.358008	+	0.933719i	$0.383456\pi$
$864$	−1.00000	−0.0340207
$865$	−4.94538	−0.168148
$866$	18.1574	0.617015
$867$	0	0
$868$	26.9394	0.914382
$869$	−1.15527	−0.0391898
$870$	9.88177	0.335023
$871$	−66.6575	−2.25860
$872$	−1.39808	−0.0473450
$873$	14.8647	0.503094
$874$	38.5013	1.30233
$875$	−3.08741	−0.104373
$876$	−9.05308	−0.305875
$877$	24.3365	0.821786	0.410893	−	0.911683i	$-0.365217\pi$
$877$	24.3365	0.821786	0.410893	+	0.911683i	$0.365217\pi$
$878$	−13.3730	−0.451318
$879$	−10.0310	−0.338338
$880$	1.80714	0.0609187
$881$	46.4859	1.56615	0.783074	−	0.621928i	$-0.213650\pi$
$881$	46.4859	1.56615	0.783074	+	0.621928i	$0.213650\pi$
$882$	−2.53209	−0.0852599
$883$	−13.5023	−0.454389	−0.227195	−	0.973849i	$-0.572955\pi$
$883$	−13.5023	−0.454389	−0.227195	+	0.973849i	$0.572955\pi$
$884$	0	0
$885$	8.71741	0.293032
$886$	17.4629	0.586678
$887$	49.4576	1.66062	0.830311	−	0.557300i	$-0.188163\pi$
$887$	49.4576	1.66062	0.830311	+	0.557300i	$0.188163\pi$
$888$	8.97729	0.301258
$889$	0.897853	0.0301130
$890$	6.44277	0.215962
$891$	1.80714	0.0605415
$892$	19.6916	0.659324
$893$	−6.37799	−0.213431
$894$	3.53675	0.118287
$895$	−3.51711	−0.117564
$896$	3.08741	0.103143
$897$	−27.5177	−0.918790
$898$	14.6209	0.487906
$899$	86.2240	2.87573
$900$	1.00000	0.0333333
$901$	0	0
$902$	−10.4618	−0.348339
$903$	−33.4091	−1.11179
$904$	−15.2757	−0.508061
$905$	22.0727	0.733722
$906$	8.82481	0.293185
$907$	−35.9267	−1.19293	−0.596464	−	0.802640i	$-0.703428\pi$
$907$	−35.9267	−1.19293	−0.596464	+	0.802640i	$0.703428\pi$
$908$	16.9010	0.560880
$909$	15.3888	0.510416
$910$	−18.3913	−0.609665
$911$	30.1120	0.997656	0.498828	−	0.866701i	$-0.333764\pi$
$911$	30.1120	0.997656	0.498828	+	0.866701i	$0.333764\pi$
$912$	−8.33452	−0.275983
$913$	23.5763	0.780261
$914$	−28.3168	−0.936638
$915$	5.38634	0.178067
$916$	7.06507	0.233437
$917$	−5.34469	−0.176497
$918$	0	0
$919$	56.0840	1.85004	0.925020	−	0.379918i	$-0.124048\pi$
$919$	56.0840	1.85004	0.925020	+	0.379918i	$0.124048\pi$
$920$	−4.61950	−0.152300
$921$	−2.74331	−0.0903952
$922$	18.2506	0.601050
$923$	−94.8823	−3.12309
$924$	−5.57938	−0.183548
$925$	−8.97729	−0.295172
$926$	−7.07454	−0.232484
$927$	−1.83745	−0.0603497
$928$	9.88177	0.324385
$929$	−2.61876	−0.0859187	−0.0429593	−	0.999077i	$-0.513679\pi$
$929$	−2.61876	−0.0859187	−0.0429593	+	0.999077i	$0.513679\pi$
$930$	8.72556	0.286123
$931$	−21.1037	−0.691647
$932$	1.92951	0.0632031
$933$	−11.2074	−0.366914
$934$	−20.9566	−0.685720
$935$	0	0
$936$	5.95687	0.194706
$937$	−9.54736	−0.311899	−0.155949	−	0.987765i	$-0.549844\pi$
$937$	−9.54736	−0.311899	−0.155949	+	0.987765i	$0.549844\pi$
$938$	34.5482	1.12804
$939$	23.4478	0.765189
$940$	0.765251	0.0249597
$941$	56.8843	1.85437	0.927187	−	0.374598i	$-0.122219\pi$
$941$	56.8843	1.85437	0.927187	+	0.374598i	$0.122219\pi$
$942$	−17.9931	−0.586248
$943$	26.7429	0.870867
$944$	8.71741	0.283727
$945$	−3.08741	−0.100433
$946$	−19.5552	−0.635795
$947$	30.6267	0.995234	0.497617	−	0.867397i	$-0.334209\pi$
$947$	30.6267	0.995234	0.497617	+	0.867397i	$0.334209\pi$
$948$	−0.639279	−0.0207628
$949$	53.9280	1.75058
$950$	8.33452	0.270407
$951$	−2.40030	−0.0778351
$952$	0	0
$953$	47.2302	1.52994	0.764968	−	0.644069i	$-0.222755\pi$
$953$	47.2302	1.52994	0.764968	+	0.644069i	$0.222755\pi$
$954$	−1.70789	−0.0552951
$955$	−17.5621	−0.568298
$956$	−11.9374	−0.386083
$957$	−17.8577	−0.577259
$958$	17.9922	0.581302
$959$	18.5810	0.600013
$960$	1.00000	0.0322749
$961$	45.1355	1.45598
$962$	−53.4765	−1.72415
$963$	14.9068	0.480364
$964$	4.77789	0.153885
$965$	−12.2609	−0.394693
$966$	14.2623	0.458881
$967$	0.125172	0.00402525	0.00201262	−	0.999998i	$-0.499359\pi$
$967$	0.125172	0.00402525	0.00201262	+	0.999998i	$0.499359\pi$
$968$	7.73425	0.248588
$969$	0	0
$970$	−14.8647	−0.477277
$971$	−28.5680	−0.916790	−0.458395	−	0.888749i	$-0.651575\pi$
$971$	−28.5680	−0.916790	−0.458395	+	0.888749i	$0.651575\pi$
$972$	1.00000	0.0320750
$973$	21.8567	0.700692
$974$	21.5579	0.690759
$975$	−5.95687	−0.190772
$976$	5.38634	0.172412
$977$	15.5312	0.496886	0.248443	−	0.968647i	$-0.420081\pi$
$977$	15.5312	0.496886	0.248443	+	0.968647i	$0.420081\pi$
$978$	4.85740	0.155323
$979$	−11.6430	−0.372112
$980$	2.53209	0.0808846
$981$	1.39808	0.0446373
$982$	5.27264	0.168257
$983$	11.8492	0.377930	0.188965	−	0.981984i	$-0.439487\pi$
$983$	11.8492	0.377930	0.188965	+	0.981984i	$0.439487\pi$
$984$	−5.78913	−0.184551
$985$	6.46320	0.205935
$986$	0	0
$987$	−2.36264	−0.0752037
$988$	49.6476	1.57950
$989$	49.9880	1.58953
$990$	−1.80714	−0.0574347
$991$	42.4253	1.34768	0.673842	−	0.738876i	$-0.264643\pi$
$991$	42.4253	1.34768	0.673842	+	0.738876i	$0.264643\pi$
$992$	8.72556	0.277037
$993$	−5.72448	−0.181661
$994$	49.1769	1.55980
$995$	−12.6004	−0.399458
$996$	13.0462	0.413384
$997$	15.2006	0.481406	0.240703	−	0.970599i	$-0.422622\pi$
$997$	15.2006	0.481406	0.240703	+	0.970599i	$0.422622\pi$
$998$	16.7839	0.531285
$999$	−8.97729	−0.284029

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8670.2.a.ce.1.1	yes	6
17.16	even	2		8670.2.a.cb.1.6	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8670.2.a.cb.1.6	✓	6	17.16	even	2
8670.2.a.ce.1.1	yes	6	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 8670 = 2 \cdot 3 \cdot 5 \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8670.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} -3.08741 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} -3.08741 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} +1.80714 q^{11} +1.00000 q^{12} -5.95687 q^{13} +3.08741 q^{14} +1.00000 q^{15} +1.00000 q^{16} -1.00000 q^{18} -8.33452 q^{19} +1.00000 q^{20} -3.08741 q^{21} -1.80714 q^{22} +4.61950 q^{23} -1.00000 q^{24} +1.00000 q^{25} +5.95687 q^{26} +1.00000 q^{27} -3.08741 q^{28} -9.88177 q^{29} -1.00000 q^{30} -8.72556 q^{31} -1.00000 q^{32} +1.80714 q^{33} -3.08741 q^{35} +1.00000 q^{36} -8.97729 q^{37} +8.33452 q^{38} -5.95687 q^{39} -1.00000 q^{40} +5.78913 q^{41} +3.08741 q^{42} +10.8211 q^{43} +1.80714 q^{44} +1.00000 q^{45} -4.61950 q^{46} +0.765251 q^{47} +1.00000 q^{48} +2.53209 q^{49} -1.00000 q^{50} -5.95687 q^{52} +1.70789 q^{53} -1.00000 q^{54} +1.80714 q^{55} +3.08741 q^{56} -8.33452 q^{57} +9.88177 q^{58} +8.71741 q^{59} +1.00000 q^{60} +5.38634 q^{61} +8.72556 q^{62} -3.08741 q^{63} +1.00000 q^{64} -5.95687 q^{65} -1.80714 q^{66} +11.1900 q^{67} +4.61950 q^{69} +3.08741 q^{70} +15.9282 q^{71} -1.00000 q^{72} -9.05308 q^{73} +8.97729 q^{74} +1.00000 q^{75} -8.33452 q^{76} -5.57938 q^{77} +5.95687 q^{78} -0.639279 q^{79} +1.00000 q^{80} +1.00000 q^{81} -5.78913 q^{82} +13.0462 q^{83} -3.08741 q^{84} -10.8211 q^{86} -9.88177 q^{87} -1.80714 q^{88} -6.44277 q^{89} -1.00000 q^{90} +18.3913 q^{91} +4.61950 q^{92} -8.72556 q^{93} -0.765251 q^{94} -8.33452 q^{95} -1.00000 q^{96} +14.8647 q^{97} -2.53209 q^{98} +1.80714 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{2} + 6 q^{3} + 6 q^{4} + 6 q^{5} - 6 q^{6} - 6 q^{8} + 6 q^{9}+O(q^{10}) \) 6 * q - 6 * q^2 + 6 * q^3 + 6 * q^4 + 6 * q^5 - 6 * q^6 - 6 * q^8 + 6 * q^9	\( 6 q - 6 q^{2} + 6 q^{3} + 6 q^{4} + 6 q^{5} - 6 q^{6} - 6 q^{8} + 6 q^{9} - 6 q^{10} + 6 q^{11} + 6 q^{12} + 6 q^{13} + 6 q^{15} + 6 q^{16} - 6 q^{18} - 6 q^{19} + 6 q^{20} - 6 q^{22} - 6 q^{24} + 6 q^{25} - 6 q^{26} + 6 q^{27} + 6 q^{29} - 6 q^{30} - 6 q^{32} + 6 q^{33} + 6 q^{36} + 6 q^{38} + 6 q^{39} - 6 q^{40} + 12 q^{41} + 24 q^{43} + 6 q^{44} + 6 q^{45} + 6 q^{47} + 6 q^{48} + 6 q^{49} - 6 q^{50} + 6 q^{52} - 6 q^{53} - 6 q^{54} + 6 q^{55} - 6 q^{57} - 6 q^{58} - 18 q^{59} + 6 q^{60} - 6 q^{61} + 6 q^{64} + 6 q^{65} - 6 q^{66} + 36 q^{67} + 30 q^{71} - 6 q^{72} + 12 q^{73} + 6 q^{75} - 6 q^{76} - 6 q^{77} - 6 q^{78} + 12 q^{79} + 6 q^{80} + 6 q^{81} - 12 q^{82} + 36 q^{83} - 24 q^{86} + 6 q^{87} - 6 q^{88} - 30 q^{89} - 6 q^{90} + 12 q^{91} - 6 q^{94} - 6 q^{95} - 6 q^{96} + 18 q^{97} - 6 q^{98} + 6 q^{99}+O(q^{100}) \) 6 * q - 6 * q^2 + 6 * q^3 + 6 * q^4 + 6 * q^5 - 6 * q^6 - 6 * q^8 + 6 * q^9 - 6 * q^10 + 6 * q^11 + 6 * q^12 + 6 * q^13 + 6 * q^15 + 6 * q^16 - 6 * q^18 - 6 * q^19 + 6 * q^20 - 6 * q^22 - 6 * q^24 + 6 * q^25 - 6 * q^26 + 6 * q^27 + 6 * q^29 - 6 * q^30 - 6 * q^32 + 6 * q^33 + 6 * q^36 + 6 * q^38 + 6 * q^39 - 6 * q^40 + 12 * q^41 + 24 * q^43 + 6 * q^44 + 6 * q^45 + 6 * q^47 + 6 * q^48 + 6 * q^49 - 6 * q^50 + 6 * q^52 - 6 * q^53 - 6 * q^54 + 6 * q^55 - 6 * q^57 - 6 * q^58 - 18 * q^59 + 6 * q^60 - 6 * q^61 + 6 * q^64 + 6 * q^65 - 6 * q^66 + 36 * q^67 + 30 * q^71 - 6 * q^72 + 12 * q^73 + 6 * q^75 - 6 * q^76 - 6 * q^77 - 6 * q^78 + 12 * q^79 + 6 * q^80 + 6 * q^81 - 12 * q^82 + 36 * q^83 - 24 * q^86 + 6 * q^87 - 6 * q^88 - 30 * q^89 - 6 * q^90 + 12 * q^91 - 6 * q^94 - 6 * q^95 - 6 * q^96 + 18 * q^97 - 6 * q^98 + 6 * q^99

Introduction

L-functions

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