LMFDB - Embedded newform 2009.2.a.u.1.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2009,2,Mod(1,2009)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2009, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("2009.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2009 = 7^{2} \cdot 41 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2009.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:	$16.0419457661$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 2 x^{19} - 27 x^{18} + 52 x^{17} + 302 x^{16} - 552 x^{15} - 1820 x^{14} + 3090 x^{13} + \cdots + 49 $ x^20 - 2x^19 - 27x^18 + 52x^17 + 302x^16 - 552x^15 - 1820x^14 + 3090x^13 + 6449x^12 - 9852x^11 - 13797x^10 + 18080x^9 + 17721x^8 - 18446x^7 - 13352x^6 + 9524x^5 + 5690x^4 - 1972x^3 - 1179x^2 + 42*x + 49
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.14
Root			$-0.874793$ of defining polynomial
Character	$\chi$	$=$	2009.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.874793 q^{2} +2.48225 q^{3} -1.23474 q^{4} +4.06589 q^{5} +2.17145 q^{6} -2.82973 q^{8} +3.16156 q^{9} +O(q^{10})$	\(q+0.874793 q^{2} +2.48225 q^{3} -1.23474 q^{4} +4.06589 q^{5} +2.17145 q^{6} -2.82973 q^{8} +3.16156 q^{9} +3.55681 q^{10} +5.27549 q^{11} -3.06493 q^{12} -3.57189 q^{13} +10.0925 q^{15} -0.00594767 q^{16} -4.81948 q^{17} +2.76571 q^{18} +5.76557 q^{19} -5.02031 q^{20} +4.61496 q^{22} -6.67829 q^{23} -7.02408 q^{24} +11.5315 q^{25} -3.12467 q^{26} +0.401022 q^{27} +2.57892 q^{29} +8.82889 q^{30} +7.55182 q^{31} +5.65425 q^{32} +13.0951 q^{33} -4.21605 q^{34} -3.90369 q^{36} +6.08024 q^{37} +5.04368 q^{38} -8.86633 q^{39} -11.5054 q^{40} +1.00000 q^{41} -0.758024 q^{43} -6.51384 q^{44} +12.8545 q^{45} -5.84212 q^{46} -4.96735 q^{47} -0.0147636 q^{48} +10.0876 q^{50} -11.9631 q^{51} +4.41035 q^{52} -10.4910 q^{53} +0.350811 q^{54} +21.4496 q^{55} +14.3116 q^{57} +2.25602 q^{58} -8.86442 q^{59} -12.4617 q^{60} -2.58111 q^{61} +6.60628 q^{62} +4.95819 q^{64} -14.5229 q^{65} +11.4555 q^{66} -4.34716 q^{67} +5.95079 q^{68} -16.5772 q^{69} +0.0215020 q^{71} -8.94633 q^{72} -4.03756 q^{73} +5.31895 q^{74} +28.6240 q^{75} -7.11897 q^{76} -7.75620 q^{78} -1.59986 q^{79} -0.0241826 q^{80} -8.48923 q^{81} +0.874793 q^{82} +2.44174 q^{83} -19.5955 q^{85} -0.663114 q^{86} +6.40153 q^{87} -14.9282 q^{88} +1.60572 q^{89} +11.2451 q^{90} +8.24594 q^{92} +18.7455 q^{93} -4.34540 q^{94} +23.4422 q^{95} +14.0352 q^{96} +12.5542 q^{97} +16.6787 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 2 q^{2} + 8 q^{3} + 18 q^{4} + 8 q^{5} + 12 q^{6} - 6 q^{8} + 16 q^{9}+O(q^{10}) $ 20 * q - 2 * q^2 + 8 * q^3 + 18 * q^4 + 8 * q^5 + 12 * q^6 - 6 * q^8 + 16 * q^9	$ 20 q - 2 q^{2} + 8 q^{3} + 18 q^{4} + 8 q^{5} + 12 q^{6} - 6 q^{8} + 16 q^{9} + 16 q^{10} + 6 q^{12} + 12 q^{13} + 14 q^{16} + 8 q^{17} - 18 q^{18} + 36 q^{19} + 24 q^{20} - 8 q^{22} - 12 q^{23} + 36 q^{24} + 20 q^{25} - 22 q^{26} + 32 q^{27} + 4 q^{29} + 28 q^{30} + 80 q^{31} + 6 q^{32} - 12 q^{33} + 48 q^{34} + 26 q^{36} + 4 q^{37} + 12 q^{38} - 28 q^{39} - 4 q^{40} + 20 q^{41} - 20 q^{44} + 40 q^{45} + 8 q^{46} + 32 q^{47} + 16 q^{48} + 6 q^{50} - 20 q^{51} + 36 q^{52} + 4 q^{53} - 50 q^{54} + 64 q^{55} - 4 q^{57} + 32 q^{59} + 20 q^{60} + 44 q^{61} - 8 q^{62} - 30 q^{64} - 8 q^{65} + 32 q^{66} - 4 q^{67} - 48 q^{68} - 24 q^{69} + 8 q^{71} - 8 q^{72} + 48 q^{73} - 38 q^{74} + 24 q^{75} + 84 q^{76} + 30 q^{78} - 4 q^{79} + 56 q^{80} - 2 q^{82} + 8 q^{83} - 12 q^{85} - 24 q^{86} + 40 q^{87} - 48 q^{88} + 20 q^{89} + 48 q^{90} - 50 q^{92} + 48 q^{93} + 26 q^{94} + 20 q^{95} + 70 q^{96} + 8 q^{97} - 8 q^{99}+O(q^{100}) $ 20 * q - 2 * q^2 + 8 * q^3 + 18 * q^4 + 8 * q^5 + 12 * q^6 - 6 * q^8 + 16 * q^9 + 16 * q^10 + 6 * q^12 + 12 * q^13 + 14 * q^16 + 8 * q^17 - 18 * q^18 + 36 * q^19 + 24 * q^20 - 8 * q^22 - 12 * q^23 + 36 * q^24 + 20 * q^25 - 22 * q^26 + 32 * q^27 + 4 * q^29 + 28 * q^30 + 80 * q^31 + 6 * q^32 - 12 * q^33 + 48 * q^34 + 26 * q^36 + 4 * q^37 + 12 * q^38 - 28 * q^39 - 4 * q^40 + 20 * q^41 - 20 * q^44 + 40 * q^45 + 8 * q^46 + 32 * q^47 + 16 * q^48 + 6 * q^50 - 20 * q^51 + 36 * q^52 + 4 * q^53 - 50 * q^54 + 64 * q^55 - 4 * q^57 + 32 * q^59 + 20 * q^60 + 44 * q^61 - 8 * q^62 - 30 * q^64 - 8 * q^65 + 32 * q^66 - 4 * q^67 - 48 * q^68 - 24 * q^69 + 8 * q^71 - 8 * q^72 + 48 * q^73 - 38 * q^74 + 24 * q^75 + 84 * q^76 + 30 * q^78 - 4 * q^79 + 56 * q^80 - 2 * q^82 + 8 * q^83 - 12 * q^85 - 24 * q^86 + 40 * q^87 - 48 * q^88 + 20 * q^89 + 48 * q^90 - 50 * q^92 + 48 * q^93 + 26 * q^94 + 20 * q^95 + 70 * q^96 + 8 * q^97 - 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 $

$2$	0.874793	0.618572	0.309286	−	0.950969i	$-0.399910\pi$
$2$	0.874793	0.618572	0.309286	+	0.950969i	$0.399910\pi$
$3$	2.48225	1.43313	0.716563	−	0.697522i	$-0.245714\pi$
$3$	2.48225	1.43313	0.716563	+	0.697522i	$0.245714\pi$
$4$	−1.23474	−0.617369
$5$	4.06589	1.81832	0.909161	−	0.416445i	$-0.136724\pi$
$5$	4.06589	1.81832	0.909161	+	0.416445i	$0.136724\pi$
$6$	2.17145	0.886492
$7$	0	0
$7$	0	0
$8$	−2.82973	−1.00046
$9$	3.16156	1.05385
$10$	3.55681	1.12476
$11$	5.27549	1.59062	0.795310	−	0.606203i	$-0.207308\pi$
$11$	5.27549	1.59062	0.795310	+	0.606203i	$0.207308\pi$
$12$	−3.06493	−0.884768
$13$	−3.57189	−0.990665	−0.495333	−	0.868703i	$-0.664954\pi$
$13$	−3.57189	−0.990665	−0.495333	+	0.868703i	$0.664954\pi$
$14$	0	0
$15$	10.0925	2.60588
$16$	−0.00594767	−0.00148692
$17$	−4.81948	−1.16890	−0.584448	−	0.811431i	$-0.698689\pi$
$17$	−4.81948	−1.16890	−0.584448	+	0.811431i	$0.698689\pi$
$18$	2.76571	0.651883
$19$	5.76557	1.32271	0.661356	−	0.750072i	$-0.269981\pi$
$19$	5.76557	1.32271	0.661356	+	0.750072i	$0.269981\pi$
$20$	−5.02031	−1.12257
$21$	0	0
$22$	4.61496	0.983912
$23$	−6.67829	−1.39252	−0.696260	−	0.717790i	$-0.745154\pi$
$23$	−6.67829	−1.39252	−0.696260	+	0.717790i	$0.745154\pi$
$24$	−7.02408	−1.43378
$25$	11.5315	2.30629
$26$	−3.12467	−0.612798
$27$	0.401022	0.0771767
$28$	0	0
$29$	2.57892	0.478894	0.239447	−	0.970909i	$-0.423034\pi$
$29$	2.57892	0.478894	0.239447	+	0.970909i	$0.423034\pi$
$30$	8.82889	1.61193
$31$	7.55182	1.35635	0.678174	−	0.734902i	$-0.262772\pi$
$31$	7.55182	1.35635	0.678174	+	0.734902i	$0.262772\pi$
$32$	5.65425	0.999539
$33$	13.0951	2.27956
$34$	−4.21605	−0.723046
$35$	0	0
$36$	−3.90369	−0.650615
$37$	6.08024	0.999585	0.499793	−	0.866145i	$-0.333410\pi$
$37$	6.08024	0.999585	0.499793	+	0.866145i	$0.333410\pi$
$38$	5.04368	0.818193
$39$	−8.86633	−1.41975
$40$	−11.5054	−1.81916
$41$	1.00000	0.156174
$41$	1.00000	0.156174
$42$	0	0
$43$	−0.758024	−0.115598	−0.0577988	−	0.998328i	$-0.518408\pi$
$43$	−0.758024	−0.115598	−0.0577988	+	0.998328i	$0.518408\pi$
$44$	−6.51384	−0.981999
$45$	12.8545	1.91624
$46$	−5.84212	−0.861374
$47$	−4.96735	−0.724563	−0.362281	−	0.932069i	$-0.618002\pi$
$47$	−4.96735	−0.724563	−0.362281	+	0.932069i	$0.618002\pi$
$48$	−0.0147636	−0.00213094
$49$	0	0
$50$	10.0876	1.42661
$51$	−11.9631	−1.67518
$52$	4.41035	0.611606
$53$	−10.4910	−1.44105	−0.720523	−	0.693431i	$-0.756098\pi$
$53$	−10.4910	−1.44105	−0.720523	+	0.693431i	$0.756098\pi$
$54$	0.350811	0.0477393
$55$	21.4496	2.89226
$56$	0	0
$57$	14.3116	1.89561
$58$	2.25602	0.296230
$59$	−8.86442	−1.15405	−0.577025	−	0.816727i	$-0.695786\pi$
$59$	−8.86442	−1.15405	−0.577025	+	0.816727i	$0.695786\pi$
$60$	−12.4617	−1.60879
$61$	−2.58111	−0.330477	−0.165238	−	0.986254i	$-0.552839\pi$
$61$	−2.58111	−0.330477	−0.165238	+	0.986254i	$0.552839\pi$
$62$	6.60628	0.838998
$63$	0	0
$64$	4.95819	0.619774
$65$	−14.5229	−1.80135
$66$	11.4555	1.41007
$67$	−4.34716	−0.531090	−0.265545	−	0.964099i	$-0.585552\pi$
$67$	−4.34716	−0.531090	−0.265545	+	0.964099i	$0.585552\pi$
$68$	5.95079	0.721640
$69$	−16.5772	−1.99566
$70$	0	0
$71$	0.0215020	0.00255182	0.00127591	−	0.999999i	$-0.499594\pi$
$71$	0.0215020	0.00255182	0.00127591	+	0.999999i	$0.499594\pi$
$72$	−8.94633	−1.05434
$73$	−4.03756	−0.472560	−0.236280	−	0.971685i	$-0.575928\pi$
$73$	−4.03756	−0.472560	−0.236280	+	0.971685i	$0.575928\pi$
$74$	5.31895	0.618315
$75$	28.6240	3.30521
$76$	−7.11897	−0.816602
$77$	0	0
$78$	−7.75620	−0.878217
$79$	−1.59986	−0.179998	−0.0899991	−	0.995942i	$-0.528686\pi$
$79$	−1.59986	−0.179998	−0.0899991	+	0.995942i	$0.528686\pi$
$80$	−0.0241826	−0.00270369
$81$	−8.48923	−0.943248
$82$	0.874793	0.0966047
$83$	2.44174	0.268016	0.134008	−	0.990980i	$-0.457215\pi$
$83$	2.44174	0.268016	0.134008	+	0.990980i	$0.457215\pi$
$84$	0	0
$85$	−19.5955	−2.12543
$86$	−0.663114	−0.0715054
$87$	6.40153	0.686316
$88$	−14.9282	−1.59135
$89$	1.60572	0.170206	0.0851029	−	0.996372i	$-0.472878\pi$
$89$	1.60572	0.170206	0.0851029	+	0.996372i	$0.472878\pi$
$90$	11.2451	1.18533
$91$	0	0
$92$	8.24594	0.859698
$93$	18.7455	1.94382
$94$	−4.34540	−0.448194
$95$	23.4422	2.40512
$96$	14.0352	1.43247
$97$	12.5542	1.27468	0.637341	−	0.770582i	$-0.280034\pi$
$97$	12.5542	1.27468	0.637341	+	0.770582i	$0.280034\pi$
$98$	0	0
$99$	16.6787	1.67628
$100$	−14.2383	−1.42383
$101$	10.0436	0.999380	0.499690	−	0.866204i	$-0.333447\pi$
$101$	10.0436	0.999380	0.499690	+	0.866204i	$0.333447\pi$
$102$	−10.4653	−1.03622
$103$	−2.34731	−0.231287	−0.115643	−	0.993291i	$-0.536893\pi$
$103$	−2.34731	−0.231287	−0.115643	+	0.993291i	$0.536893\pi$
$104$	10.1075	0.991120
$105$	0	0
$106$	−9.17743	−0.891390
$107$	−20.3131	−1.96375	−0.981873	−	0.189539i	$-0.939301\pi$
$107$	−20.3131	−1.96375	−0.981873	+	0.189539i	$0.939301\pi$
$108$	−0.495157	−0.0476465
$109$	3.57646	0.342563	0.171282	−	0.985222i	$-0.445209\pi$
$109$	3.57646	0.342563	0.171282	+	0.985222i	$0.445209\pi$
$110$	18.7639	1.78907
$111$	15.0927	1.43253
$112$	0	0
$113$	−1.75401	−0.165003	−0.0825016	−	0.996591i	$-0.526291\pi$
$113$	−1.75401	−0.165003	−0.0825016	+	0.996591i	$0.526291\pi$
$114$	12.5197	1.17257
$115$	−27.1532	−2.53205
$116$	−3.18429	−0.295654
$117$	−11.2927	−1.04401
$118$	−7.75453	−0.713863
$119$	0	0
$120$	−28.5591	−2.60708
$121$	16.8308	1.53007
$122$	−2.25793	−0.204424
$123$	2.48225	0.223817
$124$	−9.32452	−0.837367
$125$	26.5562	2.37526
$126$	0	0
$127$	12.1833	1.08110	0.540548	−	0.841313i	$-0.318217\pi$
$127$	12.1833	1.08110	0.540548	+	0.841313i	$0.318217\pi$
$128$	−6.97111	−0.616165
$129$	−1.88160	−0.165666
$130$	−12.7046	−1.11426
$131$	−17.9667	−1.56976	−0.784878	−	0.619651i	$-0.787274\pi$
$131$	−17.9667	−1.56976	−0.784878	+	0.619651i	$0.787274\pi$
$132$	−16.1690	−1.40733
$133$	0	0
$134$	−3.80286	−0.328517
$135$	1.63051	0.140332
$136$	13.6378	1.16943
$137$	2.47751	0.211668	0.105834	−	0.994384i	$-0.466249\pi$
$137$	2.47751	0.211668	0.105834	+	0.994384i	$0.466249\pi$
$138$	−14.5016	−1.23446
$139$	−14.1722	−1.20207	−0.601035	−	0.799222i	$-0.705245\pi$
$139$	−14.1722	−1.20207	−0.601035	+	0.799222i	$0.705245\pi$
$140$	0	0
$141$	−12.3302	−1.03839
$142$	0.0188098	0.00157848
$143$	−18.8435	−1.57577
$144$	−0.0188039	−0.00156699
$145$	10.4856	0.870784
$146$	−3.53203	−0.292312
$147$	0	0
$148$	−7.50750	−0.617113
$149$	1.51201	0.123869	0.0619345	−	0.998080i	$-0.480273\pi$
$149$	1.51201	0.123869	0.0619345	+	0.998080i	$0.480273\pi$
$150$	25.0400	2.04451
$151$	−21.3382	−1.73648	−0.868239	−	0.496146i	$-0.834748\pi$
$151$	−21.3382	−1.73648	−0.868239	+	0.496146i	$0.834748\pi$
$152$	−16.3150	−1.32332
$153$	−15.2371	−1.23184
$154$	0	0
$155$	30.7049	2.46628
$156$	10.9476	0.876509
$157$	3.77052	0.300920	0.150460	−	0.988616i	$-0.451924\pi$
$157$	3.77052	0.300920	0.150460	+	0.988616i	$0.451924\pi$
$158$	−1.39955	−0.111342
$159$	−26.0412	−2.06520
$160$	22.9895	1.81748
$161$	0	0
$162$	−7.42632	−0.583467
$163$	−10.3908	−0.813873	−0.406937	−	0.913456i	$-0.633403\pi$
$163$	−10.3908	−0.813873	−0.406937	+	0.913456i	$0.633403\pi$
$164$	−1.23474	−0.0964168
$165$	53.2431	4.14497
$166$	2.13602	0.165787
$167$	12.1931	0.943532	0.471766	−	0.881724i	$-0.343617\pi$
$167$	12.1931	0.943532	0.471766	+	0.881724i	$0.343617\pi$
$168$	0	0
$169$	−0.241566	−0.0185820
$170$	−17.1420	−1.31473
$171$	18.2282	1.39394
$172$	0.935961	0.0713664
$173$	−13.4181	−1.02016	−0.510081	−	0.860126i	$-0.670385\pi$
$173$	−13.4181	−1.02016	−0.510081	+	0.860126i	$0.670385\pi$
$174$	5.60001	0.424536
$175$	0	0
$176$	−0.0313768	−0.00236512
$177$	−22.0037	−1.65390
$178$	1.40467	0.105284
$179$	11.3724	0.850016	0.425008	−	0.905190i	$-0.360271\pi$
$179$	11.3724	0.850016	0.425008	+	0.905190i	$0.360271\pi$
$180$	−15.8720	−1.18303
$181$	−16.1255	−1.19860	−0.599301	−	0.800524i	$-0.704555\pi$
$181$	−16.1255	−1.19860	−0.599301	+	0.800524i	$0.704555\pi$
$182$	0	0
$183$	−6.40695	−0.473615
$184$	18.8977	1.39316
$185$	24.7216	1.81757
$186$	16.3984	1.20239
$187$	−25.4251	−1.85927
$188$	6.13338	0.447322
$189$	0	0
$190$	20.5070	1.48774
$191$	13.1574	0.952033	0.476016	−	0.879436i	$-0.342080\pi$
$191$	13.1574	0.952033	0.476016	+	0.879436i	$0.342080\pi$
$192$	12.3075	0.888214
$193$	−2.69957	−0.194319	−0.0971597	−	0.995269i	$-0.530976\pi$
$193$	−2.69957	−0.194319	−0.0971597	+	0.995269i	$0.530976\pi$
$194$	10.9823	0.788483
$195$	−36.0495	−2.58156
$196$	0	0
$197$	13.4871	0.960918	0.480459	−	0.877017i	$-0.340470\pi$
$197$	13.4871	0.960918	0.480459	+	0.877017i	$0.340470\pi$
$198$	14.5904	1.03690
$199$	20.0546	1.42163	0.710815	−	0.703379i	$-0.248326\pi$
$199$	20.0546	1.42163	0.710815	+	0.703379i	$0.248326\pi$
$200$	−32.6309	−2.30735
$201$	−10.7907	−0.761119
$202$	8.78610	0.618188
$203$	0	0
$204$	14.7713	1.03420
$205$	4.06589	0.283974
$206$	−2.05341	−0.143068
$207$	−21.1138	−1.46751
$208$	0.0212444	0.00147304
$209$	30.4162	2.10393
$210$	0	0
$211$	−13.0615	−0.899191	−0.449596	−	0.893232i	$-0.648432\pi$
$211$	−13.0615	−0.899191	−0.449596	+	0.893232i	$0.648432\pi$
$212$	12.9536	0.889657
$213$	0.0533733	0.00365708
$214$	−17.7698	−1.21472
$215$	−3.08204	−0.210194
$216$	−1.13478	−0.0772121
$217$	0	0
$218$	3.12866	0.211900
$219$	−10.0222	−0.677239
$220$	−26.4846	−1.78559
$221$	17.2147	1.15798
$222$	13.2030	0.886124
$223$	19.8995	1.33257	0.666284	−	0.745698i	$-0.267884\pi$
$223$	19.8995	1.33257	0.666284	+	0.745698i	$0.267884\pi$
$224$	0	0
$225$	36.4574	2.43049
$226$	−1.53439	−0.102066
$227$	−3.86912	−0.256803	−0.128401	−	0.991722i	$-0.540985\pi$
$227$	−3.86912	−0.256803	−0.128401	+	0.991722i	$0.540985\pi$
$228$	−17.6710	−1.17029
$229$	13.4416	0.888245	0.444122	−	0.895966i	$-0.353516\pi$
$229$	13.4416	0.888245	0.444122	+	0.895966i	$0.353516\pi$
$230$	−23.7534	−1.56625
$231$	0	0
$232$	−7.29765	−0.479114
$233$	−23.9043	−1.56602	−0.783010	−	0.622009i	$-0.786317\pi$
$233$	−23.9043	−1.56602	−0.783010	+	0.622009i	$0.786317\pi$
$234$	−9.87881	−0.645798
$235$	−20.1967	−1.31749
$236$	10.9452	0.712474
$237$	−3.97125	−0.257960
$238$	0	0
$239$	−20.9069	−1.35236	−0.676178	−	0.736738i	$-0.736365\pi$
$239$	−20.9069	−1.35236	−0.676178	+	0.736738i	$0.736365\pi$
$240$	−0.0600271	−0.00387473
$241$	17.8102	1.14726	0.573629	−	0.819115i	$-0.305535\pi$
$241$	17.8102	1.14726	0.573629	+	0.819115i	$0.305535\pi$
$242$	14.7234	0.946458
$243$	−22.2754	−1.42897
$244$	3.18699	0.204026
$245$	0	0
$246$	2.17145	0.138447
$247$	−20.5940	−1.31037
$248$	−21.3696	−1.35697
$249$	6.06100	0.384100
$250$	23.2312	1.46927
$251$	11.8981	0.751000	0.375500	−	0.926822i	$-0.377471\pi$
$251$	11.8981	0.751000	0.375500	+	0.926822i	$0.377471\pi$
$252$	0	0
$253$	−35.2312	−2.21497
$254$	10.6579	0.668735
$255$	−48.6408	−3.04601
$256$	−16.0147	−1.00092
$257$	−27.8806	−1.73915	−0.869573	−	0.493804i	$-0.835606\pi$
$257$	−27.8806	−1.73915	−0.869573	+	0.493804i	$0.835606\pi$
$258$	−1.64601	−0.102476
$259$	0	0
$260$	17.9320	1.11210
$261$	8.15341	0.504684
$262$	−15.7171	−0.971007
$263$	7.90146	0.487225	0.243613	−	0.969873i	$-0.421667\pi$
$263$	7.90146	0.487225	0.243613	+	0.969873i	$0.421667\pi$
$264$	−37.0554	−2.28060
$265$	−42.6551	−2.62028
$266$	0	0
$267$	3.98579	0.243926
$268$	5.36760	0.327878
$269$	19.0849	1.16362	0.581812	−	0.813323i	$-0.302344\pi$
$269$	19.0849	1.16362	0.581812	+	0.813323i	$0.302344\pi$
$270$	1.42636	0.0868055
$271$	26.1476	1.58836	0.794178	−	0.607686i	$-0.207902\pi$
$271$	26.1476	1.58836	0.794178	+	0.607686i	$0.207902\pi$
$272$	0.0286647	0.00173805
$273$	0	0
$274$	2.16731	0.130932
$275$	60.8341	3.66843
$276$	20.4685	1.23206
$277$	−16.2772	−0.978004	−0.489002	−	0.872283i	$-0.662639\pi$
$277$	−16.2772	−0.978004	−0.489002	+	0.872283i	$0.662639\pi$
$278$	−12.3977	−0.743567
$279$	23.8755	1.42939
$280$	0	0
$281$	18.9144	1.12834	0.564168	−	0.825660i	$-0.309197\pi$
$281$	18.9144	1.12834	0.564168	+	0.825660i	$0.309197\pi$
$282$	−10.7864	−0.642319
$283$	0.736724	0.0437937	0.0218969	−	0.999760i	$-0.493029\pi$
$283$	0.736724	0.0437937	0.0218969	+	0.999760i	$0.493029\pi$
$284$	−0.0265493	−0.00157541
$285$	58.1893	3.44684
$286$	−16.4841	−0.974728
$287$	0	0
$288$	17.8762	1.05337
$289$	6.22740	0.366317
$290$	9.17275	0.538642
$291$	31.1626	1.82678
$292$	4.98532	0.291744
$293$	−25.9031	−1.51327	−0.756636	−	0.653836i	$-0.773159\pi$
$293$	−25.9031	−1.51327	−0.756636	+	0.653836i	$0.773159\pi$
$294$	0	0
$295$	−36.0418	−2.09843
$296$	−17.2054	−1.00004
$297$	2.11559	0.122759
$298$	1.32270	0.0766218
$299$	23.8542	1.37952
$300$	−35.3431	−2.04053
$301$	0	0
$302$	−18.6665	−1.07414
$303$	24.9308	1.43224
$304$	−0.0342917	−0.00196676
$305$	−10.4945	−0.600913
$306$	−13.3293	−0.761984
$307$	−1.32384	−0.0755554	−0.0377777	−	0.999286i	$-0.512028\pi$
$307$	−1.32384	−0.0755554	−0.0377777	+	0.999286i	$0.512028\pi$
$308$	0	0
$309$	−5.82659	−0.331463
$310$	26.8604	1.52557
$311$	3.94439	0.223666	0.111833	−	0.993727i	$-0.464328\pi$
$311$	3.94439	0.223666	0.111833	+	0.993727i	$0.464328\pi$
$312$	25.0893	1.42040
$313$	−0.533508	−0.0301557	−0.0150778	−	0.999886i	$-0.504800\pi$
$313$	−0.533508	−0.0301557	−0.0150778	+	0.999886i	$0.504800\pi$
$314$	3.29843	0.186141
$315$	0	0
$316$	1.97541	0.111125
$317$	17.6596	0.991864	0.495932	−	0.868361i	$-0.334826\pi$
$317$	17.6596	0.991864	0.495932	+	0.868361i	$0.334826\pi$
$318$	−22.7806	−1.27748
$319$	13.6051	0.761738
$320$	20.1595	1.12695
$321$	−50.4223	−2.81430
$322$	0	0
$323$	−27.7871	−1.54611
$324$	10.4820	0.582332
$325$	−41.1892	−2.28476
$326$	−9.08983	−0.503439
$327$	8.87767	0.490936
$328$	−2.82973	−0.156245
$329$	0	0
$330$	46.5767	2.56396
$331$	−17.1566	−0.943012	−0.471506	−	0.881863i	$-0.656289\pi$
$331$	−17.1566	−0.943012	−0.471506	+	0.881863i	$0.656289\pi$
$332$	−3.01491	−0.165464
$333$	19.2230	1.05342
$334$	10.6664	0.583642
$335$	−17.6751	−0.965692
$336$	0	0
$337$	−9.62305	−0.524201	−0.262101	−	0.965041i	$-0.584415\pi$
$337$	−9.62305	−0.524201	−0.262101	+	0.965041i	$0.584415\pi$
$338$	−0.211321	−0.0114943
$339$	−4.35388	−0.236470
$340$	24.1953	1.31217
$341$	39.8395	2.15743
$342$	15.9459	0.862254
$343$	0	0
$344$	2.14500	0.115651
$345$	−67.4010	−3.62875
$346$	−11.7381	−0.631044
$347$	−7.96752	−0.427719	−0.213859	−	0.976864i	$-0.568603\pi$
$347$	−7.96752	−0.427719	−0.213859	+	0.976864i	$0.568603\pi$
$348$	−7.90421	−0.423710
$349$	13.1367	0.703190	0.351595	−	0.936152i	$-0.385639\pi$
$349$	13.1367	0.703190	0.351595	+	0.936152i	$0.385639\pi$
$350$	0	0
$351$	−1.43241	−0.0764563
$352$	29.8289	1.58989
$353$	−7.84742	−0.417676	−0.208838	−	0.977950i	$-0.566968\pi$
$353$	−7.84742	−0.417676	−0.208838	+	0.977950i	$0.566968\pi$
$354$	−19.2487	−1.02306
$355$	0.0874247	0.00464002
$356$	−1.98264	−0.105080
$357$	0	0
$358$	9.94853	0.525796
$359$	−10.6043	−0.559675	−0.279838	−	0.960047i	$-0.590281\pi$
$359$	−10.6043	−0.559675	−0.279838	+	0.960047i	$0.590281\pi$
$360$	−36.3748	−1.91712
$361$	14.2418	0.749569
$362$	−14.1065	−0.741421
$363$	41.7781	2.19278
$364$	0	0
$365$	−16.4163	−0.859266
$366$	−5.60475	−0.292965
$367$	19.0403	0.993898	0.496949	−	0.867780i	$-0.334454\pi$
$367$	19.0403	0.993898	0.496949	+	0.867780i	$0.334454\pi$
$368$	0.0397202	0.00207056
$369$	3.16156	0.164584
$370$	21.6263	1.12430
$371$	0	0
$372$	−23.1458	−1.20005
$373$	−12.8958	−0.667718	−0.333859	−	0.942623i	$-0.608351\pi$
$373$	−12.8958	−0.667718	−0.333859	+	0.942623i	$0.608351\pi$
$374$	−22.2417	−1.15009
$375$	65.9191	3.40405
$376$	14.0562	0.724895
$377$	−9.21165	−0.474424
$378$	0	0
$379$	4.75003	0.243993	0.121996	−	0.992531i	$-0.461070\pi$
$379$	4.75003	0.243993	0.121996	+	0.992531i	$0.461070\pi$
$380$	−28.9449	−1.48484
$381$	30.2420	1.54935
$382$	11.5100	0.588901
$383$	6.39275	0.326654	0.163327	−	0.986572i	$-0.447777\pi$
$383$	6.39275	0.326654	0.163327	+	0.986572i	$0.447777\pi$
$384$	−17.3040	−0.883042
$385$	0	0
$386$	−2.36157	−0.120201
$387$	−2.39654	−0.121823
$388$	−15.5011	−0.786949
$389$	−38.8690	−1.97074	−0.985368	−	0.170439i	$-0.945481\pi$
$389$	−38.8690	−1.97074	−0.985368	+	0.170439i	$0.945481\pi$
$390$	−31.5359	−1.59688
$391$	32.1859	1.62771
$392$	0	0
$393$	−44.5977	−2.24966
$394$	11.7984	0.594397
$395$	−6.50485	−0.327295
$396$	−20.5939	−1.03488
$397$	15.3895	0.772375	0.386188	−	0.922420i	$-0.373792\pi$
$397$	15.3895	0.772375	0.386188	+	0.922420i	$0.373792\pi$
$398$	17.5436	0.879380
$399$	0	0
$400$	−0.0685853	−0.00342926
$401$	−28.3023	−1.41335	−0.706674	−	0.707539i	$-0.749805\pi$
$401$	−28.3023	−1.41335	−0.706674	+	0.707539i	$0.749805\pi$
$402$	−9.43964	−0.470807
$403$	−26.9743	−1.34369
$404$	−12.4013	−0.616986
$405$	−34.5163	−1.71513
$406$	0	0
$407$	32.0762	1.58996
$408$	33.8524	1.67594
$409$	−1.24658	−0.0616394	−0.0308197	−	0.999525i	$-0.509812\pi$
$409$	−1.24658	−0.0616394	−0.0308197	+	0.999525i	$0.509812\pi$
$410$	3.55681	0.175658
$411$	6.14979	0.303347
$412$	2.89831	0.142789
$413$	0	0
$414$	−18.4702	−0.907760
$415$	9.92784	0.487339
$416$	−20.1964	−0.990209
$417$	−35.1789	−1.72272
$418$	26.6079	1.30143
$419$	6.37448	0.311414	0.155707	−	0.987803i	$-0.450234\pi$
$419$	6.37448	0.311414	0.155707	+	0.987803i	$0.450234\pi$
$420$	0	0
$421$	−10.9936	−0.535797	−0.267899	−	0.963447i	$-0.586329\pi$
$421$	−10.9936	−0.535797	−0.267899	+	0.963447i	$0.586329\pi$
$422$	−11.4261	−0.556215
$423$	−15.7046	−0.763582
$424$	29.6866	1.44171
$425$	−55.5757	−2.69582
$426$	0.0466906	0.00226216
$427$	0	0
$428$	25.0814	1.21236
$429$	−46.7742	−2.25828
$430$	−2.69615	−0.130020
$431$	−14.2732	−0.687514	−0.343757	−	0.939059i	$-0.611700\pi$
$431$	−14.2732	−0.687514	−0.343757	+	0.939059i	$0.611700\pi$
$432$	−0.00238514	−0.000114755 0
$433$	−3.11250	−0.149577	−0.0747887	−	0.997199i	$-0.523828\pi$
$433$	−3.11250	−0.149577	−0.0747887	+	0.997199i	$0.523828\pi$
$434$	0	0
$435$	26.0279	1.24794
$436$	−4.41599	−0.211488
$437$	−38.5042	−1.84190
$438$	−8.76736	−0.418921
$439$	27.8301	1.32826	0.664129	−	0.747618i	$-0.268802\pi$
$439$	27.8301	1.32826	0.664129	+	0.747618i	$0.268802\pi$
$440$	−60.6963	−2.89358
$441$	0	0
$442$	15.0593	0.716297
$443$	38.9695	1.85149	0.925747	−	0.378143i	$-0.123437\pi$
$443$	38.9695	1.85149	0.925747	+	0.378143i	$0.123437\pi$
$444$	−18.6355	−0.884401
$445$	6.52867	0.309489
$446$	17.4079	0.824289
$447$	3.75319	0.177520
$448$	0	0
$449$	−8.67213	−0.409263	−0.204632	−	0.978839i	$-0.565600\pi$
$449$	−8.67213	−0.409263	−0.204632	+	0.978839i	$0.565600\pi$
$450$	31.8926	1.50343
$451$	5.27549	0.248413
$452$	2.16574	0.101868
$453$	−52.9667	−2.48859
$454$	−3.38468	−0.158851
$455$	0	0
$456$	−40.4978	−1.89648
$457$	11.8500	0.554318	0.277159	−	0.960824i	$-0.410607\pi$
$457$	11.8500	0.554318	0.277159	+	0.960824i	$0.410607\pi$
$458$	11.7586	0.549443
$459$	−1.93272	−0.0902115
$460$	33.5271	1.56321
$461$	−17.8290	−0.830378	−0.415189	−	0.909735i	$-0.636285\pi$
$461$	−17.8290	−0.830378	−0.415189	+	0.909735i	$0.636285\pi$
$462$	0	0
$463$	−9.40648	−0.437156	−0.218578	−	0.975819i	$-0.570142\pi$
$463$	−9.40648	−0.437156	−0.218578	+	0.975819i	$0.570142\pi$
$464$	−0.0153386	−0.000712076 0
$465$	76.2171	3.53449
$466$	−20.9113	−0.968697
$467$	−4.14728	−0.191913	−0.0959566	−	0.995386i	$-0.530591\pi$
$467$	−4.14728	−0.191913	−0.0959566	+	0.995386i	$0.530591\pi$
$468$	13.9436	0.644542
$469$	0	0
$470$	−17.6679	−0.814961
$471$	9.35937	0.431257
$472$	25.0839	1.15458
$473$	−3.99895	−0.183872
$474$	−3.47402	−0.159567
$475$	66.4855	3.05056
$476$	0	0
$477$	−33.1678	−1.51865
$478$	−18.2892	−0.836530
$479$	15.0816	0.689096	0.344548	−	0.938769i	$-0.388032\pi$
$479$	15.0816	0.689096	0.344548	+	0.938769i	$0.388032\pi$
$480$	57.0658	2.60468
$481$	−21.7180	−0.990255
$482$	15.5803	0.709662
$483$	0	0
$484$	−20.7816	−0.944617
$485$	51.0439	2.31778
$486$	−19.4864	−0.883921
$487$	39.3537	1.78329	0.891643	−	0.452739i	$-0.149553\pi$
$487$	39.3537	1.78329	0.891643	+	0.452739i	$0.149553\pi$
$488$	7.30382	0.330629
$489$	−25.7926	−1.16638
$490$	0	0
$491$	17.8367	0.804958	0.402479	−	0.915429i	$-0.368149\pi$
$491$	17.8367	0.804958	0.402479	+	0.915429i	$0.368149\pi$
$492$	−3.06493	−0.138178
$493$	−12.4291	−0.559777
$494$	−18.0155	−0.810555
$495$	67.8140	3.04801
$496$	−0.0449157	−0.00201678
$497$	0	0
$498$	5.30212	0.237594
$499$	−13.2032	−0.591058	−0.295529	−	0.955334i	$-0.595496\pi$
$499$	−13.2032	−0.591058	−0.295529	+	0.955334i	$0.595496\pi$
$500$	−32.7900	−1.46641
$501$	30.2663	1.35220
$502$	10.4083	0.464547
$503$	9.52966	0.424907	0.212453	−	0.977171i	$-0.431855\pi$
$503$	9.52966	0.424907	0.212453	+	0.977171i	$0.431855\pi$
$504$	0	0
$505$	40.8363	1.81719
$506$	−30.8200	−1.37012
$507$	−0.599628	−0.0266304
$508$	−15.0432	−0.667435
$509$	−39.7452	−1.76168	−0.880838	−	0.473418i	$-0.843020\pi$
$509$	−39.7452	−1.76168	−0.880838	+	0.473418i	$0.843020\pi$
$510$	−42.5507	−1.88417
$511$	0	0
$512$	−0.0672901	−0.00297383
$513$	2.31212	0.102083
$514$	−24.3898	−1.07579
$515$	−9.54389	−0.420554
$516$	2.32329	0.102277
$517$	−26.2052	−1.15250
$518$	0	0
$519$	−33.3072	−1.46202
$520$	41.0959	1.80217
$521$	31.7642	1.39161	0.695806	−	0.718229i	$-0.255047\pi$
$521$	31.7642	1.39161	0.695806	+	0.718229i	$0.255047\pi$
$522$	7.13255	0.312183
$523$	−6.76873	−0.295976	−0.147988	−	0.988989i	$-0.547280\pi$
$523$	−6.76873	−0.295976	−0.147988	+	0.988989i	$0.547280\pi$
$524$	22.1841	0.969118
$525$	0	0
$526$	6.91214	0.301384
$527$	−36.3959	−1.58543
$528$	−0.0778851	−0.00338951
$529$	21.5996	0.939111
$530$	−37.3144	−1.62083
$531$	−28.0254	−1.21620
$532$	0	0
$533$	−3.57189	−0.154716
$534$	3.48674	0.150886
$535$	−82.5910	−3.57072
$536$	12.3013	0.531333
$537$	28.2292	1.21818
$538$	16.6953	0.719785
$539$	0	0
$540$	−2.01325	−0.0866366
$541$	11.3602	0.488412	0.244206	−	0.969723i	$-0.421473\pi$
$541$	11.3602	0.488412	0.244206	+	0.969723i	$0.421473\pi$
$542$	22.8738	0.982512
$543$	−40.0276	−1.71775
$544$	−27.2505	−1.16836
$545$	14.5415	0.622890
$546$	0	0
$547$	−14.2525	−0.609392	−0.304696	−	0.952450i	$-0.598555\pi$
$547$	−14.2525	−0.609392	−0.304696	+	0.952450i	$0.598555\pi$
$548$	−3.05907	−0.130677
$549$	−8.16031	−0.348274
$550$	53.2172	2.26919
$551$	14.8690	0.633439
$552$	46.9088	1.99657
$553$	0	0
$554$	−14.2392	−0.604966
$555$	61.3651	2.60480
$556$	17.4990	0.742121
$557$	−4.63299	−0.196306	−0.0981531	−	0.995171i	$-0.531293\pi$
$557$	−4.63299	−0.196306	−0.0981531	+	0.995171i	$0.531293\pi$
$558$	20.8861	0.884180
$559$	2.70758	0.114519
$560$	0	0
$561$	−63.1114	−2.66457
$562$	16.5461	0.697957
$563$	−6.30137	−0.265571	−0.132786	−	0.991145i	$-0.542392\pi$
$563$	−6.30137	−0.265571	−0.132786	+	0.991145i	$0.542392\pi$
$564$	15.2246	0.641070
$565$	−7.13160	−0.300029
$566$	0.644481	0.0270896
$567$	0	0
$568$	−0.0608447	−0.00255299
$569$	−14.8553	−0.622766	−0.311383	−	0.950284i	$-0.600792\pi$
$569$	−14.8553	−0.622766	−0.311383	+	0.950284i	$0.600792\pi$
$570$	50.9036	2.13212
$571$	0.194645	0.00814563	0.00407281	−	0.999992i	$-0.498704\pi$
$571$	0.194645	0.00814563	0.00407281	+	0.999992i	$0.498704\pi$
$572$	23.2668	0.972832
$573$	32.6598	1.36438
$574$	0	0
$575$	−77.0105	−3.21156
$576$	15.6756	0.653150
$577$	−8.63405	−0.359440	−0.179720	−	0.983718i	$-0.557519\pi$
$577$	−8.63405	−0.359440	−0.179720	+	0.983718i	$0.557519\pi$
$578$	5.44768	0.226594
$579$	−6.70101	−0.278484
$580$	−12.9470	−0.537595
$581$	0	0
$582$	27.2608	1.13000
$583$	−55.3450	−2.29216
$584$	11.4252	0.472777
$585$	−45.9151	−1.89835
$586$	−22.6598	−0.936068
$587$	6.88070	0.283997	0.141998	−	0.989867i	$-0.454647\pi$
$587$	6.88070	0.283997	0.141998	+	0.989867i	$0.454647\pi$
$588$	0	0
$589$	43.5406	1.79406
$590$	−31.5291	−1.29803
$591$	33.4784	1.37712
$592$	−0.0361632	−0.00148630
$593$	−4.08098	−0.167586	−0.0837928	−	0.996483i	$-0.526703\pi$
$593$	−4.08098	−0.167586	−0.0837928	+	0.996483i	$0.526703\pi$
$594$	1.85070	0.0759351
$595$	0	0
$596$	−1.86694	−0.0764728
$597$	49.7804	2.03738
$598$	20.8674	0.853333
$599$	40.2888	1.64616	0.823078	−	0.567929i	$-0.192255\pi$
$599$	40.2888	1.64616	0.823078	+	0.567929i	$0.192255\pi$
$600$	−80.9979	−3.30673
$601$	2.62784	0.107192	0.0535958	−	0.998563i	$-0.482932\pi$
$601$	2.62784	0.107192	0.0535958	+	0.998563i	$0.482932\pi$
$602$	0	0
$603$	−13.7438	−0.559690
$604$	26.3471	1.07205
$605$	68.4321	2.78216
$606$	21.8093	0.885942
$607$	−15.1005	−0.612909	−0.306454	−	0.951885i	$-0.599143\pi$
$607$	−15.1005	−0.612909	−0.306454	+	0.951885i	$0.599143\pi$
$608$	32.6000	1.32210
$609$	0	0
$610$	−9.18051	−0.371708
$611$	17.7429	0.717799
$612$	18.8138	0.760502
$613$	−7.10837	−0.287104	−0.143552	−	0.989643i	$-0.545852\pi$
$613$	−7.10837	−0.287104	−0.143552	+	0.989643i	$0.545852\pi$
$614$	−1.15808	−0.0467365
$615$	10.0925	0.406971
$616$	0	0
$617$	34.0305	1.37001	0.685007	−	0.728536i	$-0.259799\pi$
$617$	34.0305	1.37001	0.685007	+	0.728536i	$0.259799\pi$
$618$	−5.09706	−0.205034
$619$	11.5101	0.462632	0.231316	−	0.972879i	$-0.425697\pi$
$619$	11.5101	0.462632	0.231316	+	0.972879i	$0.425697\pi$
$620$	−37.9125	−1.52260
$621$	−2.67814	−0.107470
$622$	3.45053	0.138354
$623$	0	0
$624$	0.0527340	0.00211105
$625$	50.3174	2.01269
$626$	−0.466709	−0.0186535
$627$	75.5006	3.01520
$628$	−4.65561	−0.185779
$629$	−29.3036	−1.16841
$630$	0	0
$631$	−23.6325	−0.940796	−0.470398	−	0.882454i	$-0.655890\pi$
$631$	−23.6325	−0.940796	−0.470398	+	0.882454i	$0.655890\pi$
$632$	4.52716	0.180081
$633$	−32.4219	−1.28866
$634$	15.4485	0.613539
$635$	49.5361	1.96578
$636$	32.1540	1.27499
$637$	0	0
$638$	11.9016	0.471190
$639$	0.0679797	0.00268924
$640$	−28.3438	−1.12039
$641$	−31.1252	−1.22937	−0.614687	−	0.788771i	$-0.710718\pi$
$641$	−31.1252	−1.22937	−0.614687	+	0.788771i	$0.710718\pi$
$642$	−44.1090	−1.74085
$643$	5.84890	0.230658	0.115329	−	0.993327i	$-0.463208\pi$
$643$	5.84890	0.230658	0.115329	+	0.993327i	$0.463208\pi$
$644$	0	0
$645$	−7.65040	−0.301234
$646$	−24.3079	−0.956382
$647$	17.4255	0.685067	0.342534	−	0.939506i	$-0.388715\pi$
$647$	17.4255	0.685067	0.342534	+	0.939506i	$0.388715\pi$
$648$	24.0222	0.943681
$649$	−46.7642	−1.83565
$650$	−36.0320	−1.41329
$651$	0	0
$652$	12.8300	0.502460
$653$	3.18537	0.124653	0.0623266	−	0.998056i	$-0.480148\pi$
$653$	3.18537	0.124653	0.0623266	+	0.998056i	$0.480148\pi$
$654$	7.76612	0.303679
$655$	−73.0505	−2.85432
$656$	−0.00594767	−0.000232217 0
$657$	−12.7650	−0.498009
$658$	0	0
$659$	−12.9118	−0.502972	−0.251486	−	0.967861i	$-0.580919\pi$
$659$	−12.9118	−0.502972	−0.251486	+	0.967861i	$0.580919\pi$
$660$	−65.7413	−2.55898
$661$	23.4509	0.912134	0.456067	−	0.889945i	$-0.349258\pi$
$661$	23.4509	0.912134	0.456067	+	0.889945i	$0.349258\pi$
$662$	−15.0085	−0.583321
$663$	42.7311	1.65954
$664$	−6.90945	−0.268139
$665$	0	0
$666$	16.8162	0.651613
$667$	−17.2228	−0.666870
$668$	−15.0553	−0.582507
$669$	49.3955	1.90974
$670$	−15.4620	−0.597350
$671$	−13.6166	−0.525663
$672$	0	0
$673$	9.64226	0.371682	0.185841	−	0.982580i	$-0.440499\pi$
$673$	9.64226	0.371682	0.185841	+	0.982580i	$0.440499\pi$
$674$	−8.41818	−0.324256
$675$	4.62437	0.177992
$676$	0.298271	0.0114720
$677$	34.6286	1.33089	0.665443	−	0.746448i	$-0.268243\pi$
$677$	34.6286	1.33089	0.665443	+	0.746448i	$0.268243\pi$
$678$	−3.80874	−0.146274
$679$	0	0
$680$	55.4498	2.12640
$681$	−9.60412	−0.368031
$682$	34.8513	1.33453
$683$	26.5402	1.01553	0.507766	−	0.861495i	$-0.330472\pi$
$683$	26.5402	1.01553	0.507766	+	0.861495i	$0.330472\pi$
$684$	−22.5070	−0.860577
$685$	10.0733	0.384880
$686$	0	0
$687$	33.3653	1.27297
$688$	0.00450847	0.000171884 0
$689$	37.4727	1.42759
$690$	−58.9619	−2.24464
$691$	10.0933	0.383967	0.191983	−	0.981398i	$-0.438508\pi$
$691$	10.0933	0.383967	0.191983	+	0.981398i	$0.438508\pi$
$692$	16.5679	0.629817
$693$	0	0
$694$	−6.96993	−0.264575
$695$	−57.6226	−2.18575
$696$	−18.1146	−0.686631
$697$	−4.81948	−0.182551
$698$	11.4919	0.434973
$699$	−59.3363	−2.24431
$700$	0	0
$701$	51.1385	1.93148	0.965738	−	0.259520i	$-0.0835643\pi$
$701$	51.1385	1.93148	0.965738	+	0.259520i	$0.0835643\pi$
$702$	−1.25306	−0.0472937
$703$	35.0561	1.32216
$704$	26.1569	0.985824
$705$	−50.1332	−1.88813
$706$	−6.86486	−0.258362
$707$	0	0
$708$	27.1688	1.02107
$709$	−23.1800	−0.870543	−0.435272	−	0.900299i	$-0.643348\pi$
$709$	−23.1800	−0.870543	−0.435272	+	0.900299i	$0.643348\pi$
$710$	0.0764785	0.00287019
$711$	−5.05804	−0.189691
$712$	−4.54374	−0.170284
$713$	−50.4333	−1.88874
$714$	0	0
$715$	−76.6156	−2.86526
$716$	−14.0420	−0.524774
$717$	−51.8962	−1.93810
$718$	−9.27659	−0.346199
$719$	50.1806	1.87142	0.935710	−	0.352769i	$-0.114760\pi$
$719$	50.1806	1.87142	0.935710	+	0.352769i	$0.114760\pi$
$720$	−0.0764545	−0.00284929
$721$	0	0
$722$	12.4586	0.463662
$723$	44.2094	1.64417
$724$	19.9108	0.739979
$725$	29.7388	1.10447
$726$	36.5472	1.35639
$727$	36.5957	1.35726	0.678629	−	0.734481i	$-0.262574\pi$
$727$	36.5957	1.35726	0.678629	+	0.734481i	$0.262574\pi$
$728$	0	0
$729$	−29.8255	−1.10465
$730$	−14.3608	−0.531518
$731$	3.65328	0.135122
$732$	7.91090	0.292395
$733$	−44.0711	−1.62780	−0.813901	−	0.581003i	$-0.802660\pi$
$733$	−44.0711	−1.62780	−0.813901	+	0.581003i	$0.802660\pi$
$734$	16.6564	0.614797
$735$	0	0
$736$	−37.7607	−1.39188
$737$	−22.9334	−0.844761
$738$	2.76571	0.101807
$739$	8.85737	0.325824	0.162912	−	0.986641i	$-0.447911\pi$
$739$	8.85737	0.325824	0.162912	+	0.986641i	$0.447911\pi$
$740$	−30.5247	−1.12211
$741$	−51.1195	−1.87792
$742$	0	0
$743$	−20.3318	−0.745900	−0.372950	−	0.927851i	$-0.621654\pi$
$743$	−20.3318	−0.745900	−0.372950	+	0.927851i	$0.621654\pi$
$744$	−53.0446	−1.94471
$745$	6.14768	0.225234
$746$	−11.2811	−0.413032
$747$	7.71969	0.282449
$748$	31.3933	1.14785
$749$	0	0
$750$	57.6656	2.10565
$751$	−25.1872	−0.919093	−0.459546	−	0.888154i	$-0.651988\pi$
$751$	−25.1872	−0.919093	−0.459546	+	0.888154i	$0.651988\pi$
$752$	0.0295441	0.00107736
$753$	29.5340	1.07628
$754$	−8.05828	−0.293465
$755$	−86.7588	−3.15748
$756$	0	0
$757$	11.1450	0.405071	0.202535	−	0.979275i	$-0.435082\pi$
$757$	11.1450	0.405071	0.202535	+	0.979275i	$0.435082\pi$
$758$	4.15530	0.150927
$759$	−87.4527	−3.17433
$760$	−66.3349	−2.40622
$761$	3.59373	0.130273	0.0651363	−	0.997876i	$-0.479252\pi$
$761$	3.59373	0.130273	0.0651363	+	0.997876i	$0.479252\pi$
$762$	26.4555	0.958382
$763$	0	0
$764$	−16.2459	−0.587755
$765$	−61.9522	−2.23989
$766$	5.59233	0.202059
$767$	31.6628	1.14328
$768$	−39.7523	−1.43444
$769$	−36.2730	−1.30804	−0.654019	−	0.756478i	$-0.726918\pi$
$769$	−36.2730	−1.30804	−0.654019	+	0.756478i	$0.726918\pi$
$770$	0	0
$771$	−69.2067	−2.49242
$772$	3.33326	0.119967
$773$	−23.8496	−0.857810	−0.428905	−	0.903350i	$-0.641100\pi$
$773$	−23.8496	−0.857810	−0.428905	+	0.903350i	$0.641100\pi$
$774$	−2.09647	−0.0753561
$775$	87.0836	3.12813
$776$	−35.5249	−1.27527
$777$	0	0
$778$	−34.0023	−1.21904
$779$	5.76557	0.206573
$780$	44.5117	1.59377
$781$	0.113433	0.00405897
$782$	28.1560	1.00686
$783$	1.03421	0.0369595
$784$	0	0
$785$	15.3305	0.547170
$786$	−39.0138	−1.39158
$787$	−2.22372	−0.0792669	−0.0396335	−	0.999214i	$-0.512619\pi$
$787$	−2.22372	−0.0792669	−0.0396335	+	0.999214i	$0.512619\pi$
$788$	−16.6531	−0.593241
$789$	19.6134	0.698255
$790$	−5.69040	−0.202455
$791$	0	0
$792$	−47.1963	−1.67705
$793$	9.21944	0.327392
$794$	13.4626	0.477770
$795$	−105.881	−3.75520
$796$	−24.7621	−0.877670
$797$	−21.4585	−0.760098	−0.380049	−	0.924966i	$-0.624093\pi$
$797$	−21.4585	−0.760098	−0.380049	+	0.924966i	$0.624093\pi$
$798$	0	0
$799$	23.9401	0.846938
$800$	65.2018	2.30523
$801$	5.07657	0.179372
$802$	−24.7586	−0.874258
$803$	−21.3001	−0.751663
$804$	13.3237	0.469891
$805$	0	0
$806$	−23.5969	−0.831167
$807$	47.3733	1.66762
$808$	−28.4207	−0.999838
$809$	50.5169	1.77608	0.888040	−	0.459765i	$-0.152066\pi$
$809$	50.5169	1.77608	0.888040	+	0.459765i	$0.152066\pi$
$810$	−30.1946	−1.06093
$811$	13.1187	0.460658	0.230329	−	0.973113i	$-0.426020\pi$
$811$	13.1187	0.460658	0.230329	+	0.973113i	$0.426020\pi$
$812$	0	0
$813$	64.9049	2.27631
$814$	28.0601	0.983505
$815$	−42.2480	−1.47988
$816$	0.0711528	0.00249085
$817$	−4.37044	−0.152902
$818$	−1.09050	−0.0381284
$819$	0	0
$820$	−5.02031	−0.175317
$821$	3.08369	0.107622	0.0538108	−	0.998551i	$-0.482863\pi$
$821$	3.08369	0.107622	0.0538108	+	0.998551i	$0.482863\pi$
$822$	5.37979	0.187642
$823$	15.7478	0.548934	0.274467	−	0.961596i	$-0.411499\pi$
$823$	15.7478	0.548934	0.274467	+	0.961596i	$0.411499\pi$
$824$	6.64223	0.231393
$825$	151.005	5.25733
$826$	0	0
$827$	−41.6230	−1.44737	−0.723686	−	0.690129i	$-0.757554\pi$
$827$	−41.6230	−1.44737	−0.723686	+	0.690129i	$0.757554\pi$
$828$	26.0700	0.905995
$829$	3.50769	0.121827	0.0609136	−	0.998143i	$-0.480599\pi$
$829$	3.50769	0.121827	0.0609136	+	0.998143i	$0.480599\pi$
$830$	8.68480	0.301454
$831$	−40.4041	−1.40160
$832$	−17.7101	−0.613988
$833$	0	0
$834$	−30.7743	−1.06563
$835$	49.5759	1.71564
$836$	−37.5560	−1.29890
$837$	3.02845	0.104678
$838$	5.57635	0.192632
$839$	41.6656	1.43846	0.719228	−	0.694774i	$-0.244496\pi$
$839$	41.6656	1.43846	0.719228	+	0.694774i	$0.244496\pi$
$840$	0	0
$841$	−22.3492	−0.770660
$842$	−9.61716	−0.331429
$843$	46.9501	1.61705
$844$	16.1275	0.555133
$845$	−0.982182	−0.0337881
$846$	−13.7382	−0.472330
$847$	0	0
$848$	0.0623968	0.00214271
$849$	1.82873	0.0627619
$850$	−48.6172	−1.66756
$851$	−40.6056	−1.39194
$852$	−0.0659020	−0.00225777
$853$	7.51207	0.257208	0.128604	−	0.991696i	$-0.458950\pi$
$853$	7.51207	0.257208	0.128604	+	0.991696i	$0.458950\pi$
$854$	0	0
$855$	74.1138	2.53464
$856$	57.4806	1.96465
$857$	−39.6705	−1.35512	−0.677559	−	0.735468i	$-0.736962\pi$
$857$	−39.6705	−1.35512	−0.677559	+	0.735468i	$0.736962\pi$
$858$	−40.9177	−1.39691
$859$	−21.3641	−0.728932	−0.364466	−	0.931217i	$-0.618748\pi$
$859$	−21.3641	−0.728932	−0.364466	+	0.931217i	$0.618748\pi$
$860$	3.80551	0.129767
$861$	0	0
$862$	−12.4861	−0.425277
$863$	−25.2451	−0.859355	−0.429677	−	0.902982i	$-0.641373\pi$
$863$	−25.2451	−0.859355	−0.429677	+	0.902982i	$0.641373\pi$
$864$	2.26748	0.0771411
$865$	−54.5567	−1.85498
$866$	−2.72280	−0.0925244
$867$	15.4579	0.524979
$868$	0	0
$869$	−8.44004	−0.286309
$870$	22.7690	0.771943
$871$	15.5276	0.526132
$872$	−10.1204	−0.342720
$873$	39.6907	1.34333
$874$	−33.6832	−1.13935
$875$	0	0
$876$	12.3748	0.418106
$877$	9.13549	0.308484	0.154242	−	0.988033i	$-0.450707\pi$
$877$	9.13549	0.308484	0.154242	+	0.988033i	$0.450707\pi$
$878$	24.3456	0.821624
$879$	−64.2978	−2.16871
$880$	−0.127575	−0.00430054
$881$	38.9905	1.31362	0.656812	−	0.754055i	$-0.271905\pi$
$881$	38.9905	1.31362	0.656812	+	0.754055i	$0.271905\pi$
$882$	0	0
$883$	14.9726	0.503869	0.251934	−	0.967744i	$-0.418933\pi$
$883$	14.9726	0.503869	0.251934	+	0.967744i	$0.418933\pi$
$884$	−21.2556	−0.714904
$885$	−89.4646	−3.00732
$886$	34.0902	1.14528
$887$	−19.3530	−0.649809	−0.324905	−	0.945747i	$-0.605332\pi$
$887$	−19.3530	−0.649809	−0.324905	+	0.945747i	$0.605332\pi$
$888$	−42.7081	−1.43319
$889$	0	0
$890$	5.71124	0.191441
$891$	−44.7848	−1.50035
$892$	−24.5707	−0.822686
$893$	−28.6396	−0.958388
$894$	3.28326	0.109809
$895$	46.2391	1.54560
$896$	0	0
$897$	59.2119	1.97703
$898$	−7.58632	−0.253159
$899$	19.4756	0.649547
$900$	−45.0153	−1.50051
$901$	50.5610	1.68443
$902$	4.61496	0.153661
$903$	0	0
$904$	4.96336	0.165079
$905$	−65.5647	−2.17944
$906$	−46.3349	−1.53937
$907$	55.7307	1.85051	0.925253	−	0.379350i	$-0.123852\pi$
$907$	55.7307	1.85051	0.925253	+	0.379350i	$0.123852\pi$
$908$	4.77735	0.158542
$909$	31.7535	1.05320
$910$	0	0
$911$	14.7155	0.487546	0.243773	−	0.969832i	$-0.421615\pi$
$911$	14.7155	0.487546	0.243773	+	0.969832i	$0.421615\pi$
$912$	−0.0851205	−0.00281862
$913$	12.8814	0.426311
$914$	10.3663	0.342886
$915$	−26.0499	−0.861185
$916$	−16.5968	−0.548375
$917$	0	0
$918$	−1.69073	−0.0558023
$919$	−52.3968	−1.72841	−0.864206	−	0.503138i	$-0.832179\pi$
$919$	−52.3968	−1.72841	−0.864206	+	0.503138i	$0.832179\pi$
$920$	76.8361	2.53321
$921$	−3.28609	−0.108280
$922$	−15.5967	−0.513648
$923$	−0.0768028	−0.00252800
$924$	0	0
$925$	70.1141	2.30534
$926$	−8.22872	−0.270413
$927$	−7.42114	−0.243742
$928$	14.5819	0.478673
$929$	12.3066	0.403768	0.201884	−	0.979409i	$-0.435294\pi$
$929$	12.3066	0.403768	0.201884	+	0.979409i	$0.435294\pi$
$930$	66.6742	2.18633
$931$	0	0
$932$	29.5155	0.966812
$933$	9.79096	0.320542
$934$	−3.62801	−0.118712
$935$	−103.376	−3.38075
$936$	31.9554	1.04449
$937$	42.3242	1.38267	0.691336	−	0.722534i	$-0.257023\pi$
$937$	42.3242	1.38267	0.691336	+	0.722534i	$0.257023\pi$
$938$	0	0
$939$	−1.32430	−0.0432169
$940$	24.9376	0.813376
$941$	7.73123	0.252031	0.126015	−	0.992028i	$-0.459781\pi$
$941$	7.73123	0.252031	0.126015	+	0.992028i	$0.459781\pi$
$942$	8.18751	0.266763
$943$	−6.67829	−0.217475
$944$	0.0527226	0.00171598
$945$	0	0
$946$	−3.49825	−0.113738
$947$	56.4799	1.83535	0.917675	−	0.397332i	$-0.130064\pi$
$947$	56.4799	1.83535	0.917675	+	0.397332i	$0.130064\pi$
$948$	4.90345	0.159257
$949$	14.4217	0.468149
$950$	58.1610	1.88699
$951$	43.8356	1.42147
$952$	0	0
$953$	37.9991	1.23091	0.615455	−	0.788172i	$-0.288972\pi$
$953$	37.9991	1.23091	0.615455	+	0.788172i	$0.288972\pi$
$954$	−29.0149	−0.939394
$955$	53.4964	1.73110
$956$	25.8146	0.834903
$957$	33.7712	1.09167
$958$	13.1933	0.426256
$959$	0	0
$960$	50.0408	1.61506
$961$	26.0300	0.839678
$962$	−18.9987	−0.612544
$963$	−64.2212	−2.06950
$964$	−21.9910	−0.708281
$965$	−10.9762	−0.353335
$966$	0	0
$967$	41.9317	1.34843	0.674217	−	0.738534i	$-0.264481\pi$
$967$	41.9317	1.34843	0.674217	+	0.738534i	$0.264481\pi$
$968$	−47.6264	−1.53077
$969$	−68.9744	−2.21578
$970$	44.6528	1.43372
$971$	50.0528	1.60627	0.803136	−	0.595796i	$-0.203163\pi$
$971$	50.0528	1.60627	0.803136	+	0.595796i	$0.203163\pi$
$972$	27.5043	0.882202
$973$	0	0
$974$	34.4263	1.10309
$975$	−102.242	−3.27436
$976$	0.0153516	0.000491391 0
$977$	45.5666	1.45780	0.728902	−	0.684618i	$-0.240031\pi$
$977$	45.5666	1.45780	0.728902	+	0.684618i	$0.240031\pi$
$978$	−22.5632	−0.721492
$979$	8.47094	0.270733
$980$	0	0
$981$	11.3072	0.361011
$982$	15.6034	0.497924
$983$	−4.85339	−0.154799	−0.0773996	−	0.997000i	$-0.524662\pi$
$983$	−4.85339	−0.154799	−0.0773996	+	0.997000i	$0.524662\pi$
$984$	−7.02408	−0.223919
$985$	54.8372	1.74726
$986$	−10.8729	−0.346263
$987$	0	0
$988$	25.4282	0.808979
$989$	5.06231	0.160972
$990$	59.3232	1.88541
$991$	39.6663	1.26004	0.630021	−	0.776578i	$-0.283046\pi$
$991$	39.6663	1.26004	0.630021	+	0.776578i	$0.283046\pi$
$992$	42.6999	1.35572
$993$	−42.5870	−1.35146
$994$	0	0
$995$	81.5396	2.58498
$996$	−7.48375	−0.237132
$997$	50.0564	1.58530	0.792652	−	0.609675i	$-0.208700\pi$
$997$	50.0564	1.58530	0.792652	+	0.609675i	$0.208700\pi$
$998$	−11.5501	−0.365612
$999$	2.43831	0.0771447

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2009.2.a.u.1.14	yes	20
7.6	odd	2		2009.2.a.t.1.14	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2009.2.a.t.1.14	✓	20	7.6	odd	2
2009.2.a.u.1.14	yes	20	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2009 = 7^{2} \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2009.a (trivial)

\(f(q)\)	\(=\)	\(q+0.874793 q^{2} +2.48225 q^{3} -1.23474 q^{4} +4.06589 q^{5} +2.17145 q^{6} -2.82973 q^{8} +3.16156 q^{9} +O(q^{10})\)	\(q+0.874793 q^{2} +2.48225 q^{3} -1.23474 q^{4} +4.06589 q^{5} +2.17145 q^{6} -2.82973 q^{8} +3.16156 q^{9} +3.55681 q^{10} +5.27549 q^{11} -3.06493 q^{12} -3.57189 q^{13} +10.0925 q^{15} -0.00594767 q^{16} -4.81948 q^{17} +2.76571 q^{18} +5.76557 q^{19} -5.02031 q^{20} +4.61496 q^{22} -6.67829 q^{23} -7.02408 q^{24} +11.5315 q^{25} -3.12467 q^{26} +0.401022 q^{27} +2.57892 q^{29} +8.82889 q^{30} +7.55182 q^{31} +5.65425 q^{32} +13.0951 q^{33} -4.21605 q^{34} -3.90369 q^{36} +6.08024 q^{37} +5.04368 q^{38} -8.86633 q^{39} -11.5054 q^{40} +1.00000 q^{41} -0.758024 q^{43} -6.51384 q^{44} +12.8545 q^{45} -5.84212 q^{46} -4.96735 q^{47} -0.0147636 q^{48} +10.0876 q^{50} -11.9631 q^{51} +4.41035 q^{52} -10.4910 q^{53} +0.350811 q^{54} +21.4496 q^{55} +14.3116 q^{57} +2.25602 q^{58} -8.86442 q^{59} -12.4617 q^{60} -2.58111 q^{61} +6.60628 q^{62} +4.95819 q^{64} -14.5229 q^{65} +11.4555 q^{66} -4.34716 q^{67} +5.95079 q^{68} -16.5772 q^{69} +0.0215020 q^{71} -8.94633 q^{72} -4.03756 q^{73} +5.31895 q^{74} +28.6240 q^{75} -7.11897 q^{76} -7.75620 q^{78} -1.59986 q^{79} -0.0241826 q^{80} -8.48923 q^{81} +0.874793 q^{82} +2.44174 q^{83} -19.5955 q^{85} -0.663114 q^{86} +6.40153 q^{87} -14.9282 q^{88} +1.60572 q^{89} +11.2451 q^{90} +8.24594 q^{92} +18.7455 q^{93} -4.34540 q^{94} +23.4422 q^{95} +14.0352 q^{96} +12.5542 q^{97} +16.6787 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 2 q^{2} + 8 q^{3} + 18 q^{4} + 8 q^{5} + 12 q^{6} - 6 q^{8} + 16 q^{9}+O(q^{10}) \) 20 * q - 2 * q^2 + 8 * q^3 + 18 * q^4 + 8 * q^5 + 12 * q^6 - 6 * q^8 + 16 * q^9	\( 20 q - 2 q^{2} + 8 q^{3} + 18 q^{4} + 8 q^{5} + 12 q^{6} - 6 q^{8} + 16 q^{9} + 16 q^{10} + 6 q^{12} + 12 q^{13} + 14 q^{16} + 8 q^{17} - 18 q^{18} + 36 q^{19} + 24 q^{20} - 8 q^{22} - 12 q^{23} + 36 q^{24} + 20 q^{25} - 22 q^{26} + 32 q^{27} + 4 q^{29} + 28 q^{30} + 80 q^{31} + 6 q^{32} - 12 q^{33} + 48 q^{34} + 26 q^{36} + 4 q^{37} + 12 q^{38} - 28 q^{39} - 4 q^{40} + 20 q^{41} - 20 q^{44} + 40 q^{45} + 8 q^{46} + 32 q^{47} + 16 q^{48} + 6 q^{50} - 20 q^{51} + 36 q^{52} + 4 q^{53} - 50 q^{54} + 64 q^{55} - 4 q^{57} + 32 q^{59} + 20 q^{60} + 44 q^{61} - 8 q^{62} - 30 q^{64} - 8 q^{65} + 32 q^{66} - 4 q^{67} - 48 q^{68} - 24 q^{69} + 8 q^{71} - 8 q^{72} + 48 q^{73} - 38 q^{74} + 24 q^{75} + 84 q^{76} + 30 q^{78} - 4 q^{79} + 56 q^{80} - 2 q^{82} + 8 q^{83} - 12 q^{85} - 24 q^{86} + 40 q^{87} - 48 q^{88} + 20 q^{89} + 48 q^{90} - 50 q^{92} + 48 q^{93} + 26 q^{94} + 20 q^{95} + 70 q^{96} + 8 q^{97} - 8 q^{99}+O(q^{100}) \) 20 * q - 2 * q^2 + 8 * q^3 + 18 * q^4 + 8 * q^5 + 12 * q^6 - 6 * q^8 + 16 * q^9 + 16 * q^10 + 6 * q^12 + 12 * q^13 + 14 * q^16 + 8 * q^17 - 18 * q^18 + 36 * q^19 + 24 * q^20 - 8 * q^22 - 12 * q^23 + 36 * q^24 + 20 * q^25 - 22 * q^26 + 32 * q^27 + 4 * q^29 + 28 * q^30 + 80 * q^31 + 6 * q^32 - 12 * q^33 + 48 * q^34 + 26 * q^36 + 4 * q^37 + 12 * q^38 - 28 * q^39 - 4 * q^40 + 20 * q^41 - 20 * q^44 + 40 * q^45 + 8 * q^46 + 32 * q^47 + 16 * q^48 + 6 * q^50 - 20 * q^51 + 36 * q^52 + 4 * q^53 - 50 * q^54 + 64 * q^55 - 4 * q^57 + 32 * q^59 + 20 * q^60 + 44 * q^61 - 8 * q^62 - 30 * q^64 - 8 * q^65 + 32 * q^66 - 4 * q^67 - 48 * q^68 - 24 * q^69 + 8 * q^71 - 8 * q^72 + 48 * q^73 - 38 * q^74 + 24 * q^75 + 84 * q^76 + 30 * q^78 - 4 * q^79 + 56 * q^80 - 2 * q^82 + 8 * q^83 - 12 * q^85 - 24 * q^86 + 40 * q^87 - 48 * q^88 + 20 * q^89 + 48 * q^90 - 50 * q^92 + 48 * q^93 + 26 * q^94 + 20 * q^95 + 70 * q^96 + 8 * q^97 - 8 * q^99

Introduction

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