LMFDB - Embedded newform 5070.2.a.bo.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5070,2,Mod(1,5070)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5070, 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("5070.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5070.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:	$40.4841538248$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{14})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 2x + 1 $ x^3 - x^2 - 2*x + 1
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			$-1.24698$ of defining polynomial
Character	$\chi$	$=$	5070.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} +1.75302 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} +1.75302 q^{7} -1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} +1.19806 q^{11} +1.00000 q^{12} -1.75302 q^{14} -1.00000 q^{15} +1.00000 q^{16} -6.76271 q^{17} -1.00000 q^{18} +8.63102 q^{19} -1.00000 q^{20} +1.75302 q^{21} -1.19806 q^{22} -6.04892 q^{23} -1.00000 q^{24} +1.00000 q^{25} +1.00000 q^{27} +1.75302 q^{28} +4.07069 q^{29} +1.00000 q^{30} +6.80194 q^{31} -1.00000 q^{32} +1.19806 q^{33} +6.76271 q^{34} -1.75302 q^{35} +1.00000 q^{36} -4.34481 q^{37} -8.63102 q^{38} +1.00000 q^{40} +6.63102 q^{41} -1.75302 q^{42} +4.07606 q^{43} +1.19806 q^{44} -1.00000 q^{45} +6.04892 q^{46} -11.1957 q^{47} +1.00000 q^{48} -3.92692 q^{49} -1.00000 q^{50} -6.76271 q^{51} -1.14914 q^{53} -1.00000 q^{54} -1.19806 q^{55} -1.75302 q^{56} +8.63102 q^{57} -4.07069 q^{58} +13.7235 q^{59} -1.00000 q^{60} -1.18598 q^{61} -6.80194 q^{62} +1.75302 q^{63} +1.00000 q^{64} -1.19806 q^{66} +14.1468 q^{67} -6.76271 q^{68} -6.04892 q^{69} +1.75302 q^{70} -2.32304 q^{71} -1.00000 q^{72} +1.59850 q^{73} +4.34481 q^{74} +1.00000 q^{75} +8.63102 q^{76} +2.10023 q^{77} +3.74094 q^{79} -1.00000 q^{80} +1.00000 q^{81} -6.63102 q^{82} +8.40581 q^{83} +1.75302 q^{84} +6.76271 q^{85} -4.07606 q^{86} +4.07069 q^{87} -1.19806 q^{88} +4.49934 q^{89} +1.00000 q^{90} -6.04892 q^{92} +6.80194 q^{93} +11.1957 q^{94} -8.63102 q^{95} -1.00000 q^{96} -10.7952 q^{97} +3.92692 q^{98} +1.19806 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{2} + 3 q^{3} + 3 q^{4} - 3 q^{5} - 3 q^{6} + 10 q^{7} - 3 q^{8} + 3 q^{9}+O(q^{10}) $ 3 * q - 3 * q^2 + 3 * q^3 + 3 * q^4 - 3 * q^5 - 3 * q^6 + 10 * q^7 - 3 * q^8 + 3 * q^9	$ 3 q - 3 q^{2} + 3 q^{3} + 3 q^{4} - 3 q^{5} - 3 q^{6} + 10 q^{7} - 3 q^{8} + 3 q^{9} + 3 q^{10} + 8 q^{11} + 3 q^{12} - 10 q^{14} - 3 q^{15} + 3 q^{16} - 3 q^{17} - 3 q^{18} + 11 q^{19} - 3 q^{20} + 10 q^{21} - 8 q^{22} - 9 q^{23} - 3 q^{24} + 3 q^{25} + 3 q^{27} + 10 q^{28} + 3 q^{30} + 16 q^{31} - 3 q^{32} + 8 q^{33} + 3 q^{34} - 10 q^{35} + 3 q^{36} + 10 q^{37} - 11 q^{38} + 3 q^{40} + 5 q^{41} - 10 q^{42} - 3 q^{43} + 8 q^{44} - 3 q^{45} + 9 q^{46} + 3 q^{47} + 3 q^{48} + 17 q^{49} - 3 q^{50} - 3 q^{51} - 17 q^{53} - 3 q^{54} - 8 q^{55} - 10 q^{56} + 11 q^{57} + 11 q^{59} - 3 q^{60} + 11 q^{61} - 16 q^{62} + 10 q^{63} + 3 q^{64} - 8 q^{66} + 15 q^{67} - 3 q^{68} - 9 q^{69} + 10 q^{70} + 13 q^{71} - 3 q^{72} - q^{73} - 10 q^{74} + 3 q^{75} + 11 q^{76} + 29 q^{77} - 3 q^{79} - 3 q^{80} + 3 q^{81} - 5 q^{82} + 12 q^{83} + 10 q^{84} + 3 q^{85} + 3 q^{86} - 8 q^{88} + q^{89} + 3 q^{90} - 9 q^{92} + 16 q^{93} - 3 q^{94} - 11 q^{95} - 3 q^{96} - 6 q^{97} - 17 q^{98} + 8 q^{99}+O(q^{100}) $ 3 * q - 3 * q^2 + 3 * q^3 + 3 * q^4 - 3 * q^5 - 3 * q^6 + 10 * q^7 - 3 * q^8 + 3 * q^9 + 3 * q^10 + 8 * q^11 + 3 * q^12 - 10 * q^14 - 3 * q^15 + 3 * q^16 - 3 * q^17 - 3 * q^18 + 11 * q^19 - 3 * q^20 + 10 * q^21 - 8 * q^22 - 9 * q^23 - 3 * q^24 + 3 * q^25 + 3 * q^27 + 10 * q^28 + 3 * q^30 + 16 * q^31 - 3 * q^32 + 8 * q^33 + 3 * q^34 - 10 * q^35 + 3 * q^36 + 10 * q^37 - 11 * q^38 + 3 * q^40 + 5 * q^41 - 10 * q^42 - 3 * q^43 + 8 * q^44 - 3 * q^45 + 9 * q^46 + 3 * q^47 + 3 * q^48 + 17 * q^49 - 3 * q^50 - 3 * q^51 - 17 * q^53 - 3 * q^54 - 8 * q^55 - 10 * q^56 + 11 * q^57 + 11 * q^59 - 3 * q^60 + 11 * q^61 - 16 * q^62 + 10 * q^63 + 3 * q^64 - 8 * q^66 + 15 * q^67 - 3 * q^68 - 9 * q^69 + 10 * q^70 + 13 * q^71 - 3 * q^72 - q^73 - 10 * q^74 + 3 * q^75 + 11 * q^76 + 29 * q^77 - 3 * q^79 - 3 * q^80 + 3 * q^81 - 5 * q^82 + 12 * q^83 + 10 * q^84 + 3 * q^85 + 3 * q^86 - 8 * q^88 + q^89 + 3 * q^90 - 9 * q^92 + 16 * q^93 - 3 * q^94 - 11 * q^95 - 3 * q^96 - 6 * q^97 - 17 * q^98 + 8 * 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
$7$	1.75302	0.662579	0.331290	−	0.943529i	$-0.392516\pi$
$7$	1.75302	0.662579	0.331290	+	0.943529i	$0.392516\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	1.00000	0.316228
$11$	1.19806	0.361229	0.180615	−	0.983554i	$-0.442191\pi$
$11$	1.19806	0.361229	0.180615	+	0.983554i	$0.442191\pi$
$12$	1.00000	0.288675
$13$	0	0
$13$	0	0
$14$	−1.75302	−0.468514
$15$	−1.00000	−0.258199
$16$	1.00000	0.250000
$17$	−6.76271	−1.64020	−0.820099	−	0.572222i	$-0.806082\pi$
$17$	−6.76271	−1.64020	−0.820099	+	0.572222i	$0.806082\pi$
$18$	−1.00000	−0.235702
$19$	8.63102	1.98009	0.990046	−	0.140743i	$-0.0449491\pi$
$19$	8.63102	1.98009	0.990046	+	0.140743i	$0.0449491\pi$
$20$	−1.00000	−0.223607
$21$	1.75302	0.382540
$22$	−1.19806	−0.255428
$23$	−6.04892	−1.26129	−0.630643	−	0.776073i	$-0.717209\pi$
$23$	−6.04892	−1.26129	−0.630643	+	0.776073i	$0.717209\pi$
$24$	−1.00000	−0.204124
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	1.75302	0.331290
$29$	4.07069	0.755908	0.377954	−	0.925824i	$-0.376628\pi$
$29$	4.07069	0.755908	0.377954	+	0.925824i	$0.376628\pi$
$30$	1.00000	0.182574
$31$	6.80194	1.22166	0.610832	−	0.791760i	$-0.290835\pi$
$31$	6.80194	1.22166	0.610832	+	0.791760i	$0.290835\pi$
$32$	−1.00000	−0.176777
$33$	1.19806	0.208556
$34$	6.76271	1.15980
$35$	−1.75302	−0.296315
$36$	1.00000	0.166667
$37$	−4.34481	−0.714283	−0.357142	−	0.934050i	$-0.616249\pi$
$37$	−4.34481	−0.714283	−0.357142	+	0.934050i	$0.616249\pi$
$38$	−8.63102	−1.40014
$39$	0	0
$40$	1.00000	0.158114
$41$	6.63102	1.03559	0.517796	−	0.855504i	$-0.326753\pi$
$41$	6.63102	1.03559	0.517796	+	0.855504i	$0.326753\pi$
$42$	−1.75302	−0.270497
$43$	4.07606	0.621594	0.310797	−	0.950476i	$-0.399404\pi$
$43$	4.07606	0.621594	0.310797	+	0.950476i	$0.399404\pi$
$44$	1.19806	0.180615
$45$	−1.00000	−0.149071
$46$	6.04892	0.891864
$47$	−11.1957	−1.63306	−0.816528	−	0.577306i	$-0.804104\pi$
$47$	−11.1957	−1.63306	−0.816528	+	0.577306i	$0.804104\pi$
$48$	1.00000	0.144338
$49$	−3.92692	−0.560988
$50$	−1.00000	−0.141421
$51$	−6.76271	−0.946969
$52$	0	0
$53$	−1.14914	−0.157847	−0.0789236	−	0.996881i	$-0.525148\pi$
$53$	−1.14914	−0.157847	−0.0789236	+	0.996881i	$0.525148\pi$
$54$	−1.00000	−0.136083
$55$	−1.19806	−0.161547
$56$	−1.75302	−0.234257
$57$	8.63102	1.14321
$58$	−4.07069	−0.534507
$59$	13.7235	1.78664	0.893322	−	0.449416i	$-0.148368\pi$
$59$	13.7235	1.78664	0.893322	+	0.449416i	$0.148368\pi$
$60$	−1.00000	−0.129099
$61$	−1.18598	−0.151849	−0.0759246	−	0.997114i	$-0.524191\pi$
$61$	−1.18598	−0.151849	−0.0759246	+	0.997114i	$0.524191\pi$
$62$	−6.80194	−0.863847
$63$	1.75302	0.220860
$64$	1.00000	0.125000
$65$	0	0
$66$	−1.19806	−0.147471
$67$	14.1468	1.72830	0.864150	−	0.503234i	$-0.167856\pi$
$67$	14.1468	1.72830	0.864150	+	0.503234i	$0.167856\pi$
$68$	−6.76271	−0.820099
$69$	−6.04892	−0.728204
$70$	1.75302	0.209526
$71$	−2.32304	−0.275695	−0.137847	−	0.990453i	$-0.544018\pi$
$71$	−2.32304	−0.275695	−0.137847	+	0.990453i	$0.544018\pi$
$72$	−1.00000	−0.117851
$73$	1.59850	0.187090	0.0935451	−	0.995615i	$-0.470180\pi$
$73$	1.59850	0.187090	0.0935451	+	0.995615i	$0.470180\pi$
$74$	4.34481	0.505074
$75$	1.00000	0.115470
$76$	8.63102	0.990046
$77$	2.10023	0.239343
$78$	0	0
$79$	3.74094	0.420888	0.210444	−	0.977606i	$-0.432509\pi$
$79$	3.74094	0.420888	0.210444	+	0.977606i	$0.432509\pi$
$80$	−1.00000	−0.111803
$81$	1.00000	0.111111
$82$	−6.63102	−0.732274
$83$	8.40581	0.922658	0.461329	−	0.887229i	$-0.347373\pi$
$83$	8.40581	0.922658	0.461329	+	0.887229i	$0.347373\pi$
$84$	1.75302	0.191270
$85$	6.76271	0.733519
$86$	−4.07606	−0.439533
$87$	4.07069	0.436424
$88$	−1.19806	−0.127714
$89$	4.49934	0.476929	0.238464	−	0.971151i	$-0.423356\pi$
$89$	4.49934	0.476929	0.238464	+	0.971151i	$0.423356\pi$
$90$	1.00000	0.105409
$91$	0	0
$92$	−6.04892	−0.630643
$93$	6.80194	0.705328
$94$	11.1957	1.15475
$95$	−8.63102	−0.885524
$96$	−1.00000	−0.102062
$97$	−10.7952	−1.09609	−0.548045	−	0.836449i	$-0.684628\pi$
$97$	−10.7952	−1.09609	−0.548045	+	0.836449i	$0.684628\pi$
$98$	3.92692	0.396679
$99$	1.19806	0.120410
$100$	1.00000	0.100000
$101$	16.7875	1.67042	0.835208	−	0.549935i	$-0.185347\pi$
$101$	16.7875	1.67042	0.835208	+	0.549935i	$0.185347\pi$
$102$	6.76271	0.669608
$103$	−4.16421	−0.410312	−0.205156	−	0.978729i	$-0.565770\pi$
$103$	−4.16421	−0.410312	−0.205156	+	0.978729i	$0.565770\pi$
$104$	0	0
$105$	−1.75302	−0.171077
$106$	1.14914	0.111615
$107$	12.9661	1.25348	0.626742	−	0.779226i	$-0.284388\pi$
$107$	12.9661	1.25348	0.626742	+	0.779226i	$0.284388\pi$
$108$	1.00000	0.0962250
$109$	−19.4644	−1.86435	−0.932177	−	0.362004i	$-0.882093\pi$
$109$	−19.4644	−1.86435	−0.932177	+	0.362004i	$0.882093\pi$
$110$	1.19806	0.114231
$111$	−4.34481	−0.412392
$112$	1.75302	0.165645
$113$	0.515729	0.0485157	0.0242579	−	0.999706i	$-0.492278\pi$
$113$	0.515729	0.0485157	0.0242579	+	0.999706i	$0.492278\pi$
$114$	−8.63102	−0.808369
$115$	6.04892	0.564064
$116$	4.07069	0.377954
$117$	0	0
$118$	−13.7235	−1.26335
$119$	−11.8552	−1.08676
$120$	1.00000	0.0912871
$121$	−9.56465	−0.869513
$122$	1.18598	0.107374
$123$	6.63102	0.597899
$124$	6.80194	0.610832
$125$	−1.00000	−0.0894427
$126$	−1.75302	−0.156171
$127$	7.96077	0.706404	0.353202	−	0.935547i	$-0.385093\pi$
$127$	7.96077	0.706404	0.353202	+	0.935547i	$0.385093\pi$
$128$	−1.00000	−0.0883883
$129$	4.07606	0.358877
$130$	0	0
$131$	−5.08815	−0.444553	−0.222277	−	0.974984i	$-0.571349\pi$
$131$	−5.08815	−0.444553	−0.222277	+	0.974984i	$0.571349\pi$
$132$	1.19806	0.104278
$133$	15.1304	1.31197
$134$	−14.1468	−1.22209
$135$	−1.00000	−0.0860663
$136$	6.76271	0.579898
$137$	0.674563	0.0576318	0.0288159	−	0.999585i	$-0.490826\pi$
$137$	0.674563	0.0576318	0.0288159	+	0.999585i	$0.490826\pi$
$138$	6.04892	0.514918
$139$	8.26205	0.700778	0.350389	−	0.936604i	$-0.386049\pi$
$139$	8.26205	0.700778	0.350389	+	0.936604i	$0.386049\pi$
$140$	−1.75302	−0.148157
$141$	−11.1957	−0.942845
$142$	2.32304	0.194946
$143$	0	0
$144$	1.00000	0.0833333
$145$	−4.07069	−0.338052
$146$	−1.59850	−0.132293
$147$	−3.92692	−0.323887
$148$	−4.34481	−0.357142
$149$	−8.19136	−0.671062	−0.335531	−	0.942029i	$-0.608916\pi$
$149$	−8.19136	−0.671062	−0.335531	+	0.942029i	$0.608916\pi$
$150$	−1.00000	−0.0816497
$151$	−5.72348	−0.465770	−0.232885	−	0.972504i	$-0.574817\pi$
$151$	−5.72348	−0.465770	−0.232885	+	0.972504i	$0.574817\pi$
$152$	−8.63102	−0.700068
$153$	−6.76271	−0.546733
$154$	−2.10023	−0.169241
$155$	−6.80194	−0.546345
$156$	0	0
$157$	3.09246	0.246805	0.123403	−	0.992357i	$-0.460619\pi$
$157$	3.09246	0.246805	0.123403	+	0.992357i	$0.460619\pi$
$158$	−3.74094	−0.297613
$159$	−1.14914	−0.0911331
$160$	1.00000	0.0790569
$161$	−10.6039	−0.835702
$162$	−1.00000	−0.0785674
$163$	23.9541	1.87623	0.938114	−	0.346328i	$-0.112571\pi$
$163$	23.9541	1.87623	0.938114	+	0.346328i	$0.112571\pi$
$164$	6.63102	0.517796
$165$	−1.19806	−0.0932690
$166$	−8.40581	−0.652418
$167$	−1.19806	−0.0927088	−0.0463544	−	0.998925i	$-0.514760\pi$
$167$	−1.19806	−0.0927088	−0.0463544	+	0.998925i	$0.514760\pi$
$168$	−1.75302	−0.135248
$169$	0	0
$170$	−6.76271	−0.518676
$171$	8.63102	0.660031
$172$	4.07606	0.310797
$173$	−6.63773	−0.504657	−0.252329	−	0.967642i	$-0.581196\pi$
$173$	−6.63773	−0.504657	−0.252329	+	0.967642i	$0.581196\pi$
$174$	−4.07069	−0.308598
$175$	1.75302	0.132516
$176$	1.19806	0.0903073
$177$	13.7235	1.03152
$178$	−4.49934	−0.337239
$179$	19.6353	1.46761	0.733807	−	0.679358i	$-0.237742\pi$
$179$	19.6353	1.46761	0.733807	+	0.679358i	$0.237742\pi$
$180$	−1.00000	−0.0745356
$181$	5.25906	0.390903	0.195451	−	0.980713i	$-0.437383\pi$
$181$	5.25906	0.390903	0.195451	+	0.980713i	$0.437383\pi$
$182$	0	0
$183$	−1.18598	−0.0876702
$184$	6.04892	0.445932
$185$	4.34481	0.319437
$186$	−6.80194	−0.498742
$187$	−8.10215	−0.592488
$188$	−11.1957	−0.816528
$189$	1.75302	0.127513
$190$	8.63102	0.626160
$191$	4.45473	0.322333	0.161166	−	0.986927i	$-0.448474\pi$
$191$	4.45473	0.322333	0.161166	+	0.986927i	$0.448474\pi$
$192$	1.00000	0.0721688
$193$	−0.818331	−0.0589048	−0.0294524	−	0.999566i	$-0.509376\pi$
$193$	−0.818331	−0.0589048	−0.0294524	+	0.999566i	$0.509376\pi$
$194$	10.7952	0.775053
$195$	0	0
$196$	−3.92692	−0.280494
$197$	12.7385	0.907584	0.453792	−	0.891108i	$-0.350071\pi$
$197$	12.7385	0.907584	0.453792	+	0.891108i	$0.350071\pi$
$198$	−1.19806	−0.0851426
$199$	−4.02475	−0.285307	−0.142654	−	0.989773i	$-0.545563\pi$
$199$	−4.02475	−0.285307	−0.142654	+	0.989773i	$0.545563\pi$
$200$	−1.00000	−0.0707107
$201$	14.1468	0.997835
$202$	−16.7875	−1.18116
$203$	7.13600	0.500849
$204$	−6.76271	−0.473484
$205$	−6.63102	−0.463131
$206$	4.16421	0.290134
$207$	−6.04892	−0.420429
$208$	0	0
$209$	10.3405	0.715268
$210$	1.75302	0.120970
$211$	3.10992	0.214095	0.107048	−	0.994254i	$-0.465860\pi$
$211$	3.10992	0.214095	0.107048	+	0.994254i	$0.465860\pi$
$212$	−1.14914	−0.0789236
$213$	−2.32304	−0.159172
$214$	−12.9661	−0.886348
$215$	−4.07606	−0.277985
$216$	−1.00000	−0.0680414
$217$	11.9239	0.809449
$218$	19.4644	1.31830
$219$	1.59850	0.108017
$220$	−1.19806	−0.0807733
$221$	0	0
$222$	4.34481	0.291605
$223$	14.3787	0.962867	0.481433	−	0.876483i	$-0.340116\pi$
$223$	14.3787	0.962867	0.481433	+	0.876483i	$0.340116\pi$
$224$	−1.75302	−0.117129
$225$	1.00000	0.0666667
$226$	−0.515729	−0.0343058
$227$	−5.97716	−0.396718	−0.198359	−	0.980129i	$-0.563561\pi$
$227$	−5.97716	−0.396718	−0.198359	+	0.980129i	$0.563561\pi$
$228$	8.63102	0.571603
$229$	−15.9172	−1.05184	−0.525920	−	0.850534i	$-0.676279\pi$
$229$	−15.9172	−1.05184	−0.525920	+	0.850534i	$0.676279\pi$
$230$	−6.04892	−0.398854
$231$	2.10023	0.138185
$232$	−4.07069	−0.267254
$233$	−6.92154	−0.453445	−0.226723	−	0.973959i	$-0.572801\pi$
$233$	−6.92154	−0.453445	−0.226723	+	0.973959i	$0.572801\pi$
$234$	0	0
$235$	11.1957	0.730325
$236$	13.7235	0.893322
$237$	3.74094	0.243000
$238$	11.8552	0.768456
$239$	30.1075	1.94749	0.973747	−	0.227633i	$-0.0730988\pi$
$239$	30.1075	1.94749	0.973747	+	0.227633i	$0.0730988\pi$
$240$	−1.00000	−0.0645497
$241$	12.0562	0.776609	0.388304	−	0.921531i	$-0.373061\pi$
$241$	12.0562	0.776609	0.388304	+	0.921531i	$0.373061\pi$
$242$	9.56465	0.614839
$243$	1.00000	0.0641500
$244$	−1.18598	−0.0759246
$245$	3.92692	0.250882
$246$	−6.63102	−0.422779
$247$	0	0
$248$	−6.80194	−0.431923
$249$	8.40581	0.532697
$250$	1.00000	0.0632456
$251$	−14.8170	−0.935241	−0.467620	−	0.883929i	$-0.654889\pi$
$251$	−14.8170	−0.935241	−0.467620	+	0.883929i	$0.654889\pi$
$252$	1.75302	0.110430
$253$	−7.24698	−0.455614
$254$	−7.96077	−0.499503
$255$	6.76271	0.423497
$256$	1.00000	0.0625000
$257$	−10.5743	−0.659609	−0.329804	−	0.944049i	$-0.606983\pi$
$257$	−10.5743	−0.659609	−0.329804	+	0.944049i	$0.606983\pi$
$258$	−4.07606	−0.253765
$259$	−7.61655	−0.473269
$260$	0	0
$261$	4.07069	0.251969
$262$	5.08815	0.314347
$263$	11.9922	0.739473	0.369736	−	0.929137i	$-0.379448\pi$
$263$	11.9922	0.739473	0.369736	+	0.929137i	$0.379448\pi$
$264$	−1.19806	−0.0737356
$265$	1.14914	0.0705914
$266$	−15.1304	−0.927702
$267$	4.49934	0.275355
$268$	14.1468	0.864150
$269$	−25.4359	−1.55086	−0.775428	−	0.631435i	$-0.782466\pi$
$269$	−25.4359	−1.55086	−0.775428	+	0.631435i	$0.782466\pi$
$270$	1.00000	0.0608581
$271$	20.3666	1.23718	0.618591	−	0.785713i	$-0.287704\pi$
$271$	20.3666	1.23718	0.618591	+	0.785713i	$0.287704\pi$
$272$	−6.76271	−0.410049
$273$	0	0
$274$	−0.674563	−0.0407518
$275$	1.19806	0.0722459
$276$	−6.04892	−0.364102
$277$	20.1323	1.20963	0.604816	−	0.796366i	$-0.293247\pi$
$277$	20.1323	1.20963	0.604816	+	0.796366i	$0.293247\pi$
$278$	−8.26205	−0.495525
$279$	6.80194	0.407221
$280$	1.75302	0.104763
$281$	−29.7265	−1.77333	−0.886666	−	0.462410i	$-0.846985\pi$
$281$	−29.7265	−1.77333	−0.886666	+	0.462410i	$0.846985\pi$
$282$	11.1957	0.666692
$283$	8.84117	0.525553	0.262776	−	0.964857i	$-0.415362\pi$
$283$	8.84117	0.525553	0.262776	+	0.964857i	$0.415362\pi$
$284$	−2.32304	−0.137847
$285$	−8.63102	−0.511258
$286$	0	0
$287$	11.6243	0.686162
$288$	−1.00000	−0.0589256
$289$	28.7342	1.69025
$290$	4.07069	0.239039
$291$	−10.7952	−0.632828
$292$	1.59850	0.0935451
$293$	20.5918	1.20299	0.601493	−	0.798878i	$-0.294573\pi$
$293$	20.5918	1.20299	0.601493	+	0.798878i	$0.294573\pi$
$294$	3.92692	0.229023
$295$	−13.7235	−0.799012
$296$	4.34481	0.252537
$297$	1.19806	0.0695186
$298$	8.19136	0.474513
$299$	0	0
$300$	1.00000	0.0577350
$301$	7.14542	0.411855
$302$	5.72348	0.329349
$303$	16.7875	0.964415
$304$	8.63102	0.495023
$305$	1.18598	0.0679091
$306$	6.76271	0.386598
$307$	28.7265	1.63951	0.819753	−	0.572717i	$-0.194111\pi$
$307$	28.7265	1.63951	0.819753	+	0.572717i	$0.194111\pi$
$308$	2.10023	0.119672
$309$	−4.16421	−0.236894
$310$	6.80194	0.386324
$311$	−18.6329	−1.05658	−0.528289	−	0.849065i	$-0.677166\pi$
$311$	−18.6329	−1.05658	−0.528289	+	0.849065i	$0.677166\pi$
$312$	0	0
$313$	−29.3696	−1.66007	−0.830033	−	0.557714i	$-0.811679\pi$
$313$	−29.3696	−1.66007	−0.830033	+	0.557714i	$0.811679\pi$
$314$	−3.09246	−0.174517
$315$	−1.75302	−0.0987715
$316$	3.74094	0.210444
$317$	20.8092	1.16876	0.584381	−	0.811479i	$-0.301337\pi$
$317$	20.8092	1.16876	0.584381	+	0.811479i	$0.301337\pi$
$318$	1.14914	0.0644408
$319$	4.87694	0.273056
$320$	−1.00000	−0.0559017
$321$	12.9661	0.723700
$322$	10.6039	0.590931
$323$	−58.3691	−3.24774
$324$	1.00000	0.0555556
$325$	0	0
$326$	−23.9541	−1.32669
$327$	−19.4644	−1.07638
$328$	−6.63102	−0.366137
$329$	−19.6262	−1.08203
$330$	1.19806	0.0659512
$331$	−11.7603	−0.646405	−0.323203	−	0.946330i	$-0.604760\pi$
$331$	−11.7603	−0.646405	−0.323203	+	0.946330i	$0.604760\pi$
$332$	8.40581	0.461329
$333$	−4.34481	−0.238094
$334$	1.19806	0.0655551
$335$	−14.1468	−0.772920
$336$	1.75302	0.0956351
$337$	−26.8732	−1.46388	−0.731939	−	0.681371i	$-0.761384\pi$
$337$	−26.8732	−1.46388	−0.731939	+	0.681371i	$0.761384\pi$
$338$	0	0
$339$	0.515729	0.0280106
$340$	6.76271	0.366759
$341$	8.14914	0.441301
$342$	−8.63102	−0.466712
$343$	−19.1551	−1.03428
$344$	−4.07606	−0.219767
$345$	6.04892	0.325663
$346$	6.63773	0.356846
$347$	29.5080	1.58407	0.792035	−	0.610476i	$-0.209022\pi$
$347$	29.5080	1.58407	0.792035	+	0.610476i	$0.209022\pi$
$348$	4.07069	0.218212
$349$	17.0519	0.912767	0.456384	−	0.889783i	$-0.349144\pi$
$349$	17.0519	0.912767	0.456384	+	0.889783i	$0.349144\pi$
$350$	−1.75302	−0.0937029
$351$	0	0
$352$	−1.19806	−0.0638569
$353$	−21.8629	−1.16365	−0.581823	−	0.813315i	$-0.697660\pi$
$353$	−21.8629	−1.16365	−0.581823	+	0.813315i	$0.697660\pi$
$354$	−13.7235	−0.729395
$355$	2.32304	0.123294
$356$	4.49934	0.238464
$357$	−11.8552	−0.627442
$358$	−19.6353	−1.03776
$359$	14.3026	0.754862	0.377431	−	0.926038i	$-0.376807\pi$
$359$	14.3026	0.754862	0.377431	+	0.926038i	$0.376807\pi$
$360$	1.00000	0.0527046
$361$	55.4946	2.92077
$362$	−5.25906	−0.276410
$363$	−9.56465	−0.502014
$364$	0	0
$365$	−1.59850	−0.0836692
$366$	1.18598	0.0619922
$367$	−11.2416	−0.586807	−0.293403	−	0.955989i	$-0.594788\pi$
$367$	−11.2416	−0.586807	−0.293403	+	0.955989i	$0.594788\pi$
$368$	−6.04892	−0.315322
$369$	6.63102	0.345197
$370$	−4.34481	−0.225876
$371$	−2.01447	−0.104586
$372$	6.80194	0.352664
$373$	−9.70171	−0.502336	−0.251168	−	0.967944i	$-0.580815\pi$
$373$	−9.70171	−0.502336	−0.251168	+	0.967944i	$0.580815\pi$
$374$	8.10215	0.418952
$375$	−1.00000	−0.0516398
$376$	11.1957	0.577373
$377$	0	0
$378$	−1.75302	−0.0901656
$379$	−2.82802	−0.145266	−0.0726328	−	0.997359i	$-0.523140\pi$
$379$	−2.82802	−0.145266	−0.0726328	+	0.997359i	$0.523140\pi$
$380$	−8.63102	−0.442762
$381$	7.96077	0.407843
$382$	−4.45473	−0.227924
$383$	11.6558	0.595582	0.297791	−	0.954631i	$-0.403750\pi$
$383$	11.6558	0.595582	0.297791	+	0.954631i	$0.403750\pi$
$384$	−1.00000	−0.0510310
$385$	−2.10023	−0.107038
$386$	0.818331	0.0416520
$387$	4.07606	0.207198
$388$	−10.7952	−0.548045
$389$	12.5157	0.634573	0.317286	−	0.948330i	$-0.397228\pi$
$389$	12.5157	0.634573	0.317286	+	0.948330i	$0.397228\pi$
$390$	0	0
$391$	40.9071	2.06876
$392$	3.92692	0.198339
$393$	−5.08815	−0.256663
$394$	−12.7385	−0.641759
$395$	−3.74094	−0.188227
$396$	1.19806	0.0602049
$397$	21.6136	1.08475	0.542377	−	0.840135i	$-0.317525\pi$
$397$	21.6136	1.08475	0.542377	+	0.840135i	$0.317525\pi$
$398$	4.02475	0.201743
$399$	15.1304	0.757465
$400$	1.00000	0.0500000
$401$	−11.5157	−0.575068	−0.287534	−	0.957770i	$-0.592835\pi$
$401$	−11.5157	−0.575068	−0.287534	+	0.957770i	$0.592835\pi$
$402$	−14.1468	−0.705576
$403$	0	0
$404$	16.7875	0.835208
$405$	−1.00000	−0.0496904
$406$	−7.13600	−0.354154
$407$	−5.20536	−0.258020
$408$	6.76271	0.334804
$409$	−0.0991626	−0.00490328	−0.00245164	−	0.999997i	$-0.500780\pi$
$409$	−0.0991626	−0.00490328	−0.00245164	+	0.999997i	$0.500780\pi$
$410$	6.63102	0.327483
$411$	0.674563	0.0332737
$412$	−4.16421	−0.205156
$413$	24.0575	1.18379
$414$	6.04892	0.297288
$415$	−8.40581	−0.412625
$416$	0	0
$417$	8.26205	0.404594
$418$	−10.3405	−0.505771
$419$	31.2784	1.52805	0.764026	−	0.645186i	$-0.223220\pi$
$419$	31.2784	1.52805	0.764026	+	0.645186i	$0.223220\pi$
$420$	−1.75302	−0.0855386
$421$	−38.0452	−1.85421	−0.927105	−	0.374802i	$-0.877711\pi$
$421$	−38.0452	−1.85421	−0.927105	+	0.374802i	$0.877711\pi$
$422$	−3.10992	−0.151388
$423$	−11.1957	−0.544352
$424$	1.14914	0.0558074
$425$	−6.76271	−0.328040
$426$	2.32304	0.112552
$427$	−2.07905	−0.100612
$428$	12.9661	0.626742
$429$	0	0
$430$	4.07606	0.196565
$431$	−14.4614	−0.696583	−0.348291	−	0.937386i	$-0.613238\pi$
$431$	−14.4614	−0.696583	−0.348291	+	0.937386i	$0.613238\pi$
$432$	1.00000	0.0481125
$433$	21.6209	1.03903	0.519516	−	0.854461i	$-0.326112\pi$
$433$	21.6209	1.03903	0.519516	+	0.854461i	$0.326112\pi$
$434$	−11.9239	−0.572367
$435$	−4.07069	−0.195175
$436$	−19.4644	−0.932177
$437$	−52.2083	−2.49746
$438$	−1.59850	−0.0763792
$439$	−3.23623	−0.154457	−0.0772283	−	0.997013i	$-0.524607\pi$
$439$	−3.23623	−0.154457	−0.0772283	+	0.997013i	$0.524607\pi$
$440$	1.19806	0.0571154
$441$	−3.92692	−0.186996
$442$	0	0
$443$	−14.6649	−0.696749	−0.348375	−	0.937355i	$-0.613266\pi$
$443$	−14.6649	−0.696749	−0.348375	+	0.937355i	$0.613266\pi$
$444$	−4.34481	−0.206196
$445$	−4.49934	−0.213289
$446$	−14.3787	−0.680850
$447$	−8.19136	−0.387438
$448$	1.75302	0.0828224
$449$	4.08516	0.192791	0.0963954	−	0.995343i	$-0.469269\pi$
$449$	4.08516	0.192791	0.0963954	+	0.995343i	$0.469269\pi$
$450$	−1.00000	−0.0471405
$451$	7.94438	0.374086
$452$	0.515729	0.0242579
$453$	−5.72348	−0.268913
$454$	5.97716	0.280522
$455$	0	0
$456$	−8.63102	−0.404185
$457$	41.8678	1.95849	0.979247	−	0.202668i	$-0.0649613\pi$
$457$	41.8678	1.95849	0.979247	+	0.202668i	$0.0649613\pi$
$458$	15.9172	0.743763
$459$	−6.76271	−0.315656
$460$	6.04892	0.282032
$461$	−0.215521	−0.0100378	−0.00501890	−	0.999987i	$-0.501598\pi$
$461$	−0.215521	−0.0100378	−0.00501890	+	0.999987i	$0.501598\pi$
$462$	−2.10023	−0.0977114
$463$	23.4795	1.09118	0.545592	−	0.838051i	$-0.316305\pi$
$463$	23.4795	1.09118	0.545592	+	0.838051i	$0.316305\pi$
$464$	4.07069	0.188977
$465$	−6.80194	−0.315432
$466$	6.92154	0.320634
$467$	10.1153	0.468080	0.234040	−	0.972227i	$-0.424805\pi$
$467$	10.1153	0.468080	0.234040	+	0.972227i	$0.424805\pi$
$468$	0	0
$469$	24.7995	1.14514
$470$	−11.1957	−0.516418
$471$	3.09246	0.142493
$472$	−13.7235	−0.631674
$473$	4.88338	0.224538
$474$	−3.74094	−0.171827
$475$	8.63102	0.396018
$476$	−11.8552	−0.543381
$477$	−1.14914	−0.0526157
$478$	−30.1075	−1.37709
$479$	0.980623	0.0448058	0.0224029	−	0.999749i	$-0.492868\pi$
$479$	0.980623	0.0448058	0.0224029	+	0.999749i	$0.492868\pi$
$480$	1.00000	0.0456435
$481$	0	0
$482$	−12.0562	−0.549145
$483$	−10.6039	−0.482493
$484$	−9.56465	−0.434757
$485$	10.7952	0.490186
$486$	−1.00000	−0.0453609
$487$	7.18300	0.325493	0.162746	−	0.986668i	$-0.447965\pi$
$487$	7.18300	0.325493	0.162746	+	0.986668i	$0.447965\pi$
$488$	1.18598	0.0536868
$489$	23.9541	1.08324
$490$	−3.92692	−0.177400
$491$	24.5918	1.10981	0.554906	−	0.831913i	$-0.312754\pi$
$491$	24.5918	1.10981	0.554906	+	0.831913i	$0.312754\pi$
$492$	6.63102	0.298950
$493$	−27.5289	−1.23984
$494$	0	0
$495$	−1.19806	−0.0538489
$496$	6.80194	0.305416
$497$	−4.07234	−0.182670
$498$	−8.40581	−0.376673
$499$	30.1608	1.35018	0.675090	−	0.737735i	$-0.264105\pi$
$499$	30.1608	1.35018	0.675090	+	0.737735i	$0.264105\pi$
$500$	−1.00000	−0.0447214
$501$	−1.19806	−0.0535255
$502$	14.8170	0.661315
$503$	−33.8883	−1.51100	−0.755502	−	0.655146i	$-0.772607\pi$
$503$	−33.8883	−1.51100	−0.755502	+	0.655146i	$0.772607\pi$
$504$	−1.75302	−0.0780857
$505$	−16.7875	−0.747032
$506$	7.24698	0.322168
$507$	0	0
$508$	7.96077	0.353202
$509$	14.1089	0.625364	0.312682	−	0.949858i	$-0.398773\pi$
$509$	14.1089	0.625364	0.312682	+	0.949858i	$0.398773\pi$
$510$	−6.76271	−0.299458
$511$	2.80220	0.123962
$512$	−1.00000	−0.0441942
$513$	8.63102	0.381069
$514$	10.5743	0.466414
$515$	4.16421	0.183497
$516$	4.07606	0.179439
$517$	−13.4131	−0.589908
$518$	7.61655	0.334652
$519$	−6.63773	−0.291364
$520$	0	0
$521$	17.3937	0.762033	0.381017	−	0.924568i	$-0.375574\pi$
$521$	17.3937	0.762033	0.381017	+	0.924568i	$0.375574\pi$
$522$	−4.07069	−0.178169
$523$	−34.9879	−1.52991	−0.764957	−	0.644081i	$-0.777240\pi$
$523$	−34.9879	−1.52991	−0.764957	+	0.644081i	$0.777240\pi$
$524$	−5.08815	−0.222277
$525$	1.75302	0.0765081
$526$	−11.9922	−0.522886
$527$	−45.9995	−2.00377
$528$	1.19806	0.0521390
$529$	13.5894	0.590844
$530$	−1.14914	−0.0499157
$531$	13.7235	0.595548
$532$	15.1304	0.655984
$533$	0	0
$534$	−4.49934	−0.194705
$535$	−12.9661	−0.560575
$536$	−14.1468	−0.611047
$537$	19.6353	0.847327
$538$	25.4359	1.09662
$539$	−4.70469	−0.202646
$540$	−1.00000	−0.0430331
$541$	20.8689	0.897224	0.448612	−	0.893727i	$-0.351918\pi$
$541$	20.8689	0.897224	0.448612	+	0.893727i	$0.351918\pi$
$542$	−20.3666	−0.874820
$543$	5.25906	0.225688
$544$	6.76271	0.289949
$545$	19.4644	0.833764
$546$	0	0
$547$	5.67456	0.242627	0.121313	−	0.992614i	$-0.461289\pi$
$547$	5.67456	0.242627	0.121313	+	0.992614i	$0.461289\pi$
$548$	0.674563	0.0288159
$549$	−1.18598	−0.0506164
$550$	−1.19806	−0.0510855
$551$	35.1342	1.49677
$552$	6.04892	0.257459
$553$	6.55794	0.278872
$554$	−20.1323	−0.855339
$555$	4.34481	0.184427
$556$	8.26205	0.350389
$557$	−14.7724	−0.625927	−0.312963	−	0.949765i	$-0.601322\pi$
$557$	−14.7724	−0.625927	−0.312963	+	0.949765i	$0.601322\pi$
$558$	−6.80194	−0.287949
$559$	0	0
$560$	−1.75302	−0.0740786
$561$	−8.10215	−0.342073
$562$	29.7265	1.25394
$563$	18.0398	0.760288	0.380144	−	0.924927i	$-0.375874\pi$
$563$	18.0398	0.760288	0.380144	+	0.924927i	$0.375874\pi$
$564$	−11.1957	−0.471423
$565$	−0.515729	−0.0216969
$566$	−8.84117	−0.371622
$567$	1.75302	0.0736199
$568$	2.32304	0.0974728
$569$	−11.4886	−0.481626	−0.240813	−	0.970571i	$-0.577414\pi$
$569$	−11.4886	−0.481626	−0.240813	+	0.970571i	$0.577414\pi$
$570$	8.63102	0.361514
$571$	−44.3062	−1.85416	−0.927078	−	0.374869i	$-0.877688\pi$
$571$	−44.3062	−1.85416	−0.927078	+	0.374869i	$0.877688\pi$
$572$	0	0
$573$	4.45473	0.186099
$574$	−11.6243	−0.485190
$575$	−6.04892	−0.252257
$576$	1.00000	0.0416667
$577$	28.0000	1.16566	0.582828	−	0.812596i	$-0.301946\pi$
$577$	28.0000	1.16566	0.582828	+	0.812596i	$0.301946\pi$
$578$	−28.7342	−1.19519
$579$	−0.818331	−0.0340087
$580$	−4.07069	−0.169026
$581$	14.7356	0.611334
$582$	10.7952	0.447477
$583$	−1.37675	−0.0570190
$584$	−1.59850	−0.0661463
$585$	0	0
$586$	−20.5918	−0.850639
$587$	−35.4316	−1.46242	−0.731210	−	0.682152i	$-0.761044\pi$
$587$	−35.4316	−1.46242	−0.731210	+	0.682152i	$0.761044\pi$
$588$	−3.92692	−0.161943
$589$	58.7077	2.41901
$590$	13.7235	0.564987
$591$	12.7385	0.523994
$592$	−4.34481	−0.178571
$593$	2.19567	0.0901653	0.0450827	−	0.998983i	$-0.485645\pi$
$593$	2.19567	0.0901653	0.0450827	+	0.998983i	$0.485645\pi$
$594$	−1.19806	−0.0491571
$595$	11.8552	0.486014
$596$	−8.19136	−0.335531
$597$	−4.02475	−0.164722
$598$	0	0
$599$	−31.1855	−1.27421	−0.637103	−	0.770779i	$-0.719867\pi$
$599$	−31.1855	−1.27421	−0.637103	+	0.770779i	$0.719867\pi$
$600$	−1.00000	−0.0408248
$601$	−24.2435	−0.988914	−0.494457	−	0.869202i	$-0.664633\pi$
$601$	−24.2435	−0.988914	−0.494457	+	0.869202i	$0.664633\pi$
$602$	−7.14542	−0.291226
$603$	14.1468	0.576100
$604$	−5.72348	−0.232885
$605$	9.56465	0.388858
$606$	−16.7875	−0.681944
$607$	−17.7235	−0.719374	−0.359687	−	0.933073i	$-0.617117\pi$
$607$	−17.7235	−0.719374	−0.359687	+	0.933073i	$0.617117\pi$
$608$	−8.63102	−0.350034
$609$	7.13600	0.289165
$610$	−1.18598	−0.0480190
$611$	0	0
$612$	−6.76271	−0.273366
$613$	−1.71140	−0.0691227	−0.0345614	−	0.999403i	$-0.511003\pi$
$613$	−1.71140	−0.0691227	−0.0345614	+	0.999403i	$0.511003\pi$
$614$	−28.7265	−1.15931
$615$	−6.63102	−0.267389
$616$	−2.10023	−0.0846206
$617$	−0.569433	−0.0229245	−0.0114622	−	0.999934i	$-0.503649\pi$
$617$	−0.569433	−0.0229245	−0.0114622	+	0.999934i	$0.503649\pi$
$618$	4.16421	0.167509
$619$	−26.4282	−1.06224	−0.531119	−	0.847297i	$-0.678228\pi$
$619$	−26.4282	−1.06224	−0.531119	+	0.847297i	$0.678228\pi$
$620$	−6.80194	−0.273172
$621$	−6.04892	−0.242735
$622$	18.6329	0.747113
$623$	7.88743	0.316003
$624$	0	0
$625$	1.00000	0.0400000
$626$	29.3696	1.17384
$627$	10.3405	0.412960
$628$	3.09246	0.123403
$629$	29.3827	1.17157
$630$	1.75302	0.0698420
$631$	−38.0006	−1.51278	−0.756390	−	0.654121i	$-0.773039\pi$
$631$	−38.0006	−1.51278	−0.756390	+	0.654121i	$0.773039\pi$
$632$	−3.74094	−0.148807
$633$	3.10992	0.123608
$634$	−20.8092	−0.826440
$635$	−7.96077	−0.315914
$636$	−1.14914	−0.0455666
$637$	0	0
$638$	−4.87694	−0.193080
$639$	−2.32304	−0.0918982
$640$	1.00000	0.0395285
$641$	−25.9168	−1.02365	−0.511825	−	0.859090i	$-0.671030\pi$
$641$	−25.9168	−1.02365	−0.511825	+	0.859090i	$0.671030\pi$
$642$	−12.9661	−0.511733
$643$	−8.52243	−0.336092	−0.168046	−	0.985779i	$-0.553746\pi$
$643$	−8.52243	−0.336092	−0.168046	+	0.985779i	$0.553746\pi$
$644$	−10.6039	−0.417851
$645$	−4.07606	−0.160495
$646$	58.3691	2.29650
$647$	37.9318	1.49125	0.745627	−	0.666364i	$-0.232150\pi$
$647$	37.9318	1.49125	0.745627	+	0.666364i	$0.232150\pi$
$648$	−1.00000	−0.0392837
$649$	16.4416	0.645389
$650$	0	0
$651$	11.9239	0.467336
$652$	23.9541	0.938114
$653$	−6.07846	−0.237868	−0.118934	−	0.992902i	$-0.537948\pi$
$653$	−6.07846	−0.237868	−0.118934	+	0.992902i	$0.537948\pi$
$654$	19.4644	0.761119
$655$	5.08815	0.198810
$656$	6.63102	0.258898
$657$	1.59850	0.0623634
$658$	19.6262	0.765110
$659$	38.9162	1.51596	0.757979	−	0.652279i	$-0.226187\pi$
$659$	38.9162	1.51596	0.757979	+	0.652279i	$0.226187\pi$
$660$	−1.19806	−0.0466345
$661$	25.9191	1.00814	0.504069	−	0.863663i	$-0.331836\pi$
$661$	25.9191	1.00814	0.504069	+	0.863663i	$0.331836\pi$
$662$	11.7603	0.457078
$663$	0	0
$664$	−8.40581	−0.326209
$665$	−15.1304	−0.586730
$666$	4.34481	0.168358
$667$	−24.6233	−0.953416
$668$	−1.19806	−0.0463544
$669$	14.3787	0.555911
$670$	14.1468	0.546537
$671$	−1.42088	−0.0548524
$672$	−1.75302	−0.0676242
$673$	−25.5724	−0.985744	−0.492872	−	0.870102i	$-0.664053\pi$
$673$	−25.5724	−0.985744	−0.492872	+	0.870102i	$0.664053\pi$
$674$	26.8732	1.03512
$675$	1.00000	0.0384900
$676$	0	0
$677$	24.7453	0.951037	0.475519	−	0.879706i	$-0.342260\pi$
$677$	24.7453	0.951037	0.475519	+	0.879706i	$0.342260\pi$
$678$	−0.515729	−0.0198065
$679$	−18.9243	−0.726247
$680$	−6.76271	−0.259338
$681$	−5.97716	−0.229045
$682$	−8.14914	−0.312047
$683$	22.4045	0.857284	0.428642	−	0.903474i	$-0.358992\pi$
$683$	22.4045	0.857284	0.428642	+	0.903474i	$0.358992\pi$
$684$	8.63102	0.330015
$685$	−0.674563	−0.0257737
$686$	19.1551	0.731346
$687$	−15.9172	−0.607280
$688$	4.07606	0.155398
$689$	0	0
$690$	−6.04892	−0.230278
$691$	−16.0030	−0.608782	−0.304391	−	0.952547i	$-0.598453\pi$
$691$	−16.0030	−0.608782	−0.304391	+	0.952547i	$0.598453\pi$
$692$	−6.63773	−0.252329
$693$	2.10023	0.0797810
$694$	−29.5080	−1.12011
$695$	−8.26205	−0.313397
$696$	−4.07069	−0.154299
$697$	−44.8437	−1.69858
$698$	−17.0519	−0.645424
$699$	−6.92154	−0.261797
$700$	1.75302	0.0662579
$701$	32.7222	1.23590	0.617949	−	0.786218i	$-0.287964\pi$
$701$	32.7222	1.23590	0.617949	+	0.786218i	$0.287964\pi$
$702$	0	0
$703$	−37.5002	−1.41435
$704$	1.19806	0.0451537
$705$	11.1957	0.421653
$706$	21.8629	0.822822
$707$	29.4288	1.10678
$708$	13.7235	0.515760
$709$	14.0597	0.528022	0.264011	−	0.964520i	$-0.414954\pi$
$709$	14.0597	0.528022	0.264011	+	0.964520i	$0.414954\pi$
$710$	−2.32304	−0.0871823
$711$	3.74094	0.140296
$712$	−4.49934	−0.168620
$713$	−41.1444	−1.54087
$714$	11.8552	0.443668
$715$	0	0
$716$	19.6353	0.733807
$717$	30.1075	1.12439
$718$	−14.3026	−0.533768
$719$	−7.15585	−0.266868	−0.133434	−	0.991058i	$-0.542600\pi$
$719$	−7.15585	−0.266868	−0.133434	+	0.991058i	$0.542600\pi$
$720$	−1.00000	−0.0372678
$721$	−7.29995	−0.271864
$722$	−55.4946	−2.06529
$723$	12.0562	0.448375
$724$	5.25906	0.195451
$725$	4.07069	0.151182
$726$	9.56465	0.354977
$727$	36.4704	1.35261	0.676306	−	0.736621i	$-0.263580\pi$
$727$	36.4704	1.35261	0.676306	+	0.736621i	$0.263580\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	1.59850	0.0591631
$731$	−27.5652	−1.01954
$732$	−1.18598	−0.0438351
$733$	−38.0646	−1.40595	−0.702974	−	0.711216i	$-0.748145\pi$
$733$	−38.0646	−1.40595	−0.702974	+	0.711216i	$0.748145\pi$
$734$	11.2416	0.414935
$735$	3.92692	0.144847
$736$	6.04892	0.222966
$737$	16.9487	0.624313
$738$	−6.63102	−0.244091
$739$	35.5114	1.30631	0.653154	−	0.757225i	$-0.273445\pi$
$739$	35.5114	1.30631	0.653154	+	0.757225i	$0.273445\pi$
$740$	4.34481	0.159719
$741$	0	0
$742$	2.01447	0.0739537
$743$	20.3696	0.747287	0.373643	−	0.927572i	$-0.378108\pi$
$743$	20.3696	0.747287	0.373643	+	0.927572i	$0.378108\pi$
$744$	−6.80194	−0.249371
$745$	8.19136	0.300108
$746$	9.70171	0.355205
$747$	8.40581	0.307553
$748$	−8.10215	−0.296244
$749$	22.7299	0.830533
$750$	1.00000	0.0365148
$751$	−5.58104	−0.203655	−0.101828	−	0.994802i	$-0.532469\pi$
$751$	−5.58104	−0.203655	−0.101828	+	0.994802i	$0.532469\pi$
$752$	−11.1957	−0.408264
$753$	−14.8170	−0.539962
$754$	0	0
$755$	5.72348	0.208299
$756$	1.75302	0.0637567
$757$	12.8267	0.466194	0.233097	−	0.972453i	$-0.425114\pi$
$757$	12.8267	0.466194	0.233097	+	0.972453i	$0.425114\pi$
$758$	2.82802	0.102718
$759$	−7.24698	−0.263049
$760$	8.63102	0.313080
$761$	−14.0084	−0.507803	−0.253901	−	0.967230i	$-0.581714\pi$
$761$	−14.0084	−0.507803	−0.253901	+	0.967230i	$0.581714\pi$
$762$	−7.96077	−0.288388
$763$	−34.1215	−1.23528
$764$	4.45473	0.161166
$765$	6.76271	0.244506
$766$	−11.6558	−0.421140
$767$	0	0
$768$	1.00000	0.0360844
$769$	−50.4051	−1.81765	−0.908827	−	0.417174i	$-0.863021\pi$
$769$	−50.4051	−1.81765	−0.908827	+	0.417174i	$0.863021\pi$
$770$	2.10023	0.0756869
$771$	−10.5743	−0.380825
$772$	−0.818331	−0.0294524
$773$	−42.5532	−1.53053	−0.765265	−	0.643715i	$-0.777392\pi$
$773$	−42.5532	−1.53053	−0.765265	+	0.643715i	$0.777392\pi$
$774$	−4.07606	−0.146511
$775$	6.80194	0.244333
$776$	10.7952	0.387526
$777$	−7.61655	−0.273242
$778$	−12.5157	−0.448711
$779$	57.2325	2.05057
$780$	0	0
$781$	−2.78315	−0.0995890
$782$	−40.9071	−1.46283
$783$	4.07069	0.145475
$784$	−3.92692	−0.140247
$785$	−3.09246	−0.110375
$786$	5.08815	0.181488
$787$	12.1927	0.434622	0.217311	−	0.976102i	$-0.430271\pi$
$787$	12.1927	0.434622	0.217311	+	0.976102i	$0.430271\pi$
$788$	12.7385	0.453792
$789$	11.9922	0.426935
$790$	3.74094	0.133097
$791$	0.904084	0.0321455
$792$	−1.19806	−0.0425713
$793$	0	0
$794$	−21.6136	−0.767037
$795$	1.14914	0.0407560
$796$	−4.02475	−0.142654
$797$	36.4534	1.29125	0.645623	−	0.763656i	$-0.276598\pi$
$797$	36.4534	1.29125	0.645623	+	0.763656i	$0.276598\pi$
$798$	−15.1304	−0.535609
$799$	75.7131	2.67854
$800$	−1.00000	−0.0353553
$801$	4.49934	0.158976
$802$	11.5157	0.406635
$803$	1.91510	0.0675824
$804$	14.1468	0.498917
$805$	10.6039	0.373738
$806$	0	0
$807$	−25.4359	−0.895388
$808$	−16.7875	−0.590581
$809$	9.70278	0.341131	0.170566	−	0.985346i	$-0.445440\pi$
$809$	9.70278	0.341131	0.170566	+	0.985346i	$0.445440\pi$
$810$	1.00000	0.0351364
$811$	−26.6547	−0.935974	−0.467987	−	0.883735i	$-0.655021\pi$
$811$	−26.6547	−0.935974	−0.467987	+	0.883735i	$0.655021\pi$
$812$	7.13600	0.250424
$813$	20.3666	0.714287
$814$	5.20536	0.182448
$815$	−23.9541	−0.839074
$816$	−6.76271	−0.236742
$817$	35.1806	1.23081
$818$	0.0991626	0.00346714
$819$	0	0
$820$	−6.63102	−0.231565
$821$	−2.58343	−0.0901624	−0.0450812	−	0.998983i	$-0.514355\pi$
$821$	−2.58343	−0.0901624	−0.0450812	+	0.998983i	$0.514355\pi$
$822$	−0.674563	−0.0235281
$823$	19.9855	0.696652	0.348326	−	0.937374i	$-0.386750\pi$
$823$	19.9855	0.696652	0.348326	+	0.937374i	$0.386750\pi$
$824$	4.16421	0.145067
$825$	1.19806	0.0417112
$826$	−24.0575	−0.837069
$827$	−1.39506	−0.0485110	−0.0242555	−	0.999706i	$-0.507722\pi$
$827$	−1.39506	−0.0485110	−0.0242555	+	0.999706i	$0.507722\pi$
$828$	−6.04892	−0.210214
$829$	−27.6179	−0.959208	−0.479604	−	0.877485i	$-0.659220\pi$
$829$	−27.6179	−0.959208	−0.479604	+	0.877485i	$0.659220\pi$
$830$	8.40581	0.291770
$831$	20.1323	0.698381
$832$	0	0
$833$	26.5566	0.920132
$834$	−8.26205	−0.286091
$835$	1.19806	0.0414607
$836$	10.3405	0.357634
$837$	6.80194	0.235109
$838$	−31.2784	−1.08050
$839$	54.3357	1.87588	0.937939	−	0.346801i	$-0.112732\pi$
$839$	54.3357	1.87588	0.937939	+	0.346801i	$0.112732\pi$
$840$	1.75302	0.0604850
$841$	−12.4295	−0.428604
$842$	38.0452	1.31112
$843$	−29.7265	−1.02383
$844$	3.10992	0.107048
$845$	0	0
$846$	11.1957	0.384915
$847$	−16.7670	−0.576122
$848$	−1.14914	−0.0394618
$849$	8.84117	0.303428
$850$	6.76271	0.231959
$851$	26.2814	0.900916
$852$	−2.32304	−0.0795862
$853$	13.4553	0.460701	0.230351	−	0.973108i	$-0.426013\pi$
$853$	13.4553	0.460701	0.230351	+	0.973108i	$0.426013\pi$
$854$	2.07905	0.0711436
$855$	−8.63102	−0.295175
$856$	−12.9661	−0.443174
$857$	42.7200	1.45929	0.729644	−	0.683827i	$-0.239686\pi$
$857$	42.7200	1.45929	0.729644	+	0.683827i	$0.239686\pi$
$858$	0	0
$859$	44.3038	1.51163	0.755813	−	0.654788i	$-0.227242\pi$
$859$	44.3038	1.51163	0.755813	+	0.654788i	$0.227242\pi$
$860$	−4.07606	−0.138993
$861$	11.6243	0.396156
$862$	14.4614	0.492558
$863$	5.47783	0.186467	0.0932337	−	0.995644i	$-0.470280\pi$
$863$	5.47783	0.186467	0.0932337	+	0.995644i	$0.470280\pi$
$864$	−1.00000	−0.0340207
$865$	6.63773	0.225689
$866$	−21.6209	−0.734707
$867$	28.7342	0.975866
$868$	11.9239	0.404725
$869$	4.48188	0.152037
$870$	4.07069	0.138009
$871$	0	0
$872$	19.4644	0.659148
$873$	−10.7952	−0.365363
$874$	52.2083	1.76597
$875$	−1.75302	−0.0592629
$876$	1.59850	0.0540083
$877$	41.3726	1.39705	0.698526	−	0.715585i	$-0.253840\pi$
$877$	41.3726	1.39705	0.698526	+	0.715585i	$0.253840\pi$
$878$	3.23623	0.109217
$879$	20.5918	0.694544
$880$	−1.19806	−0.0403867
$881$	−42.3919	−1.42822	−0.714110	−	0.700033i	$-0.753168\pi$
$881$	−42.3919	−1.42822	−0.714110	+	0.700033i	$0.753168\pi$
$882$	3.92692	0.132226
$883$	−40.8412	−1.37441	−0.687207	−	0.726461i	$-0.741164\pi$
$883$	−40.8412	−1.37441	−0.687207	+	0.726461i	$0.741164\pi$
$884$	0	0
$885$	−13.7235	−0.461310
$886$	14.6649	0.492676
$887$	−35.0810	−1.17790	−0.588952	−	0.808168i	$-0.700459\pi$
$887$	−35.0810	−1.17790	−0.588952	+	0.808168i	$0.700459\pi$
$888$	4.34481	0.145802
$889$	13.9554	0.468049
$890$	4.49934	0.150818
$891$	1.19806	0.0401366
$892$	14.3787	0.481433
$893$	−96.6301	−3.23360
$894$	8.19136	0.273960
$895$	−19.6353	−0.656337
$896$	−1.75302	−0.0585643
$897$	0	0
$898$	−4.08516	−0.136324
$899$	27.6886	0.923465
$900$	1.00000	0.0333333
$901$	7.77133	0.258901
$902$	−7.94438	−0.264519
$903$	7.14542	0.237785
$904$	−0.515729	−0.0171529
$905$	−5.25906	−0.174817
$906$	5.72348	0.190150
$907$	−45.4916	−1.51052	−0.755261	−	0.655424i	$-0.772490\pi$
$907$	−45.4916	−1.51052	−0.755261	+	0.655424i	$0.772490\pi$
$908$	−5.97716	−0.198359
$909$	16.7875	0.556805
$910$	0	0
$911$	47.4349	1.57159	0.785794	−	0.618489i	$-0.212255\pi$
$911$	47.4349	1.57159	0.785794	+	0.618489i	$0.212255\pi$
$912$	8.63102	0.285802
$913$	10.0707	0.333291
$914$	−41.8678	−1.38487
$915$	1.18598	0.0392073
$916$	−15.9172	−0.525920
$917$	−8.91962	−0.294552
$918$	6.76271	0.223203
$919$	−20.3907	−0.672629	−0.336314	−	0.941750i	$-0.609180\pi$
$919$	−20.3907	−0.672629	−0.336314	+	0.941750i	$0.609180\pi$
$920$	−6.04892	−0.199427
$921$	28.7265	0.946569
$922$	0.215521	0.00709779
$923$	0	0
$924$	2.10023	0.0690924
$925$	−4.34481	−0.142857
$926$	−23.4795	−0.771584
$927$	−4.16421	−0.136771
$928$	−4.07069	−0.133627
$929$	−1.26636	−0.0415478	−0.0207739	−	0.999784i	$-0.506613\pi$
$929$	−1.26636	−0.0415478	−0.0207739	+	0.999784i	$0.506613\pi$
$930$	6.80194	0.223044
$931$	−33.8933	−1.11081
$932$	−6.92154	−0.226723
$933$	−18.6329	−0.610015
$934$	−10.1153	−0.330983
$935$	8.10215	0.264969
$936$	0	0
$937$	27.3558	0.893676	0.446838	−	0.894615i	$-0.352550\pi$
$937$	27.3558	0.893676	0.446838	+	0.894615i	$0.352550\pi$
$938$	−24.7995	−0.809734
$939$	−29.3696	−0.958440
$940$	11.1957	0.365162
$941$	−60.6620	−1.97752	−0.988762	−	0.149495i	$-0.952235\pi$
$941$	−60.6620	−1.97752	−0.988762	+	0.149495i	$0.952235\pi$
$942$	−3.09246	−0.100758
$943$	−40.1105	−1.30618
$944$	13.7235	0.446661
$945$	−1.75302	−0.0570258
$946$	−4.88338	−0.158772
$947$	−12.3797	−0.402287	−0.201144	−	0.979562i	$-0.564466\pi$
$947$	−12.3797	−0.402287	−0.201144	+	0.979562i	$0.564466\pi$
$948$	3.74094	0.121500
$949$	0	0
$950$	−8.63102	−0.280027
$951$	20.8092	0.674786
$952$	11.8552	0.384228
$953$	28.7784	0.932223	0.466111	−	0.884726i	$-0.345655\pi$
$953$	28.7784	0.932223	0.466111	+	0.884726i	$0.345655\pi$
$954$	1.14914	0.0372049
$955$	−4.45473	−0.144152
$956$	30.1075	0.973747
$957$	4.87694	0.157649
$958$	−0.980623	−0.0316825
$959$	1.18252	0.0381857
$960$	−1.00000	−0.0322749
$961$	15.2664	0.492463
$962$	0	0
$963$	12.9661	0.417828
$964$	12.0562	0.388304
$965$	0.818331	0.0263430
$966$	10.6039	0.341174
$967$	−23.1545	−0.744599	−0.372300	−	0.928113i	$-0.621431\pi$
$967$	−23.1545	−0.744599	−0.372300	+	0.928113i	$0.621431\pi$
$968$	9.56465	0.307419
$969$	−58.3691	−1.87509
$970$	−10.7952	−0.346614
$971$	−2.89572	−0.0929282	−0.0464641	−	0.998920i	$-0.514795\pi$
$971$	−2.89572	−0.0929282	−0.0464641	+	0.998920i	$0.514795\pi$
$972$	1.00000	0.0320750
$973$	14.4835	0.464321
$974$	−7.18300	−0.230158
$975$	0	0
$976$	−1.18598	−0.0379623
$977$	−4.86294	−0.155579	−0.0777896	−	0.996970i	$-0.524786\pi$
$977$	−4.86294	−0.155579	−0.0777896	+	0.996970i	$0.524786\pi$
$978$	−23.9541	−0.765967
$979$	5.39048	0.172281
$980$	3.92692	0.125441
$981$	−19.4644	−0.621451
$982$	−24.5918	−0.784756
$983$	−52.7506	−1.68248	−0.841242	−	0.540659i	$-0.818175\pi$
$983$	−52.7506	−1.68248	−0.841242	+	0.540659i	$0.818175\pi$
$984$	−6.63102	−0.211389
$985$	−12.7385	−0.405884
$986$	27.5289	0.876698
$987$	−19.6262	−0.624710
$988$	0	0
$989$	−24.6558	−0.784008
$990$	1.19806	0.0380769
$991$	7.15538	0.227298	0.113649	−	0.993521i	$-0.463746\pi$
$991$	7.15538	0.227298	0.113649	+	0.993521i	$0.463746\pi$
$992$	−6.80194	−0.215962
$993$	−11.7603	−0.373202
$994$	4.07234	0.129167
$995$	4.02475	0.127593
$996$	8.40581	0.266348
$997$	39.3870	1.24740	0.623700	−	0.781664i	$-0.285629\pi$
$997$	39.3870	1.24740	0.623700	+	0.781664i	$0.285629\pi$
$998$	−30.1608	−0.954722
$999$	−4.34481	−0.137464

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5070.2.a.bo.1.1	✓	3
13.5	odd	4		5070.2.b.w.1351.4		6
13.8	odd	4		5070.2.b.w.1351.3		6
13.12	even	2		5070.2.a.bx.1.3	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5070.2.a.bo.1.1	✓	3	1.1	even	1	trivial
5070.2.a.bx.1.3	yes	3	13.12	even	2
5070.2.b.w.1351.3		6	13.8	odd	4
5070.2.b.w.1351.4		6	13.5	odd	4

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:	\( N \)	\(=\)	\( 5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5070.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} +1.75302 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} +1.75302 q^{7} -1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} +1.19806 q^{11} +1.00000 q^{12} -1.75302 q^{14} -1.00000 q^{15} +1.00000 q^{16} -6.76271 q^{17} -1.00000 q^{18} +8.63102 q^{19} -1.00000 q^{20} +1.75302 q^{21} -1.19806 q^{22} -6.04892 q^{23} -1.00000 q^{24} +1.00000 q^{25} +1.00000 q^{27} +1.75302 q^{28} +4.07069 q^{29} +1.00000 q^{30} +6.80194 q^{31} -1.00000 q^{32} +1.19806 q^{33} +6.76271 q^{34} -1.75302 q^{35} +1.00000 q^{36} -4.34481 q^{37} -8.63102 q^{38} +1.00000 q^{40} +6.63102 q^{41} -1.75302 q^{42} +4.07606 q^{43} +1.19806 q^{44} -1.00000 q^{45} +6.04892 q^{46} -11.1957 q^{47} +1.00000 q^{48} -3.92692 q^{49} -1.00000 q^{50} -6.76271 q^{51} -1.14914 q^{53} -1.00000 q^{54} -1.19806 q^{55} -1.75302 q^{56} +8.63102 q^{57} -4.07069 q^{58} +13.7235 q^{59} -1.00000 q^{60} -1.18598 q^{61} -6.80194 q^{62} +1.75302 q^{63} +1.00000 q^{64} -1.19806 q^{66} +14.1468 q^{67} -6.76271 q^{68} -6.04892 q^{69} +1.75302 q^{70} -2.32304 q^{71} -1.00000 q^{72} +1.59850 q^{73} +4.34481 q^{74} +1.00000 q^{75} +8.63102 q^{76} +2.10023 q^{77} +3.74094 q^{79} -1.00000 q^{80} +1.00000 q^{81} -6.63102 q^{82} +8.40581 q^{83} +1.75302 q^{84} +6.76271 q^{85} -4.07606 q^{86} +4.07069 q^{87} -1.19806 q^{88} +4.49934 q^{89} +1.00000 q^{90} -6.04892 q^{92} +6.80194 q^{93} +11.1957 q^{94} -8.63102 q^{95} -1.00000 q^{96} -10.7952 q^{97} +3.92692 q^{98} +1.19806 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{2} + 3 q^{3} + 3 q^{4} - 3 q^{5} - 3 q^{6} + 10 q^{7} - 3 q^{8} + 3 q^{9}+O(q^{10}) \) 3 * q - 3 * q^2 + 3 * q^3 + 3 * q^4 - 3 * q^5 - 3 * q^6 + 10 * q^7 - 3 * q^8 + 3 * q^9	\( 3 q - 3 q^{2} + 3 q^{3} + 3 q^{4} - 3 q^{5} - 3 q^{6} + 10 q^{7} - 3 q^{8} + 3 q^{9} + 3 q^{10} + 8 q^{11} + 3 q^{12} - 10 q^{14} - 3 q^{15} + 3 q^{16} - 3 q^{17} - 3 q^{18} + 11 q^{19} - 3 q^{20} + 10 q^{21} - 8 q^{22} - 9 q^{23} - 3 q^{24} + 3 q^{25} + 3 q^{27} + 10 q^{28} + 3 q^{30} + 16 q^{31} - 3 q^{32} + 8 q^{33} + 3 q^{34} - 10 q^{35} + 3 q^{36} + 10 q^{37} - 11 q^{38} + 3 q^{40} + 5 q^{41} - 10 q^{42} - 3 q^{43} + 8 q^{44} - 3 q^{45} + 9 q^{46} + 3 q^{47} + 3 q^{48} + 17 q^{49} - 3 q^{50} - 3 q^{51} - 17 q^{53} - 3 q^{54} - 8 q^{55} - 10 q^{56} + 11 q^{57} + 11 q^{59} - 3 q^{60} + 11 q^{61} - 16 q^{62} + 10 q^{63} + 3 q^{64} - 8 q^{66} + 15 q^{67} - 3 q^{68} - 9 q^{69} + 10 q^{70} + 13 q^{71} - 3 q^{72} - q^{73} - 10 q^{74} + 3 q^{75} + 11 q^{76} + 29 q^{77} - 3 q^{79} - 3 q^{80} + 3 q^{81} - 5 q^{82} + 12 q^{83} + 10 q^{84} + 3 q^{85} + 3 q^{86} - 8 q^{88} + q^{89} + 3 q^{90} - 9 q^{92} + 16 q^{93} - 3 q^{94} - 11 q^{95} - 3 q^{96} - 6 q^{97} - 17 q^{98} + 8 q^{99}+O(q^{100}) \) 3 * q - 3 * q^2 + 3 * q^3 + 3 * q^4 - 3 * q^5 - 3 * q^6 + 10 * q^7 - 3 * q^8 + 3 * q^9 + 3 * q^10 + 8 * q^11 + 3 * q^12 - 10 * q^14 - 3 * q^15 + 3 * q^16 - 3 * q^17 - 3 * q^18 + 11 * q^19 - 3 * q^20 + 10 * q^21 - 8 * q^22 - 9 * q^23 - 3 * q^24 + 3 * q^25 + 3 * q^27 + 10 * q^28 + 3 * q^30 + 16 * q^31 - 3 * q^32 + 8 * q^33 + 3 * q^34 - 10 * q^35 + 3 * q^36 + 10 * q^37 - 11 * q^38 + 3 * q^40 + 5 * q^41 - 10 * q^42 - 3 * q^43 + 8 * q^44 - 3 * q^45 + 9 * q^46 + 3 * q^47 + 3 * q^48 + 17 * q^49 - 3 * q^50 - 3 * q^51 - 17 * q^53 - 3 * q^54 - 8 * q^55 - 10 * q^56 + 11 * q^57 + 11 * q^59 - 3 * q^60 + 11 * q^61 - 16 * q^62 + 10 * q^63 + 3 * q^64 - 8 * q^66 + 15 * q^67 - 3 * q^68 - 9 * q^69 + 10 * q^70 + 13 * q^71 - 3 * q^72 - q^73 - 10 * q^74 + 3 * q^75 + 11 * q^76 + 29 * q^77 - 3 * q^79 - 3 * q^80 + 3 * q^81 - 5 * q^82 + 12 * q^83 + 10 * q^84 + 3 * q^85 + 3 * q^86 - 8 * q^88 + q^89 + 3 * q^90 - 9 * q^92 + 16 * q^93 - 3 * q^94 - 11 * q^95 - 3 * q^96 - 6 * q^97 - 17 * q^98 + 8 * q^99

Introduction

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