LMFDB - Embedded newform 8670.2.a.cb.1.2

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:	$1$
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			1.2
Root			$-2.88917$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} -2.88917 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} -2.88917 q^{7} -1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} +5.96195 q^{11} -1.00000 q^{12} +5.41487 q^{13} +2.88917 q^{14} +1.00000 q^{15} +1.00000 q^{16} -1.00000 q^{18} -5.77376 q^{19} -1.00000 q^{20} +2.88917 q^{21} -5.96195 q^{22} +2.54187 q^{23} +1.00000 q^{24} +1.00000 q^{25} -5.41487 q^{26} -1.00000 q^{27} -2.88917 q^{28} -4.19797 q^{29} -1.00000 q^{30} -7.48544 q^{31} -1.00000 q^{32} -5.96195 q^{33} +2.88917 q^{35} +1.00000 q^{36} +6.65953 q^{37} +5.77376 q^{38} -5.41487 q^{39} +1.00000 q^{40} -10.0038 q^{41} -2.88917 q^{42} +6.37207 q^{43} +5.96195 q^{44} -1.00000 q^{45} -2.54187 q^{46} -0.323794 q^{47} -1.00000 q^{48} +1.34730 q^{49} -1.00000 q^{50} +5.41487 q^{52} -8.33615 q^{53} +1.00000 q^{54} -5.96195 q^{55} +2.88917 q^{56} +5.77376 q^{57} +4.19797 q^{58} +2.64975 q^{59} +1.00000 q^{60} +1.87078 q^{61} +7.48544 q^{62} -2.88917 q^{63} +1.00000 q^{64} -5.41487 q^{65} +5.96195 q^{66} -0.0859626 q^{67} -2.54187 q^{69} -2.88917 q^{70} +5.36550 q^{71} -1.00000 q^{72} -9.52783 q^{73} -6.65953 q^{74} -1.00000 q^{75} -5.77376 q^{76} -17.2251 q^{77} +5.41487 q^{78} -6.37886 q^{79} -1.00000 q^{80} +1.00000 q^{81} +10.0038 q^{82} +1.96571 q^{83} +2.88917 q^{84} -6.37207 q^{86} +4.19797 q^{87} -5.96195 q^{88} -4.96671 q^{89} +1.00000 q^{90} -15.6445 q^{91} +2.54187 q^{92} +7.48544 q^{93} +0.323794 q^{94} +5.77376 q^{95} +1.00000 q^{96} -4.85131 q^{97} -1.34730 q^{98} +5.96195 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$	−2.88917	−1.09200	−0.546002	−	0.837784i	$-0.683851\pi$
$7$	−2.88917	−1.09200	−0.546002	+	0.837784i	$0.683851\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	1.00000	0.316228
$11$	5.96195	1.79760	0.898798	−	0.438363i	$-0.144442\pi$
$11$	5.96195	1.79760	0.898798	+	0.438363i	$0.144442\pi$
$12$	−1.00000	−0.288675
$13$	5.41487	1.50182	0.750908	−	0.660407i	$-0.229616\pi$
$13$	5.41487	1.50182	0.750908	+	0.660407i	$0.229616\pi$
$14$	2.88917	0.772163
$15$	1.00000	0.258199
$16$	1.00000	0.250000
$17$	0	0
$17$	0	0
$18$	−1.00000	−0.235702
$19$	−5.77376	−1.32459	−0.662296	−	0.749243i	$-0.730418\pi$
$19$	−5.77376	−1.32459	−0.662296	+	0.749243i	$0.730418\pi$
$20$	−1.00000	−0.223607
$21$	2.88917	0.630468
$22$	−5.96195	−1.27109
$23$	2.54187	0.530017	0.265009	−	0.964246i	$-0.414625\pi$
$23$	2.54187	0.530017	0.265009	+	0.964246i	$0.414625\pi$
$24$	1.00000	0.204124
$25$	1.00000	0.200000
$26$	−5.41487	−1.06194
$27$	−1.00000	−0.192450
$28$	−2.88917	−0.546002
$29$	−4.19797	−0.779544	−0.389772	−	0.920911i	$-0.627446\pi$
$29$	−4.19797	−0.779544	−0.389772	+	0.920911i	$0.627446\pi$
$30$	−1.00000	−0.182574
$31$	−7.48544	−1.34442	−0.672212	−	0.740359i	$-0.734656\pi$
$31$	−7.48544	−1.34442	−0.672212	+	0.740359i	$0.734656\pi$
$32$	−1.00000	−0.176777
$33$	−5.96195	−1.03784
$34$	0	0
$35$	2.88917	0.488359
$36$	1.00000	0.166667
$37$	6.65953	1.09482	0.547410	−	0.836865i	$-0.315614\pi$
$37$	6.65953	1.09482	0.547410	+	0.836865i	$0.315614\pi$
$38$	5.77376	0.936628
$39$	−5.41487	−0.867073
$40$	1.00000	0.158114
$41$	−10.0038	−1.56234	−0.781169	−	0.624320i	$-0.785376\pi$
$41$	−10.0038	−1.56234	−0.781169	+	0.624320i	$0.785376\pi$
$42$	−2.88917	−0.445808
$43$	6.37207	0.971731	0.485866	−	0.874033i	$-0.338504\pi$
$43$	6.37207	0.971731	0.485866	+	0.874033i	$0.338504\pi$
$44$	5.96195	0.898798
$45$	−1.00000	−0.149071
$46$	−2.54187	−0.374779
$47$	−0.323794	−0.0472303	−0.0236151	−	0.999721i	$-0.507518\pi$
$47$	−0.323794	−0.0472303	−0.0236151	+	0.999721i	$0.507518\pi$
$48$	−1.00000	−0.144338
$49$	1.34730	0.192471
$50$	−1.00000	−0.141421
$51$	0	0
$52$	5.41487	0.750908
$53$	−8.33615	−1.14506	−0.572529	−	0.819885i	$-0.694038\pi$
$53$	−8.33615	−1.14506	−0.572529	+	0.819885i	$0.694038\pi$
$54$	1.00000	0.136083
$55$	−5.96195	−0.803909
$56$	2.88917	0.386081
$57$	5.77376	0.764753
$58$	4.19797	0.551221
$59$	2.64975	0.344968	0.172484	−	0.985012i	$-0.444821\pi$
$59$	2.64975	0.344968	0.172484	+	0.985012i	$0.444821\pi$
$60$	1.00000	0.129099
$61$	1.87078	0.239529	0.119765	−	0.992802i	$-0.461786\pi$
$61$	1.87078	0.239529	0.119765	+	0.992802i	$0.461786\pi$
$62$	7.48544	0.950651
$63$	−2.88917	−0.364001
$64$	1.00000	0.125000
$65$	−5.41487	−0.671632
$66$	5.96195	0.733865
$67$	−0.0859626	−0.0105020	−0.00525100	−	0.999986i	$-0.501671\pi$
$67$	−0.0859626	−0.0105020	−0.00525100	+	0.999986i	$0.501671\pi$
$68$	0	0
$69$	−2.54187	−0.306005
$70$	−2.88917	−0.345322
$71$	5.36550	0.636767	0.318384	−	0.947962i	$-0.396860\pi$
$71$	5.36550	0.636767	0.318384	+	0.947962i	$0.396860\pi$
$72$	−1.00000	−0.117851
$73$	−9.52783	−1.11515	−0.557574	−	0.830127i	$-0.688268\pi$
$73$	−9.52783	−1.11515	−0.557574	+	0.830127i	$0.688268\pi$
$74$	−6.65953	−0.774155
$75$	−1.00000	−0.115470
$76$	−5.77376	−0.662296
$77$	−17.2251	−1.96298
$78$	5.41487	0.613114
$79$	−6.37886	−0.717678	−0.358839	−	0.933399i	$-0.616827\pi$
$79$	−6.37886	−0.717678	−0.358839	+	0.933399i	$0.616827\pi$
$80$	−1.00000	−0.111803
$81$	1.00000	0.111111
$82$	10.0038	1.10474
$83$	1.96571	0.215765	0.107882	−	0.994164i	$-0.465593\pi$
$83$	1.96571	0.215765	0.107882	+	0.994164i	$0.465593\pi$
$84$	2.88917	0.315234
$85$	0	0
$86$	−6.37207	−0.687118
$87$	4.19797	0.450070
$88$	−5.96195	−0.635546
$89$	−4.96671	−0.526471	−0.263235	−	0.964732i	$-0.584790\pi$
$89$	−4.96671	−0.526471	−0.263235	+	0.964732i	$0.584790\pi$
$90$	1.00000	0.105409
$91$	−15.6445	−1.63999
$92$	2.54187	0.265009
$93$	7.48544	0.776203
$94$	0.323794	0.0333968
$95$	5.77376	0.592375
$96$	1.00000	0.102062
$97$	−4.85131	−0.492575	−0.246288	−	0.969197i	$-0.579211\pi$
$97$	−4.85131	−0.492575	−0.246288	+	0.969197i	$0.579211\pi$
$98$	−1.34730	−0.136097
$99$	5.96195	0.599199
$100$	1.00000	0.100000
$101$	12.2990	1.22379	0.611896	−	0.790938i	$-0.290407\pi$
$101$	12.2990	1.22379	0.611896	+	0.790938i	$0.290407\pi$
$102$	0	0
$103$	5.90534	0.581870	0.290935	−	0.956743i	$-0.406034\pi$
$103$	5.90534	0.581870	0.290935	+	0.956743i	$0.406034\pi$
$104$	−5.41487	−0.530972
$105$	−2.88917	−0.281954
$106$	8.33615	0.809678
$107$	−9.34706	−0.903615	−0.451807	−	0.892116i	$-0.649221\pi$
$107$	−9.34706	−0.903615	−0.451807	+	0.892116i	$0.649221\pi$
$108$	−1.00000	−0.0962250
$109$	−19.2630	−1.84506	−0.922532	−	0.385921i	$-0.873884\pi$
$109$	−19.2630	−1.84506	−0.922532	+	0.385921i	$0.873884\pi$
$110$	5.96195	0.568450
$111$	−6.65953	−0.632095
$112$	−2.88917	−0.273001
$113$	14.1039	1.32679	0.663393	−	0.748271i	$-0.269116\pi$
$113$	14.1039	1.32679	0.663393	+	0.748271i	$0.269116\pi$
$114$	−5.77376	−0.540762
$115$	−2.54187	−0.237031
$116$	−4.19797	−0.389772
$117$	5.41487	0.500605
$118$	−2.64975	−0.243929
$119$	0	0
$120$	−1.00000	−0.0912871
$121$	24.5448	2.23135
$122$	−1.87078	−0.169373
$123$	10.0038	0.902016
$124$	−7.48544	−0.672212
$125$	−1.00000	−0.0894427
$126$	2.88917	0.257388
$127$	18.7283	1.66187	0.830933	−	0.556372i	$-0.187807\pi$
$127$	18.7283	1.66187	0.830933	+	0.556372i	$0.187807\pi$
$128$	−1.00000	−0.0883883
$129$	−6.37207	−0.561029
$130$	5.41487	0.474916
$131$	11.9199	1.04145	0.520724	−	0.853725i	$-0.325662\pi$
$131$	11.9199	1.04145	0.520724	+	0.853725i	$0.325662\pi$
$132$	−5.96195	−0.518921
$133$	16.6814	1.44646
$134$	0.0859626	0.00742604
$135$	1.00000	0.0860663
$136$	0	0
$137$	11.9326	1.01947	0.509734	−	0.860332i	$-0.329744\pi$
$137$	11.9326	1.01947	0.509734	+	0.860332i	$0.329744\pi$
$138$	2.54187	0.216379
$139$	−16.6358	−1.41103	−0.705517	−	0.708693i	$-0.749285\pi$
$139$	−16.6358	−1.41103	−0.705517	+	0.708693i	$0.749285\pi$
$140$	2.88917	0.244179
$141$	0.323794	0.0272684
$142$	−5.36550	−0.450262
$143$	32.2832	2.69966
$144$	1.00000	0.0833333
$145$	4.19797	0.348623
$146$	9.52783	0.788529
$147$	−1.34730	−0.111123
$148$	6.65953	0.547410
$149$	15.0525	1.23315	0.616575	−	0.787296i	$-0.288520\pi$
$149$	15.0525	1.23315	0.616575	+	0.787296i	$0.288520\pi$
$150$	1.00000	0.0816497
$151$	5.11124	0.415947	0.207973	−	0.978134i	$-0.433313\pi$
$151$	5.11124	0.415947	0.207973	+	0.978134i	$0.433313\pi$
$152$	5.77376	0.468314
$153$	0	0
$154$	17.2251	1.38804
$155$	7.48544	0.601245
$156$	−5.41487	−0.433537
$157$	−13.6300	−1.08779	−0.543895	−	0.839154i	$-0.683051\pi$
$157$	−13.6300	−1.08779	−0.543895	+	0.839154i	$0.683051\pi$
$158$	6.37886	0.507475
$159$	8.33615	0.661099
$160$	1.00000	0.0790569
$161$	−7.34390	−0.578780
$162$	−1.00000	−0.0785674
$163$	23.1047	1.80970	0.904850	−	0.425731i	$-0.139983\pi$
$163$	23.1047	1.80970	0.904850	+	0.425731i	$0.139983\pi$
$164$	−10.0038	−0.781169
$165$	5.96195	0.464137
$166$	−1.96571	−0.152569
$167$	−7.94026	−0.614435	−0.307218	−	0.951639i	$-0.599398\pi$
$167$	−7.94026	−0.614435	−0.307218	+	0.951639i	$0.599398\pi$
$168$	−2.88917	−0.222904
$169$	16.3208	1.25545
$170$	0	0
$171$	−5.77376	−0.441530
$172$	6.37207	0.485866
$173$	−16.5906	−1.26136	−0.630678	−	0.776044i	$-0.717223\pi$
$173$	−16.5906	−1.26136	−0.630678	+	0.776044i	$0.717223\pi$
$174$	−4.19797	−0.318248
$175$	−2.88917	−0.218401
$176$	5.96195	0.449399
$177$	−2.64975	−0.199167
$178$	4.96671	0.372271
$179$	−23.8002	−1.77891	−0.889457	−	0.457020i	$-0.848917\pi$
$179$	−23.8002	−1.77891	−0.889457	+	0.457020i	$0.848917\pi$
$180$	−1.00000	−0.0745356
$181$	−9.49423	−0.705700	−0.352850	−	0.935680i	$-0.614787\pi$
$181$	−9.49423	−0.705700	−0.352850	+	0.935680i	$0.614787\pi$
$182$	15.6445	1.15965
$183$	−1.87078	−0.138292
$184$	−2.54187	−0.187389
$185$	−6.65953	−0.489618
$186$	−7.48544	−0.548859
$187$	0	0
$188$	−0.323794	−0.0236151
$189$	2.88917	0.210156
$190$	−5.77376	−0.418873
$191$	−1.44578	−0.104613	−0.0523063	−	0.998631i	$-0.516657\pi$
$191$	−1.44578	−0.104613	−0.0523063	+	0.998631i	$0.516657\pi$
$192$	−1.00000	−0.0721688
$193$	24.8560	1.78917	0.894587	−	0.446894i	$-0.147470\pi$
$193$	24.8560	1.78917	0.894587	+	0.446894i	$0.147470\pi$
$194$	4.85131	0.348303
$195$	5.41487	0.387767
$196$	1.34730	0.0962355
$197$	−14.0411	−1.00039	−0.500194	−	0.865913i	$-0.666738\pi$
$197$	−14.0411	−1.00039	−0.500194	+	0.865913i	$0.666738\pi$
$198$	−5.96195	−0.423697
$199$	14.2065	1.00707	0.503537	−	0.863974i	$-0.332032\pi$
$199$	14.2065	1.00707	0.503537	+	0.863974i	$0.332032\pi$
$200$	−1.00000	−0.0707107
$201$	0.0859626	0.00606334
$202$	−12.2990	−0.865352
$203$	12.1287	0.851265
$204$	0	0
$205$	10.0038	0.698698
$206$	−5.90534	−0.411444
$207$	2.54187	0.176672
$208$	5.41487	0.375454
$209$	−34.4229	−2.38108
$210$	2.88917	0.199372
$211$	−10.6852	−0.735601	−0.367800	−	0.929905i	$-0.619889\pi$
$211$	−10.6852	−0.735601	−0.367800	+	0.929905i	$0.619889\pi$
$212$	−8.33615	−0.572529
$213$	−5.36550	−0.367638
$214$	9.34706	0.638952
$215$	−6.37207	−0.434572
$216$	1.00000	0.0680414
$217$	21.6267	1.46812
$218$	19.2630	1.30466
$219$	9.52783	0.643831
$220$	−5.96195	−0.401955
$221$	0	0
$222$	6.65953	0.446958
$223$	22.7470	1.52325	0.761627	−	0.648016i	$-0.224401\pi$
$223$	22.7470	1.52325	0.761627	+	0.648016i	$0.224401\pi$
$224$	2.88917	0.193041
$225$	1.00000	0.0666667
$226$	−14.1039	−0.938179
$227$	2.35238	0.156133	0.0780664	−	0.996948i	$-0.475125\pi$
$227$	2.35238	0.156133	0.0780664	+	0.996948i	$0.475125\pi$
$228$	5.77376	0.382377
$229$	−14.2333	−0.940564	−0.470282	−	0.882516i	$-0.655848\pi$
$229$	−14.2333	−0.940564	−0.470282	+	0.882516i	$0.655848\pi$
$230$	2.54187	0.167606
$231$	17.2251	1.13333
$232$	4.19797	0.275610
$233$	−30.3936	−1.99115	−0.995576	−	0.0939627i	$-0.970047\pi$
$233$	−30.3936	−1.99115	−0.995576	+	0.0939627i	$0.970047\pi$
$234$	−5.41487	−0.353981
$235$	0.323794	0.0211220
$236$	2.64975	0.172484
$237$	6.37886	0.414352
$238$	0	0
$239$	−5.69699	−0.368507	−0.184254	−	0.982879i	$-0.558987\pi$
$239$	−5.69699	−0.368507	−0.184254	+	0.982879i	$0.558987\pi$
$240$	1.00000	0.0645497
$241$	−9.88211	−0.636562	−0.318281	−	0.947996i	$-0.603106\pi$
$241$	−9.88211	−0.636562	−0.318281	+	0.947996i	$0.603106\pi$
$242$	−24.5448	−1.57780
$243$	−1.00000	−0.0641500
$244$	1.87078	0.119765
$245$	−1.34730	−0.0860756
$246$	−10.0038	−0.637821
$247$	−31.2642	−1.98929
$248$	7.48544	0.475326
$249$	−1.96571	−0.124572
$250$	1.00000	0.0632456
$251$	4.05353	0.255857	0.127928	−	0.991783i	$-0.459167\pi$
$251$	4.05353	0.255857	0.127928	+	0.991783i	$0.459167\pi$
$252$	−2.88917	−0.182001
$253$	15.1545	0.952756
$254$	−18.7283	−1.17512
$255$	0	0
$256$	1.00000	0.0625000
$257$	−10.0712	−0.628227	−0.314113	−	0.949385i	$-0.601707\pi$
$257$	−10.0712	−0.628227	−0.314113	+	0.949385i	$0.601707\pi$
$258$	6.37207	0.396708
$259$	−19.2405	−1.19555
$260$	−5.41487	−0.335816
$261$	−4.19797	−0.259848
$262$	−11.9199	−0.736414
$263$	−0.195470	−0.0120532	−0.00602661	−	0.999982i	$-0.501918\pi$
$263$	−0.195470	−0.0120532	−0.00602661	+	0.999982i	$0.501918\pi$
$264$	5.96195	0.366933
$265$	8.33615	0.512085
$266$	−16.6814	−1.02280
$267$	4.96671	0.303958
$268$	−0.0859626	−0.00525100
$269$	−28.2941	−1.72512	−0.862561	−	0.505953i	$-0.831141\pi$
$269$	−28.2941	−1.72512	−0.862561	+	0.505953i	$0.831141\pi$
$270$	−1.00000	−0.0608581
$271$	20.5289	1.24704	0.623521	−	0.781806i	$-0.285701\pi$
$271$	20.5289	1.24704	0.623521	+	0.781806i	$0.285701\pi$
$272$	0	0
$273$	15.6445	0.946847
$274$	−11.9326	−0.720873
$275$	5.96195	0.359519
$276$	−2.54187	−0.153003
$277$	21.1035	1.26799	0.633994	−	0.773338i	$-0.281414\pi$
$277$	21.1035	1.26799	0.633994	+	0.773338i	$0.281414\pi$
$278$	16.6358	0.997752
$279$	−7.48544	−0.448141
$280$	−2.88917	−0.172661
$281$	−12.6436	−0.754257	−0.377128	−	0.926161i	$-0.623088\pi$
$281$	−12.6436	−0.754257	−0.377128	+	0.926161i	$0.623088\pi$
$282$	−0.323794	−0.0192817
$283$	20.9475	1.24520	0.622600	−	0.782540i	$-0.286076\pi$
$283$	20.9475	1.24520	0.622600	+	0.782540i	$0.286076\pi$
$284$	5.36550	0.318384
$285$	−5.77376	−0.342008
$286$	−32.2832	−1.90895
$287$	28.9028	1.70608
$288$	−1.00000	−0.0589256
$289$	0	0
$290$	−4.19797	−0.246514
$291$	4.85131	0.284389
$292$	−9.52783	−0.557574
$293$	−9.28474	−0.542420	−0.271210	−	0.962520i	$-0.587424\pi$
$293$	−9.28474	−0.542420	−0.271210	+	0.962520i	$0.587424\pi$
$294$	1.34730	0.0785759
$295$	−2.64975	−0.154274
$296$	−6.65953	−0.387077
$297$	−5.96195	−0.345947
$298$	−15.0525	−0.871969
$299$	13.7639	0.795988
$300$	−1.00000	−0.0577350
$301$	−18.4100	−1.06113
$302$	−5.11124	−0.294119
$303$	−12.2990	−0.706557
$304$	−5.77376	−0.331148
$305$	−1.87078	−0.107121
$306$	0	0
$307$	−23.0828	−1.31740	−0.658701	−	0.752405i	$-0.728894\pi$
$307$	−23.0828	−1.31740	−0.658701	+	0.752405i	$0.728894\pi$
$308$	−17.2251	−0.981490
$309$	−5.90534	−0.335943
$310$	−7.48544	−0.425144
$311$	−32.2048	−1.82617	−0.913083	−	0.407774i	$-0.866305\pi$
$311$	−32.2048	−1.82617	−0.913083	+	0.407774i	$0.866305\pi$
$312$	5.41487	0.306557
$313$	−22.7803	−1.28762	−0.643808	−	0.765187i	$-0.722647\pi$
$313$	−22.7803	−1.28762	−0.643808	+	0.765187i	$0.722647\pi$
$314$	13.6300	0.769183
$315$	2.88917	0.162786
$316$	−6.37886	−0.358839
$317$	−22.2314	−1.24864	−0.624321	−	0.781168i	$-0.714624\pi$
$317$	−22.2314	−1.24864	−0.624321	+	0.781168i	$0.714624\pi$
$318$	−8.33615	−0.467468
$319$	−25.0281	−1.40131
$320$	−1.00000	−0.0559017
$321$	9.34706	0.521702
$322$	7.34390	0.409259
$323$	0	0
$324$	1.00000	0.0555556
$325$	5.41487	0.300363
$326$	−23.1047	−1.27965
$327$	19.2630	1.06525
$328$	10.0038	0.552370
$329$	0.935497	0.0515756
$330$	−5.96195	−0.328195
$331$	−3.29012	−0.180841	−0.0904207	−	0.995904i	$-0.528821\pi$
$331$	−3.29012	−0.180841	−0.0904207	+	0.995904i	$0.528821\pi$
$332$	1.96571	0.107882
$333$	6.65953	0.364940
$334$	7.94026	0.434471
$335$	0.0859626	0.00469664
$336$	2.88917	0.157617
$337$	18.3957	1.00208	0.501039	−	0.865425i	$-0.332951\pi$
$337$	18.3957	1.00208	0.501039	+	0.865425i	$0.332951\pi$
$338$	−16.3208	−0.887737
$339$	−14.1039	−0.766020
$340$	0	0
$341$	−44.6278	−2.41673
$342$	5.77376	0.312209
$343$	16.3316	0.881824
$344$	−6.37207	−0.343559
$345$	2.54187	0.136850
$346$	16.5906	0.891914
$347$	18.1251	0.973008	0.486504	−	0.873678i	$-0.338272\pi$
$347$	18.1251	0.973008	0.486504	+	0.873678i	$0.338272\pi$
$348$	4.19797	0.225035
$349$	24.6083	1.31725	0.658626	−	0.752470i	$-0.271138\pi$
$349$	24.6083	1.31725	0.658626	+	0.752470i	$0.271138\pi$
$350$	2.88917	0.154433
$351$	−5.41487	−0.289024
$352$	−5.96195	−0.317773
$353$	−23.2908	−1.23964	−0.619821	−	0.784743i	$-0.712795\pi$
$353$	−23.2908	−1.23964	−0.619821	+	0.784743i	$0.712795\pi$
$354$	2.64975	0.140832
$355$	−5.36550	−0.284771
$356$	−4.96671	−0.263235
$357$	0	0
$358$	23.8002	1.25788
$359$	−22.8475	−1.20584	−0.602922	−	0.797800i	$-0.705997\pi$
$359$	−22.8475	−1.20584	−0.602922	+	0.797800i	$0.705997\pi$
$360$	1.00000	0.0527046
$361$	14.3363	0.754542
$362$	9.49423	0.499005
$363$	−24.5448	−1.28827
$364$	−15.6445	−0.819994
$365$	9.52783	0.498709
$366$	1.87078	0.0977873
$367$	8.00957	0.418096	0.209048	−	0.977905i	$-0.432964\pi$
$367$	8.00957	0.418096	0.209048	+	0.977905i	$0.432964\pi$
$368$	2.54187	0.132504
$369$	−10.0038	−0.520779
$370$	6.65953	0.346213
$371$	24.0845	1.25041
$372$	7.48544	0.388102
$373$	28.9776	1.50040	0.750201	−	0.661210i	$-0.229957\pi$
$373$	28.9776	1.50040	0.750201	+	0.661210i	$0.229957\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	0.323794	0.0166984
$377$	−22.7315	−1.17073
$378$	−2.88917	−0.148603
$379$	16.4566	0.845319	0.422660	−	0.906289i	$-0.361097\pi$
$379$	16.4566	0.845319	0.422660	+	0.906289i	$0.361097\pi$
$380$	5.77376	0.296188
$381$	−18.7283	−0.959479
$382$	1.44578	0.0739723
$383$	−3.06852	−0.156794	−0.0783971	−	0.996922i	$-0.524980\pi$
$383$	−3.06852	−0.156794	−0.0783971	+	0.996922i	$0.524980\pi$
$384$	1.00000	0.0510310
$385$	17.2251	0.877871
$386$	−24.8560	−1.26514
$387$	6.37207	0.323910
$388$	−4.85131	−0.246288
$389$	−17.2970	−0.876991	−0.438496	−	0.898733i	$-0.644489\pi$
$389$	−17.2970	−0.876991	−0.438496	+	0.898733i	$0.644489\pi$
$390$	−5.41487	−0.274193
$391$	0	0
$392$	−1.34730	−0.0680487
$393$	−11.9199	−0.601280
$394$	14.0411	0.707381
$395$	6.37886	0.320956
$396$	5.96195	0.299599
$397$	−13.9806	−0.701668	−0.350834	−	0.936438i	$-0.614102\pi$
$397$	−13.9806	−0.701668	−0.350834	+	0.936438i	$0.614102\pi$
$398$	−14.2065	−0.712108
$399$	−16.6814	−0.835113
$400$	1.00000	0.0500000
$401$	−0.985042	−0.0491907	−0.0245953	−	0.999697i	$-0.507830\pi$
$401$	−0.985042	−0.0491907	−0.0245953	+	0.999697i	$0.507830\pi$
$402$	−0.0859626	−0.00428743
$403$	−40.5327	−2.01908
$404$	12.2990	0.611896
$405$	−1.00000	−0.0496904
$406$	−12.1287	−0.601935
$407$	39.7038	1.96804
$408$	0	0
$409$	−15.7878	−0.780657	−0.390328	−	0.920676i	$-0.627639\pi$
$409$	−15.7878	−0.780657	−0.390328	+	0.920676i	$0.627639\pi$
$410$	−10.0038	−0.494054
$411$	−11.9326	−0.588590
$412$	5.90534	0.290935
$413$	−7.65557	−0.376706
$414$	−2.54187	−0.124926
$415$	−1.96571	−0.0964929
$416$	−5.41487	−0.265486
$417$	16.6358	0.814661
$418$	34.4229	1.68368
$419$	−29.6744	−1.44969	−0.724845	−	0.688912i	$-0.758089\pi$
$419$	−29.6744	−1.44969	−0.724845	+	0.688912i	$0.758089\pi$
$420$	−2.88917	−0.140977
$421$	−27.2130	−1.32628	−0.663140	−	0.748496i	$-0.730777\pi$
$421$	−27.2130	−1.32628	−0.663140	+	0.748496i	$0.730777\pi$
$422$	10.6852	0.520148
$423$	−0.323794	−0.0157434
$424$	8.33615	0.404839
$425$	0	0
$426$	5.36550	0.259959
$427$	−5.40500	−0.261566
$428$	−9.34706	−0.451807
$429$	−32.2832	−1.55865
$430$	6.37207	0.307288
$431$	19.2546	0.927459	0.463730	−	0.885977i	$-0.346511\pi$
$431$	19.2546	0.927459	0.463730	+	0.885977i	$0.346511\pi$
$432$	−1.00000	−0.0481125
$433$	−17.4955	−0.840782	−0.420391	−	0.907343i	$-0.638107\pi$
$433$	−17.4955	−0.840782	−0.420391	+	0.907343i	$0.638107\pi$
$434$	−21.6267	−1.03811
$435$	−4.19797	−0.201277
$436$	−19.2630	−0.922532
$437$	−14.6762	−0.702056
$438$	−9.52783	−0.455257
$439$	−2.10329	−0.100384	−0.0501922	−	0.998740i	$-0.515983\pi$
$439$	−2.10329	−0.100384	−0.0501922	+	0.998740i	$0.515983\pi$
$440$	5.96195	0.284225
$441$	1.34730	0.0641570
$442$	0	0
$443$	−27.1559	−1.29022	−0.645108	−	0.764091i	$-0.723188\pi$
$443$	−27.1559	−1.29022	−0.645108	+	0.764091i	$0.723188\pi$
$444$	−6.65953	−0.316047
$445$	4.96671	0.235445
$446$	−22.7470	−1.07710
$447$	−15.0525	−0.711960
$448$	−2.88917	−0.136500
$449$	−23.3756	−1.10316	−0.551581	−	0.834122i	$-0.685975\pi$
$449$	−23.3756	−1.10316	−0.551581	+	0.834122i	$0.685975\pi$
$450$	−1.00000	−0.0471405
$451$	−59.6424	−2.80845
$452$	14.1039	0.663393
$453$	−5.11124	−0.240147
$454$	−2.35238	−0.110403
$455$	15.6445	0.733424
$456$	−5.77376	−0.270381
$457$	17.0597	0.798017	0.399008	−	0.916947i	$-0.369354\pi$
$457$	17.0597	0.798017	0.399008	+	0.916947i	$0.369354\pi$
$458$	14.2333	0.665079
$459$	0	0
$460$	−2.54187	−0.118515
$461$	41.6718	1.94085	0.970424	−	0.241407i	$-0.0776088\pi$
$461$	41.6718	1.94085	0.970424	+	0.241407i	$0.0776088\pi$
$462$	−17.2251	−0.801383
$463$	11.6529	0.541555	0.270777	−	0.962642i	$-0.412719\pi$
$463$	11.6529	0.541555	0.270777	+	0.962642i	$0.412719\pi$
$464$	−4.19797	−0.194886
$465$	−7.48544	−0.347129
$466$	30.3936	1.40796
$467$	−12.4802	−0.577516	−0.288758	−	0.957402i	$-0.593242\pi$
$467$	−12.4802	−0.577516	−0.288758	+	0.957402i	$0.593242\pi$
$468$	5.41487	0.250303
$469$	0.248361	0.0114682
$470$	−0.323794	−0.0149355
$471$	13.6300	0.628035
$472$	−2.64975	−0.121965
$473$	37.9900	1.74678
$474$	−6.37886	−0.292991
$475$	−5.77376	−0.264918
$476$	0	0
$477$	−8.33615	−0.381686
$478$	5.69699	0.260574
$479$	42.7831	1.95481	0.977405	−	0.211374i	$-0.0677938\pi$
$479$	42.7831	1.95481	0.977405	+	0.211374i	$0.0677938\pi$
$480$	−1.00000	−0.0456435
$481$	36.0605	1.64422
$482$	9.88211	0.450118
$483$	7.34390	0.334159
$484$	24.5448	1.11567
$485$	4.85131	0.220286
$486$	1.00000	0.0453609
$487$	−38.8849	−1.76204	−0.881021	−	0.473076i	$-0.843143\pi$
$487$	−38.8849	−1.76204	−0.881021	+	0.473076i	$0.843143\pi$
$488$	−1.87078	−0.0846863
$489$	−23.1047	−1.04483
$490$	1.34730	0.0608646
$491$	−36.0329	−1.62614	−0.813070	−	0.582166i	$-0.802206\pi$
$491$	−36.0329	−1.62614	−0.813070	+	0.582166i	$0.802206\pi$
$492$	10.0038	0.451008
$493$	0	0
$494$	31.2642	1.40664
$495$	−5.96195	−0.267970
$496$	−7.48544	−0.336106
$497$	−15.5018	−0.695352
$498$	1.96571	0.0880856
$499$	33.3165	1.49145	0.745725	−	0.666254i	$-0.232103\pi$
$499$	33.3165	1.49145	0.745725	+	0.666254i	$0.232103\pi$
$500$	−1.00000	−0.0447214
$501$	7.94026	0.354744
$502$	−4.05353	−0.180918
$503$	−10.2222	−0.455785	−0.227892	−	0.973686i	$-0.573183\pi$
$503$	−10.2222	−0.455785	−0.227892	+	0.973686i	$0.573183\pi$
$504$	2.88917	0.128694
$505$	−12.2990	−0.547296
$506$	−15.1545	−0.673700
$507$	−16.3208	−0.724834
$508$	18.7283	0.830933
$509$	−14.8276	−0.657223	−0.328611	−	0.944465i	$-0.606581\pi$
$509$	−14.8276	−0.657223	−0.328611	+	0.944465i	$0.606581\pi$
$510$	0	0
$511$	27.5275	1.21775
$512$	−1.00000	−0.0441942
$513$	5.77376	0.254918
$514$	10.0712	0.444223
$515$	−5.90534	−0.260220
$516$	−6.37207	−0.280515
$517$	−1.93045	−0.0849009
$518$	19.2405	0.845379
$519$	16.5906	0.728245
$520$	5.41487	0.237458
$521$	12.8173	0.561534	0.280767	−	0.959776i	$-0.409411\pi$
$521$	12.8173	0.561534	0.280767	+	0.959776i	$0.409411\pi$
$522$	4.19797	0.183740
$523$	1.94776	0.0851698	0.0425849	−	0.999093i	$-0.486441\pi$
$523$	1.94776	0.0851698	0.0425849	+	0.999093i	$0.486441\pi$
$524$	11.9199	0.520724
$525$	2.88917	0.126094
$526$	0.195470	0.00852291
$527$	0	0
$528$	−5.96195	−0.259461
$529$	−16.5389	−0.719082
$530$	−8.33615	−0.362099
$531$	2.64975	0.114989
$532$	16.6814	0.723229
$533$	−54.1695	−2.34634
$534$	−4.96671	−0.214931
$535$	9.34706	0.404109
$536$	0.0859626	0.00371302
$537$	23.8002	1.02706
$538$	28.2941	1.21985
$539$	8.03251	0.345985
$540$	1.00000	0.0430331
$541$	14.8022	0.636398	0.318199	−	0.948024i	$-0.396922\pi$
$541$	14.8022	0.636398	0.318199	+	0.948024i	$0.396922\pi$
$542$	−20.5289	−0.881792
$543$	9.49423	0.407436
$544$	0	0
$545$	19.2630	0.825138
$546$	−15.6445	−0.669522
$547$	−18.5261	−0.792119	−0.396059	−	0.918225i	$-0.629623\pi$
$547$	−18.5261	−0.792119	−0.396059	+	0.918225i	$0.629623\pi$
$548$	11.9326	0.509734
$549$	1.87078	0.0798430
$550$	−5.96195	−0.254218
$551$	24.2381	1.03258
$552$	2.54187	0.108189
$553$	18.4296	0.783707
$554$	−21.1035	−0.896603
$555$	6.65953	0.282681
$556$	−16.6358	−0.705517
$557$	−11.4643	−0.485760	−0.242880	−	0.970056i	$-0.578092\pi$
$557$	−11.4643	−0.485760	−0.242880	+	0.970056i	$0.578092\pi$
$558$	7.48544	0.316884
$559$	34.5039	1.45936
$560$	2.88917	0.122090
$561$	0	0
$562$	12.6436	0.533340
$563$	−18.0101	−0.759034	−0.379517	−	0.925185i	$-0.623910\pi$
$563$	−18.0101	−0.759034	−0.379517	+	0.925185i	$0.623910\pi$
$564$	0.323794	0.0136342
$565$	−14.1039	−0.593357
$566$	−20.9475	−0.880490
$567$	−2.88917	−0.121334
$568$	−5.36550	−0.225131
$569$	11.8655	0.497427	0.248714	−	0.968577i	$-0.419992\pi$
$569$	11.8655	0.497427	0.248714	+	0.968577i	$0.419992\pi$
$570$	5.77376	0.241836
$571$	−24.4785	−1.02439	−0.512196	−	0.858869i	$-0.671168\pi$
$571$	−24.4785	−1.02439	−0.512196	+	0.858869i	$0.671168\pi$
$572$	32.2832	1.34983
$573$	1.44578	0.0603981
$574$	−28.9028	−1.20638
$575$	2.54187	0.106003
$576$	1.00000	0.0416667
$577$	19.3275	0.804612	0.402306	−	0.915505i	$-0.368209\pi$
$577$	19.3275	0.804612	0.402306	+	0.915505i	$0.368209\pi$
$578$	0	0
$579$	−24.8560	−1.03298
$580$	4.19797	0.174311
$581$	−5.67927	−0.235616
$582$	−4.85131	−0.201093
$583$	−49.6997	−2.05835
$584$	9.52783	0.394264
$585$	−5.41487	−0.223877
$586$	9.28474	0.383549
$587$	−8.86777	−0.366012	−0.183006	−	0.983112i	$-0.558583\pi$
$587$	−8.86777	−0.366012	−0.183006	+	0.983112i	$0.558583\pi$
$588$	−1.34730	−0.0555616
$589$	43.2191	1.78081
$590$	2.64975	0.109088
$591$	14.0411	0.577574
$592$	6.65953	0.273705
$593$	22.9878	0.943996	0.471998	−	0.881600i	$-0.343533\pi$
$593$	22.9878	0.943996	0.471998	+	0.881600i	$0.343533\pi$
$594$	5.96195	0.244622
$595$	0	0
$596$	15.0525	0.616575
$597$	−14.2065	−0.581434
$598$	−13.7639	−0.562848
$599$	−3.27023	−0.133618	−0.0668091	−	0.997766i	$-0.521282\pi$
$599$	−3.27023	−0.133618	−0.0668091	+	0.997766i	$0.521282\pi$
$600$	1.00000	0.0408248
$601$	−20.1776	−0.823062	−0.411531	−	0.911396i	$-0.635006\pi$
$601$	−20.1776	−0.823062	−0.411531	+	0.911396i	$0.635006\pi$
$602$	18.4100	0.750335
$603$	−0.0859626	−0.00350067
$604$	5.11124	0.207973
$605$	−24.5448	−0.997890
$606$	12.2990	0.499611
$607$	−0.979205	−0.0397447	−0.0198724	−	0.999803i	$-0.506326\pi$
$607$	−0.979205	−0.0397447	−0.0198724	+	0.999803i	$0.506326\pi$
$608$	5.77376	0.234157
$609$	−12.1287	−0.491478
$610$	1.87078	0.0757457
$611$	−1.75331	−0.0709311
$612$	0	0
$613$	9.64038	0.389371	0.194686	−	0.980866i	$-0.437631\pi$
$613$	9.64038	0.389371	0.194686	+	0.980866i	$0.437631\pi$
$614$	23.0828	0.931544
$615$	−10.0038	−0.403394
$616$	17.2251	0.694018
$617$	−36.0013	−1.44936	−0.724680	−	0.689086i	$-0.758012\pi$
$617$	−36.0013	−1.44936	−0.724680	+	0.689086i	$0.758012\pi$
$618$	5.90534	0.237548
$619$	−19.8582	−0.798168	−0.399084	−	0.916914i	$-0.630672\pi$
$619$	−19.8582	−0.798168	−0.399084	+	0.916914i	$0.630672\pi$
$620$	7.48544	0.300622
$621$	−2.54187	−0.102002
$622$	32.2048	1.29129
$623$	14.3497	0.574908
$624$	−5.41487	−0.216768
$625$	1.00000	0.0400000
$626$	22.7803	0.910482
$627$	34.4229	1.37472
$628$	−13.6300	−0.543895
$629$	0	0
$630$	−2.88917	−0.115107
$631$	−7.43942	−0.296159	−0.148079	−	0.988975i	$-0.547309\pi$
$631$	−7.43942	−0.296159	−0.148079	+	0.988975i	$0.547309\pi$
$632$	6.37886	0.253738
$633$	10.6852	0.424699
$634$	22.2314	0.882923
$635$	−18.7283	−0.743209
$636$	8.33615	0.330550
$637$	7.29544	0.289056
$638$	25.0281	0.990872
$639$	5.36550	0.212256
$640$	1.00000	0.0395285
$641$	40.3573	1.59402	0.797009	−	0.603968i	$-0.206415\pi$
$641$	40.3573	1.59402	0.797009	+	0.603968i	$0.206415\pi$
$642$	−9.34706	−0.368899
$643$	−6.02690	−0.237678	−0.118839	−	0.992914i	$-0.537917\pi$
$643$	−6.02690	−0.237678	−0.118839	+	0.992914i	$0.537917\pi$
$644$	−7.34390	−0.289390
$645$	6.37207	0.250900
$646$	0	0
$647$	10.1173	0.397751	0.198876	−	0.980025i	$-0.436271\pi$
$647$	10.1173	0.397751	0.198876	+	0.980025i	$0.436271\pi$
$648$	−1.00000	−0.0392837
$649$	15.7977	0.620112
$650$	−5.41487	−0.212389
$651$	−21.6267	−0.847617
$652$	23.1047	0.904850
$653$	−25.7232	−1.00663	−0.503313	−	0.864104i	$-0.667886\pi$
$653$	−25.7232	−1.00663	−0.503313	+	0.864104i	$0.667886\pi$
$654$	−19.2630	−0.753244
$655$	−11.9199	−0.465749
$656$	−10.0038	−0.390584
$657$	−9.52783	−0.371716
$658$	−0.935497	−0.0364695
$659$	−27.2503	−1.06152	−0.530760	−	0.847522i	$-0.678093\pi$
$659$	−27.2503	−1.06152	−0.530760	+	0.847522i	$0.678093\pi$
$660$	5.96195	0.232069
$661$	−35.8755	−1.39540	−0.697698	−	0.716392i	$-0.745792\pi$
$661$	−35.8755	−1.39540	−0.697698	+	0.716392i	$0.745792\pi$
$662$	3.29012	0.127874
$663$	0	0
$664$	−1.96571	−0.0762843
$665$	−16.6814	−0.646876
$666$	−6.65953	−0.258052
$667$	−10.6707	−0.413172
$668$	−7.94026	−0.307218
$669$	−22.7470	−0.879451
$670$	−0.0859626	−0.00332103
$671$	11.1535	0.430576
$672$	−2.88917	−0.111452
$673$	−33.5591	−1.29361	−0.646804	−	0.762656i	$-0.723895\pi$
$673$	−33.5591	−1.29361	−0.646804	+	0.762656i	$0.723895\pi$
$674$	−18.3957	−0.708576
$675$	−1.00000	−0.0384900
$676$	16.3208	0.627725
$677$	19.7749	0.760011	0.380006	−	0.924984i	$-0.375922\pi$
$677$	19.7749	0.760011	0.380006	+	0.924984i	$0.375922\pi$
$678$	14.1039	0.541658
$679$	14.0162	0.537894
$680$	0	0
$681$	−2.35238	−0.0901433
$682$	44.6278	1.70889
$683$	1.13606	0.0434701	0.0217351	−	0.999764i	$-0.493081\pi$
$683$	1.13606	0.0434701	0.0217351	+	0.999764i	$0.493081\pi$
$684$	−5.77376	−0.220765
$685$	−11.9326	−0.455920
$686$	−16.3316	−0.623544
$687$	14.2333	0.543035
$688$	6.37207	0.242933
$689$	−45.1392	−1.71967
$690$	−2.54187	−0.0967674
$691$	40.6921	1.54800	0.774001	−	0.633185i	$-0.218253\pi$
$691$	40.6921	1.54800	0.774001	+	0.633185i	$0.218253\pi$
$692$	−16.5906	−0.630678
$693$	−17.2251	−0.654327
$694$	−18.1251	−0.688021
$695$	16.6358	0.631033
$696$	−4.19797	−0.159124
$697$	0	0
$698$	−24.6083	−0.931438
$699$	30.3936	1.14959
$700$	−2.88917	−0.109200
$701$	−9.49403	−0.358584	−0.179292	−	0.983796i	$-0.557381\pi$
$701$	−9.49403	−0.358584	−0.179292	+	0.983796i	$0.557381\pi$
$702$	5.41487	0.204371
$703$	−38.4505	−1.45019
$704$	5.96195	0.224699
$705$	−0.323794	−0.0121948
$706$	23.2908	0.876559
$707$	−35.5338	−1.33638
$708$	−2.64975	−0.0995836
$709$	12.0013	0.450719	0.225360	−	0.974276i	$-0.427644\pi$
$709$	12.0013	0.450719	0.225360	+	0.974276i	$0.427644\pi$
$710$	5.36550	0.201364
$711$	−6.37886	−0.239226
$712$	4.96671	0.186135
$713$	−19.0270	−0.712568
$714$	0	0
$715$	−32.2832	−1.20732
$716$	−23.8002	−0.889457
$717$	5.69699	0.212758
$718$	22.8475	0.852661
$719$	21.4920	0.801516	0.400758	−	0.916184i	$-0.368747\pi$
$719$	21.4920	0.801516	0.400758	+	0.916184i	$0.368747\pi$
$720$	−1.00000	−0.0372678
$721$	−17.0615	−0.635404
$722$	−14.3363	−0.533542
$723$	9.88211	0.367520
$724$	−9.49423	−0.352850
$725$	−4.19797	−0.155909
$726$	24.5448	0.910945
$727$	−35.2780	−1.30839	−0.654195	−	0.756326i	$-0.726992\pi$
$727$	−35.2780	−1.30839	−0.654195	+	0.756326i	$0.726992\pi$
$728$	15.6445	0.579823
$729$	1.00000	0.0370370
$730$	−9.52783	−0.352641
$731$	0	0
$732$	−1.87078	−0.0691461
$733$	−6.39701	−0.236279	−0.118139	−	0.992997i	$-0.537693\pi$
$733$	−6.39701	−0.236279	−0.118139	+	0.992997i	$0.537693\pi$
$734$	−8.00957	−0.295639
$735$	1.34730	0.0496958
$736$	−2.54187	−0.0936947
$737$	−0.512505	−0.0188784
$738$	10.0038	0.368246
$739$	50.1030	1.84307	0.921535	−	0.388296i	$-0.126936\pi$
$739$	50.1030	1.84307	0.921535	+	0.388296i	$0.126936\pi$
$740$	−6.65953	−0.244809
$741$	31.2642	1.14852
$742$	−24.0845	−0.884171
$743$	−52.1973	−1.91493	−0.957467	−	0.288544i	$-0.906829\pi$
$743$	−52.1973	−1.91493	−0.957467	+	0.288544i	$0.906829\pi$
$744$	−7.48544	−0.274429
$745$	−15.0525	−0.551482
$746$	−28.9776	−1.06094
$747$	1.96571	0.0719216
$748$	0	0
$749$	27.0052	0.986750
$750$	−1.00000	−0.0365148
$751$	−35.0908	−1.28048	−0.640241	−	0.768174i	$-0.721166\pi$
$751$	−35.0908	−1.28048	−0.640241	+	0.768174i	$0.721166\pi$
$752$	−0.323794	−0.0118076
$753$	−4.05353	−0.147719
$754$	22.7315	0.827832
$755$	−5.11124	−0.186017
$756$	2.88917	0.105078
$757$	43.7124	1.58875	0.794377	−	0.607425i	$-0.207797\pi$
$757$	43.7124	1.58875	0.794377	+	0.607425i	$0.207797\pi$
$758$	−16.4566	−0.597731
$759$	−15.1545	−0.550074
$760$	−5.77376	−0.209436
$761$	34.2723	1.24237	0.621185	−	0.783664i	$-0.286652\pi$
$761$	34.2723	1.24237	0.621185	+	0.783664i	$0.286652\pi$
$762$	18.7283	0.678454
$763$	55.6542	2.01482
$764$	−1.44578	−0.0523063
$765$	0	0
$766$	3.06852	0.110870
$767$	14.3480	0.518078
$768$	−1.00000	−0.0360844
$769$	−34.9799	−1.26141	−0.630703	−	0.776024i	$-0.717233\pi$
$769$	−34.9799	−1.26141	−0.630703	+	0.776024i	$0.717233\pi$
$770$	−17.2251	−0.620749
$771$	10.0712	0.362707
$772$	24.8560	0.894587
$773$	17.5539	0.631369	0.315684	−	0.948864i	$-0.397766\pi$
$773$	17.5539	0.631369	0.315684	+	0.948864i	$0.397766\pi$
$774$	−6.37207	−0.229039
$775$	−7.48544	−0.268885
$776$	4.85131	0.174152
$777$	19.2405	0.690249
$778$	17.2970	0.620127
$779$	57.7598	2.06946
$780$	5.41487	0.193884
$781$	31.9888	1.14465
$782$	0	0
$783$	4.19797	0.150023
$784$	1.34730	0.0481177
$785$	13.6300	0.486474
$786$	11.9199	0.425169
$787$	53.0908	1.89248	0.946242	−	0.323459i	$-0.104846\pi$
$787$	53.0908	1.89248	0.946242	+	0.323459i	$0.104846\pi$
$788$	−14.0411	−0.500194
$789$	0.195470	0.00695893
$790$	−6.37886	−0.226950
$791$	−40.7486	−1.44885
$792$	−5.96195	−0.211849
$793$	10.1300	0.359728
$794$	13.9806	0.496154
$795$	−8.33615	−0.295653
$796$	14.2065	0.503537
$797$	−16.2390	−0.575216	−0.287608	−	0.957748i	$-0.592860\pi$
$797$	−16.2390	−0.575216	−0.287608	+	0.957748i	$0.592860\pi$
$798$	16.6814	0.590514
$799$	0	0
$800$	−1.00000	−0.0353553
$801$	−4.96671	−0.175490
$802$	0.985042	0.0347831
$803$	−56.8044	−2.00459
$804$	0.0859626	0.00303167
$805$	7.34390	0.258838
$806$	40.5327	1.42770
$807$	28.2941	0.995999
$808$	−12.2990	−0.432676
$809$	25.3462	0.891124	0.445562	−	0.895251i	$-0.353004\pi$
$809$	25.3462	0.891124	0.445562	+	0.895251i	$0.353004\pi$
$810$	1.00000	0.0351364
$811$	−7.67005	−0.269332	−0.134666	−	0.990891i	$-0.542996\pi$
$811$	−7.67005	−0.269332	−0.134666	+	0.990891i	$0.542996\pi$
$812$	12.1287	0.425632
$813$	−20.5289	−0.719980
$814$	−39.7038	−1.39162
$815$	−23.1047	−0.809322
$816$	0	0
$817$	−36.7908	−1.28715
$818$	15.7878	0.552008
$819$	−15.6445	−0.546662
$820$	10.0038	0.349349
$821$	−13.7940	−0.481413	−0.240707	−	0.970598i	$-0.577379\pi$
$821$	−13.7940	−0.481413	−0.240707	+	0.970598i	$0.577379\pi$
$822$	11.9326	0.416196
$823$	−13.3213	−0.464350	−0.232175	−	0.972674i	$-0.574584\pi$
$823$	−13.3213	−0.464350	−0.232175	+	0.972674i	$0.574584\pi$
$824$	−5.90534	−0.205722
$825$	−5.96195	−0.207568
$826$	7.65557	0.266371
$827$	−16.8942	−0.587468	−0.293734	−	0.955887i	$-0.594898\pi$
$827$	−16.8942	−0.587468	−0.293734	+	0.955887i	$0.594898\pi$
$828$	2.54187	0.0883362
$829$	−20.7635	−0.721146	−0.360573	−	0.932731i	$-0.617419\pi$
$829$	−20.7635	−0.721146	−0.360573	+	0.932731i	$0.617419\pi$
$830$	1.96571	0.0682308
$831$	−21.1035	−0.732073
$832$	5.41487	0.187727
$833$	0	0
$834$	−16.6358	−0.576052
$835$	7.94026	0.274784
$836$	−34.4229	−1.19054
$837$	7.48544	0.258734
$838$	29.6744	1.02509
$839$	22.7270	0.784623	0.392311	−	0.919833i	$-0.371676\pi$
$839$	22.7270	0.784623	0.392311	+	0.919833i	$0.371676\pi$
$840$	2.88917	0.0996858
$841$	−11.3770	−0.392311
$842$	27.2130	0.937821
$843$	12.6436	0.435470
$844$	−10.6852	−0.367800
$845$	−16.3208	−0.561454
$846$	0.323794	0.0111323
$847$	−70.9142	−2.43664
$848$	−8.33615	−0.286264
$849$	−20.9475	−0.718917
$850$	0	0
$851$	16.9277	0.580273
$852$	−5.36550	−0.183819
$853$	7.38871	0.252984	0.126492	−	0.991968i	$-0.459628\pi$
$853$	7.38871	0.252984	0.126492	+	0.991968i	$0.459628\pi$
$854$	5.40500	0.184955
$855$	5.77376	0.197458
$856$	9.34706	0.319476
$857$	−2.97047	−0.101469	−0.0507347	−	0.998712i	$-0.516156\pi$
$857$	−2.97047	−0.101469	−0.0507347	+	0.998712i	$0.516156\pi$
$858$	32.2832	1.10213
$859$	17.0534	0.581856	0.290928	−	0.956745i	$-0.406036\pi$
$859$	17.0534	0.581856	0.290928	+	0.956745i	$0.406036\pi$
$860$	−6.37207	−0.217286
$861$	−28.9028	−0.985004
$862$	−19.2546	−0.655813
$863$	40.6756	1.38461	0.692306	−	0.721604i	$-0.256595\pi$
$863$	40.6756	1.38461	0.692306	+	0.721604i	$0.256595\pi$
$864$	1.00000	0.0340207
$865$	16.5906	0.564096
$866$	17.4955	0.594523
$867$	0	0
$868$	21.6267	0.734058
$869$	−38.0305	−1.29010
$870$	4.19797	0.142325
$871$	−0.465477	−0.0157721
$872$	19.2630	0.652329
$873$	−4.85131	−0.164192
$874$	14.6762	0.496429
$875$	2.88917	0.0976717
$876$	9.52783	0.321915
$877$	−6.06325	−0.204741	−0.102371	−	0.994746i	$-0.532643\pi$
$877$	−6.06325	−0.204741	−0.102371	+	0.994746i	$0.532643\pi$
$878$	2.10329	0.0709825
$879$	9.28474	0.313167
$880$	−5.96195	−0.200977
$881$	5.33371	0.179697	0.0898486	−	0.995955i	$-0.471362\pi$
$881$	5.33371	0.179697	0.0898486	+	0.995955i	$0.471362\pi$
$882$	−1.34730	−0.0453658
$883$	4.44248	0.149501	0.0747507	−	0.997202i	$-0.476184\pi$
$883$	4.44248	0.149501	0.0747507	+	0.997202i	$0.476184\pi$
$884$	0	0
$885$	2.64975	0.0890703
$886$	27.1559	0.912320
$887$	−39.4097	−1.32325	−0.661625	−	0.749835i	$-0.730133\pi$
$887$	−39.4097	−1.32325	−0.661625	+	0.749835i	$0.730133\pi$
$888$	6.65953	0.223479
$889$	−54.1092	−1.81476
$890$	−4.96671	−0.166485
$891$	5.96195	0.199733
$892$	22.7470	0.761627
$893$	1.86951	0.0625608
$894$	15.0525	0.503432
$895$	23.8002	0.795554
$896$	2.88917	0.0965204
$897$	−13.7639	−0.459564
$898$	23.3756	0.780053
$899$	31.4237	1.04804
$900$	1.00000	0.0333333
$901$	0	0
$902$	59.6424	1.98587
$903$	18.4100	0.612646
$904$	−14.1039	−0.469090
$905$	9.49423	0.315599
$906$	5.11124	0.169810
$907$	9.70767	0.322338	0.161169	−	0.986927i	$-0.448474\pi$
$907$	9.70767	0.322338	0.161169	+	0.986927i	$0.448474\pi$
$908$	2.35238	0.0780664
$909$	12.2990	0.407931
$910$	−15.6445	−0.518609
$911$	−56.7509	−1.88024	−0.940121	−	0.340841i	$-0.889288\pi$
$911$	−56.7509	−1.88024	−0.940121	+	0.340841i	$0.889288\pi$
$912$	5.77376	0.191188
$913$	11.7195	0.387858
$914$	−17.0597	−0.564283
$915$	1.87078	0.0618461
$916$	−14.2333	−0.470282
$917$	−34.4386	−1.13726
$918$	0	0
$919$	2.92347	0.0964364	0.0482182	−	0.998837i	$-0.484646\pi$
$919$	2.92347	0.0964364	0.0482182	+	0.998837i	$0.484646\pi$
$920$	2.54187	0.0838031
$921$	23.0828	0.760603
$922$	−41.6718	−1.37239
$923$	29.0535	0.956307
$924$	17.2251	0.566664
$925$	6.65953	0.218964
$926$	−11.6529	−0.382937
$927$	5.90534	0.193957
$928$	4.19797	0.137805
$929$	−7.82446	−0.256712	−0.128356	−	0.991728i	$-0.540970\pi$
$929$	−7.82446	−0.256712	−0.128356	+	0.991728i	$0.540970\pi$
$930$	7.48544	0.245457
$931$	−7.77897	−0.254945
$932$	−30.3936	−0.995576
$933$	32.2048	1.05434
$934$	12.4802	0.408366
$935$	0	0
$936$	−5.41487	−0.176991
$937$	−5.33796	−0.174383	−0.0871917	−	0.996192i	$-0.527789\pi$
$937$	−5.33796	−0.174383	−0.0871917	+	0.996192i	$0.527789\pi$
$938$	−0.248361	−0.00810926
$939$	22.7803	0.743406
$940$	0.323794	0.0105610
$941$	−11.4137	−0.372076	−0.186038	−	0.982543i	$-0.559565\pi$
$941$	−11.4137	−0.372076	−0.186038	+	0.982543i	$0.559565\pi$
$942$	−13.6300	−0.444088
$943$	−25.4285	−0.828065
$944$	2.64975	0.0862419
$945$	−2.88917	−0.0939847
$946$	−37.9900	−1.23516
$947$	−2.47188	−0.0803252	−0.0401626	−	0.999193i	$-0.512788\pi$
$947$	−2.47188	−0.0803252	−0.0401626	+	0.999193i	$0.512788\pi$
$948$	6.37886	0.207176
$949$	−51.5920	−1.67475
$950$	5.77376	0.187326
$951$	22.2314	0.720904
$952$	0	0
$953$	−39.1057	−1.26676	−0.633378	−	0.773842i	$-0.718332\pi$
$953$	−39.1057	−1.26676	−0.633378	+	0.773842i	$0.718332\pi$
$954$	8.33615	0.269893
$955$	1.44578	0.0467842
$956$	−5.69699	−0.184254
$957$	25.0281	0.809044
$958$	−42.7831	−1.38226
$959$	−34.4752	−1.11326
$960$	1.00000	0.0322749
$961$	25.0317	0.807476
$962$	−36.0605	−1.16264
$963$	−9.34706	−0.301205
$964$	−9.88211	−0.318281
$965$	−24.8560	−0.800143
$966$	−7.34390	−0.236286
$967$	−49.3980	−1.58853	−0.794267	−	0.607569i	$-0.792145\pi$
$967$	−49.3980	−1.58853	−0.794267	+	0.607569i	$0.792145\pi$
$968$	−24.5448	−0.788901
$969$	0	0
$970$	−4.85131	−0.155766
$971$	−27.3231	−0.876839	−0.438420	−	0.898770i	$-0.644462\pi$
$971$	−27.3231	−0.876839	−0.438420	+	0.898770i	$0.644462\pi$
$972$	−1.00000	−0.0320750
$973$	48.0638	1.54085
$974$	38.8849	1.24595
$975$	−5.41487	−0.173415
$976$	1.87078	0.0598823
$977$	−9.67251	−0.309451	−0.154725	−	0.987958i	$-0.549449\pi$
$977$	−9.67251	−0.309451	−0.154725	+	0.987958i	$0.549449\pi$
$978$	23.1047	0.738807
$979$	−29.6113	−0.946381
$980$	−1.34730	−0.0430378
$981$	−19.2630	−0.615021
$982$	36.0329	1.14985
$983$	4.11594	0.131278	0.0656390	−	0.997843i	$-0.479091\pi$
$983$	4.11594	0.131278	0.0656390	+	0.997843i	$0.479091\pi$
$984$	−10.0038	−0.318911
$985$	14.0411	0.447387
$986$	0	0
$987$	−0.935497	−0.0297772
$988$	−31.2642	−0.994646
$989$	16.1970	0.515034
$990$	5.96195	0.189483
$991$	22.6157	0.718410	0.359205	−	0.933259i	$-0.383048\pi$
$991$	22.6157	0.718410	0.359205	+	0.933259i	$0.383048\pi$
$992$	7.48544	0.237663
$993$	3.29012	0.104409
$994$	15.5018	0.491688
$995$	−14.2065	−0.450377
$996$	−1.96571	−0.0622859
$997$	−48.6825	−1.54179	−0.770894	−	0.636963i	$-0.780190\pi$
$997$	−48.6825	−1.54179	−0.770894	+	0.636963i	$0.780190\pi$
$998$	−33.3165	−1.05461
$999$	−6.65953	−0.210698

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8670.2.a.cb.1.2	✓	6
17.16	even	2		8670.2.a.ce.1.5	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8670.2.a.cb.1.2	✓	6	1.1	even	1	trivial
8670.2.a.ce.1.5	yes	6	17.16	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} -2.88917 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} -2.88917 q^{7} -1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} +5.96195 q^{11} -1.00000 q^{12} +5.41487 q^{13} +2.88917 q^{14} +1.00000 q^{15} +1.00000 q^{16} -1.00000 q^{18} -5.77376 q^{19} -1.00000 q^{20} +2.88917 q^{21} -5.96195 q^{22} +2.54187 q^{23} +1.00000 q^{24} +1.00000 q^{25} -5.41487 q^{26} -1.00000 q^{27} -2.88917 q^{28} -4.19797 q^{29} -1.00000 q^{30} -7.48544 q^{31} -1.00000 q^{32} -5.96195 q^{33} +2.88917 q^{35} +1.00000 q^{36} +6.65953 q^{37} +5.77376 q^{38} -5.41487 q^{39} +1.00000 q^{40} -10.0038 q^{41} -2.88917 q^{42} +6.37207 q^{43} +5.96195 q^{44} -1.00000 q^{45} -2.54187 q^{46} -0.323794 q^{47} -1.00000 q^{48} +1.34730 q^{49} -1.00000 q^{50} +5.41487 q^{52} -8.33615 q^{53} +1.00000 q^{54} -5.96195 q^{55} +2.88917 q^{56} +5.77376 q^{57} +4.19797 q^{58} +2.64975 q^{59} +1.00000 q^{60} +1.87078 q^{61} +7.48544 q^{62} -2.88917 q^{63} +1.00000 q^{64} -5.41487 q^{65} +5.96195 q^{66} -0.0859626 q^{67} -2.54187 q^{69} -2.88917 q^{70} +5.36550 q^{71} -1.00000 q^{72} -9.52783 q^{73} -6.65953 q^{74} -1.00000 q^{75} -5.77376 q^{76} -17.2251 q^{77} +5.41487 q^{78} -6.37886 q^{79} -1.00000 q^{80} +1.00000 q^{81} +10.0038 q^{82} +1.96571 q^{83} +2.88917 q^{84} -6.37207 q^{86} +4.19797 q^{87} -5.96195 q^{88} -4.96671 q^{89} +1.00000 q^{90} -15.6445 q^{91} +2.54187 q^{92} +7.48544 q^{93} +0.323794 q^{94} +5.77376 q^{95} +1.00000 q^{96} -4.85131 q^{97} -1.34730 q^{98} +5.96195 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{2} - 6 q^{3} + 6 q^{4} - 6 q^{5} + 6 q^{6} - 6 q^{8} + 6 q^{9}+O(q^{10}) \) 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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