LMFDB - Embedded newform 3751.2.a.a.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3751,2,Mod(1,3751)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3751, 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("3751.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3751 = 11^{2} \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3751.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:	$29.9518857982$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 341)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	3751.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.618034 q^{2} -1.00000 q^{3} -1.61803 q^{4} -0.381966 q^{5} -0.618034 q^{6} -3.85410 q^{7} -2.23607 q^{8} -2.00000 q^{9} +O(q^{10})$	\(q+0.618034 q^{2} -1.00000 q^{3} -1.61803 q^{4} -0.381966 q^{5} -0.618034 q^{6} -3.85410 q^{7} -2.23607 q^{8} -2.00000 q^{9} -0.236068 q^{10} +1.61803 q^{12} +5.47214 q^{13} -2.38197 q^{14} +0.381966 q^{15} +1.85410 q^{16} +4.23607 q^{17} -1.23607 q^{18} +7.23607 q^{19} +0.618034 q^{20} +3.85410 q^{21} -3.23607 q^{23} +2.23607 q^{24} -4.85410 q^{25} +3.38197 q^{26} +5.00000 q^{27} +6.23607 q^{28} +0.236068 q^{30} +1.00000 q^{31} +5.61803 q^{32} +2.61803 q^{34} +1.47214 q^{35} +3.23607 q^{36} +0.236068 q^{37} +4.47214 q^{38} -5.47214 q^{39} +0.854102 q^{40} -10.0902 q^{41} +2.38197 q^{42} +7.38197 q^{43} +0.763932 q^{45} -2.00000 q^{46} +1.09017 q^{47} -1.85410 q^{48} +7.85410 q^{49} -3.00000 q^{50} -4.23607 q^{51} -8.85410 q^{52} -4.09017 q^{53} +3.09017 q^{54} +8.61803 q^{56} -7.23607 q^{57} +2.23607 q^{59} -0.618034 q^{60} +11.0902 q^{61} +0.618034 q^{62} +7.70820 q^{63} -0.236068 q^{64} -2.09017 q^{65} -7.00000 q^{67} -6.85410 q^{68} +3.23607 q^{69} +0.909830 q^{70} -11.0902 q^{71} +4.47214 q^{72} -10.7082 q^{73} +0.145898 q^{74} +4.85410 q^{75} -11.7082 q^{76} -3.38197 q^{78} +5.00000 q^{79} -0.708204 q^{80} +1.00000 q^{81} -6.23607 q^{82} -10.3820 q^{83} -6.23607 q^{84} -1.61803 q^{85} +4.56231 q^{86} -10.3262 q^{89} +0.472136 q^{90} -21.0902 q^{91} +5.23607 q^{92} -1.00000 q^{93} +0.673762 q^{94} -2.76393 q^{95} -5.61803 q^{96} -0.944272 q^{97} +4.85410 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{2} - 2 q^{3} - q^{4} - 3 q^{5} + q^{6} - q^{7} - 4 q^{9}+O(q^{10}) $ 2 * q - q^2 - 2 * q^3 - q^4 - 3 * q^5 + q^6 - q^7 - 4 * q^9	$ 2 q - q^{2} - 2 q^{3} - q^{4} - 3 q^{5} + q^{6} - q^{7} - 4 q^{9} + 4 q^{10} + q^{12} + 2 q^{13} - 7 q^{14} + 3 q^{15} - 3 q^{16} + 4 q^{17} + 2 q^{18} + 10 q^{19} - q^{20} + q^{21} - 2 q^{23} - 3 q^{25} + 9 q^{26} + 10 q^{27} + 8 q^{28} - 4 q^{30} + 2 q^{31} + 9 q^{32} + 3 q^{34} - 6 q^{35} + 2 q^{36} - 4 q^{37} - 2 q^{39} - 5 q^{40} - 9 q^{41} + 7 q^{42} + 17 q^{43} + 6 q^{45} - 4 q^{46} - 9 q^{47} + 3 q^{48} + 9 q^{49} - 6 q^{50} - 4 q^{51} - 11 q^{52} + 3 q^{53} - 5 q^{54} + 15 q^{56} - 10 q^{57} + q^{60} + 11 q^{61} - q^{62} + 2 q^{63} + 4 q^{64} + 7 q^{65} - 14 q^{67} - 7 q^{68} + 2 q^{69} + 13 q^{70} - 11 q^{71} - 8 q^{73} + 7 q^{74} + 3 q^{75} - 10 q^{76} - 9 q^{78} + 10 q^{79} + 12 q^{80} + 2 q^{81} - 8 q^{82} - 23 q^{83} - 8 q^{84} - q^{85} - 11 q^{86} - 5 q^{89} - 8 q^{90} - 31 q^{91} + 6 q^{92} - 2 q^{93} + 17 q^{94} - 10 q^{95} - 9 q^{96} + 16 q^{97} + 3 q^{98}+O(q^{100}) $ 2 * q - q^2 - 2 * q^3 - q^4 - 3 * q^5 + q^6 - q^7 - 4 * q^9 + 4 * q^10 + q^12 + 2 * q^13 - 7 * q^14 + 3 * q^15 - 3 * q^16 + 4 * q^17 + 2 * q^18 + 10 * q^19 - q^20 + q^21 - 2 * q^23 - 3 * q^25 + 9 * q^26 + 10 * q^27 + 8 * q^28 - 4 * q^30 + 2 * q^31 + 9 * q^32 + 3 * q^34 - 6 * q^35 + 2 * q^36 - 4 * q^37 - 2 * q^39 - 5 * q^40 - 9 * q^41 + 7 * q^42 + 17 * q^43 + 6 * q^45 - 4 * q^46 - 9 * q^47 + 3 * q^48 + 9 * q^49 - 6 * q^50 - 4 * q^51 - 11 * q^52 + 3 * q^53 - 5 * q^54 + 15 * q^56 - 10 * q^57 + q^60 + 11 * q^61 - q^62 + 2 * q^63 + 4 * q^64 + 7 * q^65 - 14 * q^67 - 7 * q^68 + 2 * q^69 + 13 * q^70 - 11 * q^71 - 8 * q^73 + 7 * q^74 + 3 * q^75 - 10 * q^76 - 9 * q^78 + 10 * q^79 + 12 * q^80 + 2 * q^81 - 8 * q^82 - 23 * q^83 - 8 * q^84 - q^85 - 11 * q^86 - 5 * q^89 - 8 * q^90 - 31 * q^91 + 6 * q^92 - 2 * q^93 + 17 * q^94 - 10 * q^95 - 9 * q^96 + 16 * q^97 + 3 * q^98

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 $

$2$	0.618034	0.437016	0.218508	−	0.975835i	$-0.429881\pi$
$2$	0.618034	0.437016	0.218508	+	0.975835i	$0.429881\pi$
$3$	−1.00000	−0.577350	−0.288675	−	0.957427i	$-0.593215\pi$
$3$	−1.00000	−0.577350	−0.288675	+	0.957427i	$0.593215\pi$
$4$	−1.61803	−0.809017
$5$	−0.381966	−0.170820	−0.0854102	−	0.996346i	$-0.527220\pi$
$5$	−0.381966	−0.170820	−0.0854102	+	0.996346i	$0.527220\pi$
$6$	−0.618034	−0.252311
$7$	−3.85410	−1.45671	−0.728357	−	0.685198i	$-0.759716\pi$
$7$	−3.85410	−1.45671	−0.728357	+	0.685198i	$0.759716\pi$
$8$	−2.23607	−0.790569
$9$	−2.00000	−0.666667
$10$	−0.236068	−0.0746512
$11$	0	0
$11$	0	0
$12$	1.61803	0.467086
$13$	5.47214	1.51770	0.758849	−	0.651267i	$-0.225762\pi$
$13$	5.47214	1.51770	0.758849	+	0.651267i	$0.225762\pi$
$14$	−2.38197	−0.636607
$15$	0.381966	0.0986232
$16$	1.85410	0.463525
$17$	4.23607	1.02740	0.513699	−	0.857971i	$-0.328275\pi$
$17$	4.23607	1.02740	0.513699	+	0.857971i	$0.328275\pi$
$18$	−1.23607	−0.291344
$19$	7.23607	1.66007	0.830034	−	0.557713i	$-0.188321\pi$
$19$	7.23607	1.66007	0.830034	+	0.557713i	$0.188321\pi$
$20$	0.618034	0.138197
$21$	3.85410	0.841034
$22$	0	0
$23$	−3.23607	−0.674767	−0.337383	−	0.941367i	$-0.609542\pi$
$23$	−3.23607	−0.674767	−0.337383	+	0.941367i	$0.609542\pi$
$24$	2.23607	0.456435
$25$	−4.85410	−0.970820
$26$	3.38197	0.663258
$27$	5.00000	0.962250
$28$	6.23607	1.17851
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0.236068	0.0430999
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	5.61803	0.993137
$33$	0	0
$34$	2.61803	0.448989
$35$	1.47214	0.248836
$36$	3.23607	0.539345
$37$	0.236068	0.0388093	0.0194047	−	0.999812i	$-0.493823\pi$
$37$	0.236068	0.0388093	0.0194047	+	0.999812i	$0.493823\pi$
$38$	4.47214	0.725476
$39$	−5.47214	−0.876243
$40$	0.854102	0.135045
$41$	−10.0902	−1.57582	−0.787910	−	0.615791i	$-0.788837\pi$
$41$	−10.0902	−1.57582	−0.787910	+	0.615791i	$0.788837\pi$
$42$	2.38197	0.367545
$43$	7.38197	1.12574	0.562870	−	0.826546i	$-0.309697\pi$
$43$	7.38197	1.12574	0.562870	+	0.826546i	$0.309697\pi$
$44$	0	0
$45$	0.763932	0.113880
$46$	−2.00000	−0.294884
$47$	1.09017	0.159018	0.0795088	−	0.996834i	$-0.474665\pi$
$47$	1.09017	0.159018	0.0795088	+	0.996834i	$0.474665\pi$
$48$	−1.85410	−0.267617
$49$	7.85410	1.12201
$50$	−3.00000	−0.424264
$51$	−4.23607	−0.593168
$52$	−8.85410	−1.22784
$53$	−4.09017	−0.561828	−0.280914	−	0.959733i	$-0.590638\pi$
$53$	−4.09017	−0.561828	−0.280914	+	0.959733i	$0.590638\pi$
$54$	3.09017	0.420519
$55$	0	0
$56$	8.61803	1.15163
$57$	−7.23607	−0.958441
$58$	0	0
$59$	2.23607	0.291111	0.145556	−	0.989350i	$-0.453503\pi$
$59$	2.23607	0.291111	0.145556	+	0.989350i	$0.453503\pi$
$60$	−0.618034	−0.0797878
$61$	11.0902	1.41995	0.709975	−	0.704226i	$-0.248706\pi$
$61$	11.0902	1.41995	0.709975	+	0.704226i	$0.248706\pi$
$62$	0.618034	0.0784904
$63$	7.70820	0.971142
$64$	−0.236068	−0.0295085
$65$	−2.09017	−0.259254
$66$	0	0
$67$	−7.00000	−0.855186	−0.427593	−	0.903971i	$-0.640638\pi$
$67$	−7.00000	−0.855186	−0.427593	+	0.903971i	$0.640638\pi$
$68$	−6.85410	−0.831182
$69$	3.23607	0.389577
$70$	0.909830	0.108745
$71$	−11.0902	−1.31616	−0.658081	−	0.752948i	$-0.728631\pi$
$71$	−11.0902	−1.31616	−0.658081	+	0.752948i	$0.728631\pi$
$72$	4.47214	0.527046
$73$	−10.7082	−1.25330	−0.626650	−	0.779301i	$-0.715575\pi$
$73$	−10.7082	−1.25330	−0.626650	+	0.779301i	$0.715575\pi$
$74$	0.145898	0.0169603
$75$	4.85410	0.560503
$76$	−11.7082	−1.34302
$77$	0	0
$78$	−3.38197	−0.382932
$79$	5.00000	0.562544	0.281272	−	0.959628i	$-0.409244\pi$
$79$	5.00000	0.562544	0.281272	+	0.959628i	$0.409244\pi$
$80$	−0.708204	−0.0791796
$81$	1.00000	0.111111
$82$	−6.23607	−0.688659
$83$	−10.3820	−1.13957	−0.569784	−	0.821794i	$-0.692973\pi$
$83$	−10.3820	−1.13957	−0.569784	+	0.821794i	$0.692973\pi$
$84$	−6.23607	−0.680411
$85$	−1.61803	−0.175500
$86$	4.56231	0.491966
$87$	0	0
$88$	0	0
$89$	−10.3262	−1.09458	−0.547290	−	0.836943i	$-0.684340\pi$
$89$	−10.3262	−1.09458	−0.547290	+	0.836943i	$0.684340\pi$
$90$	0.472136	0.0497675
$91$	−21.0902	−2.21085
$92$	5.23607	0.545898
$93$	−1.00000	−0.103695
$94$	0.673762	0.0694933
$95$	−2.76393	−0.283573
$96$	−5.61803	−0.573388
$97$	−0.944272	−0.0958763	−0.0479381	−	0.998850i	$-0.515265\pi$
$97$	−0.944272	−0.0958763	−0.0479381	+	0.998850i	$0.515265\pi$
$98$	4.85410	0.490338
$99$	0	0
$100$	7.85410	0.785410
$101$	−18.1803	−1.80901	−0.904506	−	0.426461i	$-0.859760\pi$
$101$	−18.1803	−1.80901	−0.904506	+	0.426461i	$0.859760\pi$
$102$	−2.61803	−0.259224
$103$	9.00000	0.886796	0.443398	−	0.896325i	$-0.353773\pi$
$103$	9.00000	0.886796	0.443398	+	0.896325i	$0.353773\pi$
$104$	−12.2361	−1.19985
$105$	−1.47214	−0.143666
$106$	−2.52786	−0.245528
$107$	−8.32624	−0.804928	−0.402464	−	0.915436i	$-0.631846\pi$
$107$	−8.32624	−0.804928	−0.402464	+	0.915436i	$0.631846\pi$
$108$	−8.09017	−0.778477
$109$	3.29180	0.315297	0.157648	−	0.987495i	$-0.449609\pi$
$109$	3.29180	0.315297	0.157648	+	0.987495i	$0.449609\pi$
$110$	0	0
$111$	−0.236068	−0.0224066
$112$	−7.14590	−0.675224
$113$	12.9443	1.21769	0.608847	−	0.793287i	$-0.291632\pi$
$113$	12.9443	1.21769	0.608847	+	0.793287i	$0.291632\pi$
$114$	−4.47214	−0.418854
$115$	1.23607	0.115264
$116$	0	0
$117$	−10.9443	−1.01180
$118$	1.38197	0.127220
$119$	−16.3262	−1.49662
$120$	−0.854102	−0.0779685
$121$	0	0
$122$	6.85410	0.620541
$123$	10.0902	0.909800
$124$	−1.61803	−0.145304
$125$	3.76393	0.336656
$126$	4.76393	0.424405
$127$	5.29180	0.469571	0.234785	−	0.972047i	$-0.424561\pi$
$127$	5.29180	0.469571	0.234785	+	0.972047i	$0.424561\pi$
$128$	−11.3820	−1.00603
$129$	−7.38197	−0.649946
$130$	−1.29180	−0.113298
$131$	16.0902	1.40580	0.702902	−	0.711286i	$-0.251887\pi$
$131$	16.0902	1.40580	0.702902	+	0.711286i	$0.251887\pi$
$132$	0	0
$133$	−27.8885	−2.41824
$134$	−4.32624	−0.373730
$135$	−1.90983	−0.164372
$136$	−9.47214	−0.812229
$137$	9.70820	0.829428	0.414714	−	0.909952i	$-0.363882\pi$
$137$	9.70820	0.829428	0.414714	+	0.909952i	$0.363882\pi$
$138$	2.00000	0.170251
$139$	−8.41641	−0.713870	−0.356935	−	0.934129i	$-0.616178\pi$
$139$	−8.41641	−0.713870	−0.356935	+	0.934129i	$0.616178\pi$
$140$	−2.38197	−0.201313
$141$	−1.09017	−0.0918089
$142$	−6.85410	−0.575183
$143$	0	0
$144$	−3.70820	−0.309017
$145$	0	0
$146$	−6.61803	−0.547712
$147$	−7.85410	−0.647795
$148$	−0.381966	−0.0313974
$149$	−2.23607	−0.183186	−0.0915929	−	0.995797i	$-0.529196\pi$
$149$	−2.23607	−0.183186	−0.0915929	+	0.995797i	$0.529196\pi$
$150$	3.00000	0.244949
$151$	17.2705	1.40545	0.702727	−	0.711460i	$-0.251966\pi$
$151$	17.2705	1.40545	0.702727	+	0.711460i	$0.251966\pi$
$152$	−16.1803	−1.31240
$153$	−8.47214	−0.684932
$154$	0	0
$155$	−0.381966	−0.0306802
$156$	8.85410	0.708896
$157$	−9.56231	−0.763155	−0.381578	−	0.924337i	$-0.624619\pi$
$157$	−9.56231	−0.763155	−0.381578	+	0.924337i	$0.624619\pi$
$158$	3.09017	0.245841
$159$	4.09017	0.324372
$160$	−2.14590	−0.169648
$161$	12.4721	0.982942
$162$	0.618034	0.0485573
$163$	−11.0000	−0.861586	−0.430793	−	0.902451i	$-0.641766\pi$
$163$	−11.0000	−0.861586	−0.430793	+	0.902451i	$0.641766\pi$
$164$	16.3262	1.27486
$165$	0	0
$166$	−6.41641	−0.498010
$167$	−9.38197	−0.725998	−0.362999	−	0.931789i	$-0.618247\pi$
$167$	−9.38197	−0.725998	−0.362999	+	0.931789i	$0.618247\pi$
$168$	−8.61803	−0.664896
$169$	16.9443	1.30341
$170$	−1.00000	−0.0766965
$171$	−14.4721	−1.10671
$172$	−11.9443	−0.910742
$173$	−10.9098	−0.829459	−0.414730	−	0.909945i	$-0.636124\pi$
$173$	−10.9098	−0.829459	−0.414730	+	0.909945i	$0.636124\pi$
$174$	0	0
$175$	18.7082	1.41421
$176$	0	0
$177$	−2.23607	−0.168073
$178$	−6.38197	−0.478349
$179$	−17.7639	−1.32774	−0.663869	−	0.747849i	$-0.731087\pi$
$179$	−17.7639	−1.32774	−0.663869	+	0.747849i	$0.731087\pi$
$180$	−1.23607	−0.0921311
$181$	−6.09017	−0.452679	−0.226339	−	0.974049i	$-0.572676\pi$
$181$	−6.09017	−0.452679	−0.226339	+	0.974049i	$0.572676\pi$
$182$	−13.0344	−0.966177
$183$	−11.0902	−0.819809
$184$	7.23607	0.533450
$185$	−0.0901699	−0.00662943
$186$	−0.618034	−0.0453165
$187$	0	0
$188$	−1.76393	−0.128648
$189$	−19.2705	−1.40172
$190$	−1.70820	−0.123926
$191$	8.18034	0.591909	0.295954	−	0.955202i	$-0.404362\pi$
$191$	8.18034	0.591909	0.295954	+	0.955202i	$0.404362\pi$
$192$	0.236068	0.0170367
$193$	−2.81966	−0.202964	−0.101482	−	0.994837i	$-0.532358\pi$
$193$	−2.81966	−0.202964	−0.101482	+	0.994837i	$0.532358\pi$
$194$	−0.583592	−0.0418995
$195$	2.09017	0.149680
$196$	−12.7082	−0.907729
$197$	−9.70820	−0.691681	−0.345840	−	0.938293i	$-0.612406\pi$
$197$	−9.70820	−0.691681	−0.345840	+	0.938293i	$0.612406\pi$
$198$	0	0
$199$	22.8885	1.62253	0.811263	−	0.584682i	$-0.198781\pi$
$199$	22.8885	1.62253	0.811263	+	0.584682i	$0.198781\pi$
$200$	10.8541	0.767501
$201$	7.00000	0.493742
$202$	−11.2361	−0.790567
$203$	0	0
$204$	6.85410	0.479883
$205$	3.85410	0.269182
$206$	5.56231	0.387544
$207$	6.47214	0.449845
$208$	10.1459	0.703491
$209$	0	0
$210$	−0.909830	−0.0627842
$211$	18.0000	1.23917	0.619586	−	0.784929i	$-0.287301\pi$
$211$	18.0000	1.23917	0.619586	+	0.784929i	$0.287301\pi$
$212$	6.61803	0.454528
$213$	11.0902	0.759886
$214$	−5.14590	−0.351766
$215$	−2.81966	−0.192299
$216$	−11.1803	−0.760726
$217$	−3.85410	−0.261633
$218$	2.03444	0.137790
$219$	10.7082	0.723593
$220$	0	0
$221$	23.1803	1.55928
$222$	−0.145898	−0.00979203
$223$	−9.29180	−0.622225	−0.311112	−	0.950373i	$-0.600702\pi$
$223$	−9.29180	−0.622225	−0.311112	+	0.950373i	$0.600702\pi$
$224$	−21.6525	−1.44672
$225$	9.70820	0.647214
$226$	8.00000	0.532152
$227$	29.3607	1.94874	0.974368	−	0.224958i	$-0.0722246\pi$
$227$	29.3607	1.94874	0.974368	+	0.224958i	$0.0722246\pi$
$228$	11.7082	0.775395
$229$	−25.8541	−1.70849	−0.854244	−	0.519873i	$-0.825979\pi$
$229$	−25.8541	−1.70849	−0.854244	+	0.519873i	$0.825979\pi$
$230$	0.763932	0.0503722
$231$	0	0
$232$	0	0
$233$	20.4721	1.34117	0.670587	−	0.741831i	$-0.266042\pi$
$233$	20.4721	1.34117	0.670587	+	0.741831i	$0.266042\pi$
$234$	−6.76393	−0.442172
$235$	−0.416408	−0.0271635
$236$	−3.61803	−0.235514
$237$	−5.00000	−0.324785
$238$	−10.0902	−0.654049
$239$	−18.6180	−1.20430	−0.602150	−	0.798383i	$-0.705689\pi$
$239$	−18.6180	−1.20430	−0.602150	+	0.798383i	$0.705689\pi$
$240$	0.708204	0.0457144
$241$	−3.90983	−0.251854	−0.125927	−	0.992039i	$-0.540191\pi$
$241$	−3.90983	−0.251854	−0.125927	+	0.992039i	$0.540191\pi$
$242$	0	0
$243$	−16.0000	−1.02640
$244$	−17.9443	−1.14876
$245$	−3.00000	−0.191663
$246$	6.23607	0.397597
$247$	39.5967	2.51948
$248$	−2.23607	−0.141990
$249$	10.3820	0.657930
$250$	2.32624	0.147124
$251$	15.0902	0.952483	0.476242	−	0.879315i	$-0.341999\pi$
$251$	15.0902	0.952483	0.476242	+	0.879315i	$0.341999\pi$
$252$	−12.4721	−0.785671
$253$	0	0
$254$	3.27051	0.205210
$255$	1.61803	0.101325
$256$	−6.56231	−0.410144
$257$	27.0689	1.68851	0.844255	−	0.535941i	$-0.180043\pi$
$257$	27.0689	1.68851	0.844255	+	0.535941i	$0.180043\pi$
$258$	−4.56231	−0.284037
$259$	−0.909830	−0.0565341
$260$	3.38197	0.209741
$261$	0	0
$262$	9.94427	0.614359
$263$	−20.3820	−1.25681	−0.628403	−	0.777888i	$-0.716291\pi$
$263$	−20.3820	−1.25681	−0.628403	+	0.777888i	$0.716291\pi$
$264$	0	0
$265$	1.56231	0.0959717
$266$	−17.2361	−1.05681
$267$	10.3262	0.631955
$268$	11.3262	0.691860
$269$	−13.7426	−0.837904	−0.418952	−	0.908008i	$-0.637602\pi$
$269$	−13.7426	−0.837904	−0.418952	+	0.908008i	$0.637602\pi$
$270$	−1.18034	−0.0718332
$271$	22.2705	1.35284	0.676419	−	0.736517i	$-0.263531\pi$
$271$	22.2705	1.35284	0.676419	+	0.736517i	$0.263531\pi$
$272$	7.85410	0.476225
$273$	21.0902	1.27644
$274$	6.00000	0.362473
$275$	0	0
$276$	−5.23607	−0.315174
$277$	13.7082	0.823646	0.411823	−	0.911264i	$-0.364892\pi$
$277$	13.7082	0.823646	0.411823	+	0.911264i	$0.364892\pi$
$278$	−5.20163	−0.311973
$279$	−2.00000	−0.119737
$280$	−3.29180	−0.196722
$281$	−12.0000	−0.715860	−0.357930	−	0.933748i	$-0.616517\pi$
$281$	−12.0000	−0.715860	−0.357930	+	0.933748i	$0.616517\pi$
$282$	−0.673762	−0.0401219
$283$	−22.2148	−1.32053	−0.660266	−	0.751032i	$-0.729556\pi$
$283$	−22.2148	−1.32053	−0.660266	+	0.751032i	$0.729556\pi$
$284$	17.9443	1.06480
$285$	2.76393	0.163721
$286$	0	0
$287$	38.8885	2.29552
$288$	−11.2361	−0.662092
$289$	0.944272	0.0555454
$290$	0	0
$291$	0.944272	0.0553542
$292$	17.3262	1.01394
$293$	−0.0557281	−0.00325567	−0.00162783	−	0.999999i	$-0.500518\pi$
$293$	−0.0557281	−0.00325567	−0.00162783	+	0.999999i	$0.500518\pi$
$294$	−4.85410	−0.283097
$295$	−0.854102	−0.0497277
$296$	−0.527864	−0.0306815
$297$	0	0
$298$	−1.38197	−0.0800551
$299$	−17.7082	−1.02409
$300$	−7.85410	−0.453457
$301$	−28.4508	−1.63988
$302$	10.6738	0.614206
$303$	18.1803	1.04443
$304$	13.4164	0.769484
$305$	−4.23607	−0.242557
$306$	−5.23607	−0.299326
$307$	−25.6869	−1.46603	−0.733015	−	0.680213i	$-0.761887\pi$
$307$	−25.6869	−1.46603	−0.733015	+	0.680213i	$0.761887\pi$
$308$	0	0
$309$	−9.00000	−0.511992
$310$	−0.236068	−0.0134078
$311$	−28.4508	−1.61330	−0.806650	−	0.591030i	$-0.798722\pi$
$311$	−28.4508	−1.61330	−0.806650	+	0.591030i	$0.798722\pi$
$312$	12.2361	0.692731
$313$	−11.1246	−0.628800	−0.314400	−	0.949291i	$-0.601803\pi$
$313$	−11.1246	−0.628800	−0.314400	+	0.949291i	$0.601803\pi$
$314$	−5.90983	−0.333511
$315$	−2.94427	−0.165891
$316$	−8.09017	−0.455108
$317$	−17.1246	−0.961814	−0.480907	−	0.876772i	$-0.659693\pi$
$317$	−17.1246	−0.961814	−0.480907	+	0.876772i	$0.659693\pi$
$318$	2.52786	0.141756
$319$	0	0
$320$	0.0901699	0.00504065
$321$	8.32624	0.464725
$322$	7.70820	0.429561
$323$	30.6525	1.70555
$324$	−1.61803	−0.0898908
$325$	−26.5623	−1.47341
$326$	−6.79837	−0.376527
$327$	−3.29180	−0.182037
$328$	22.5623	1.24579
$329$	−4.20163	−0.231643
$330$	0	0
$331$	−11.8197	−0.649667	−0.324834	−	0.945771i	$-0.605308\pi$
$331$	−11.8197	−0.649667	−0.324834	+	0.945771i	$0.605308\pi$
$332$	16.7984	0.921931
$333$	−0.472136	−0.0258729
$334$	−5.79837	−0.317273
$335$	2.67376	0.146083
$336$	7.14590	0.389841
$337$	−28.8541	−1.57178	−0.785892	−	0.618364i	$-0.787796\pi$
$337$	−28.8541	−1.57178	−0.785892	+	0.618364i	$0.787796\pi$
$338$	10.4721	0.569609
$339$	−12.9443	−0.703036
$340$	2.61803	0.141983
$341$	0	0
$342$	−8.94427	−0.483651
$343$	−3.29180	−0.177740
$344$	−16.5066	−0.889975
$345$	−1.23607	−0.0665477
$346$	−6.74265	−0.362487
$347$	4.88854	0.262431	0.131215	−	0.991354i	$-0.458112\pi$
$347$	4.88854	0.262431	0.131215	+	0.991354i	$0.458112\pi$
$348$	0	0
$349$	−25.9787	−1.39061	−0.695304	−	0.718715i	$-0.744730\pi$
$349$	−25.9787	−1.39061	−0.695304	+	0.718715i	$0.744730\pi$
$350$	11.5623	0.618031
$351$	27.3607	1.46041
$352$	0	0
$353$	−29.7426	−1.58304	−0.791521	−	0.611142i	$-0.790710\pi$
$353$	−29.7426	−1.58304	−0.791521	+	0.611142i	$0.790710\pi$
$354$	−1.38197	−0.0734507
$355$	4.23607	0.224827
$356$	16.7082	0.885533
$357$	16.3262	0.864076
$358$	−10.9787	−0.580243
$359$	20.1246	1.06214	0.531068	−	0.847329i	$-0.321791\pi$
$359$	20.1246	1.06214	0.531068	+	0.847329i	$0.321791\pi$
$360$	−1.70820	−0.0900303
$361$	33.3607	1.75583
$362$	−3.76393	−0.197828
$363$	0	0
$364$	34.1246	1.78862
$365$	4.09017	0.214089
$366$	−6.85410	−0.358270
$367$	7.14590	0.373013	0.186506	−	0.982454i	$-0.440283\pi$
$367$	7.14590	0.373013	0.186506	+	0.982454i	$0.440283\pi$
$368$	−6.00000	−0.312772
$369$	20.1803	1.05055
$370$	−0.0557281	−0.00289717
$371$	15.7639	0.818423
$372$	1.61803	0.0838912
$373$	−17.0902	−0.884895	−0.442448	−	0.896794i	$-0.645890\pi$
$373$	−17.0902	−0.884895	−0.442448	+	0.896794i	$0.645890\pi$
$374$	0	0
$375$	−3.76393	−0.194369
$376$	−2.43769	−0.125714
$377$	0	0
$378$	−11.9098	−0.612576
$379$	14.2705	0.733027	0.366513	−	0.930413i	$-0.380551\pi$
$379$	14.2705	0.733027	0.366513	+	0.930413i	$0.380551\pi$
$380$	4.47214	0.229416
$381$	−5.29180	−0.271107
$382$	5.05573	0.258674
$383$	−6.52786	−0.333558	−0.166779	−	0.985994i	$-0.553337\pi$
$383$	−6.52786	−0.333558	−0.166779	+	0.985994i	$0.553337\pi$
$384$	11.3820	0.580834
$385$	0	0
$386$	−1.74265	−0.0886983
$387$	−14.7639	−0.750493
$388$	1.52786	0.0775655
$389$	−32.2361	−1.63443	−0.817217	−	0.576330i	$-0.804484\pi$
$389$	−32.2361	−1.63443	−0.817217	+	0.576330i	$0.804484\pi$
$390$	1.29180	0.0654126
$391$	−13.7082	−0.693254
$392$	−17.5623	−0.887030
$393$	−16.0902	−0.811642
$394$	−6.00000	−0.302276
$395$	−1.90983	−0.0960940
$396$	0	0
$397$	13.0000	0.652451	0.326226	−	0.945292i	$-0.394223\pi$
$397$	13.0000	0.652451	0.326226	+	0.945292i	$0.394223\pi$
$398$	14.1459	0.709070
$399$	27.8885	1.39617
$400$	−9.00000	−0.450000
$401$	−29.1803	−1.45720	−0.728598	−	0.684941i	$-0.759828\pi$
$401$	−29.1803	−1.45720	−0.728598	+	0.684941i	$0.759828\pi$
$402$	4.32624	0.215773
$403$	5.47214	0.272587
$404$	29.4164	1.46352
$405$	−0.381966	−0.0189800
$406$	0	0
$407$	0	0
$408$	9.47214	0.468941
$409$	−30.1246	−1.48957	−0.744783	−	0.667307i	$-0.767447\pi$
$409$	−30.1246	−1.48957	−0.744783	+	0.667307i	$0.767447\pi$
$410$	2.38197	0.117637
$411$	−9.70820	−0.478870
$412$	−14.5623	−0.717433
$413$	−8.61803	−0.424066
$414$	4.00000	0.196589
$415$	3.96556	0.194662
$416$	30.7426	1.50728
$417$	8.41641	0.412153
$418$	0	0
$419$	−14.4721	−0.707010	−0.353505	−	0.935433i	$-0.615010\pi$
$419$	−14.4721	−0.707010	−0.353505	+	0.935433i	$0.615010\pi$
$420$	2.38197	0.116228
$421$	−3.00000	−0.146211	−0.0731055	−	0.997324i	$-0.523291\pi$
$421$	−3.00000	−0.146211	−0.0731055	+	0.997324i	$0.523291\pi$
$422$	11.1246	0.541538
$423$	−2.18034	−0.106012
$424$	9.14590	0.444164
$425$	−20.5623	−0.997418
$426$	6.85410	0.332082
$427$	−42.7426	−2.06846
$428$	13.4721	0.651200
$429$	0	0
$430$	−1.74265	−0.0840378
$431$	26.0902	1.25672	0.628360	−	0.777923i	$-0.283727\pi$
$431$	26.0902	1.25672	0.628360	+	0.777923i	$0.283727\pi$
$432$	9.27051	0.446028
$433$	−8.43769	−0.405490	−0.202745	−	0.979232i	$-0.564986\pi$
$433$	−8.43769	−0.405490	−0.202745	+	0.979232i	$0.564986\pi$
$434$	−2.38197	−0.114338
$435$	0	0
$436$	−5.32624	−0.255081
$437$	−23.4164	−1.12016
$438$	6.61803	0.316222
$439$	−6.18034	−0.294972	−0.147486	−	0.989064i	$-0.547118\pi$
$439$	−6.18034	−0.294972	−0.147486	+	0.989064i	$0.547118\pi$
$440$	0	0
$441$	−15.7082	−0.748010
$442$	14.3262	0.681430
$443$	20.8328	0.989797	0.494898	−	0.868951i	$-0.335205\pi$
$443$	20.8328	0.989797	0.494898	+	0.868951i	$0.335205\pi$
$444$	0.381966	0.0181273
$445$	3.94427	0.186976
$446$	−5.74265	−0.271922
$447$	2.23607	0.105762
$448$	0.909830	0.0429854
$449$	−19.7984	−0.934343	−0.467172	−	0.884167i	$-0.654727\pi$
$449$	−19.7984	−0.934343	−0.467172	+	0.884167i	$0.654727\pi$
$450$	6.00000	0.282843
$451$	0	0
$452$	−20.9443	−0.985136
$453$	−17.2705	−0.811439
$454$	18.1459	0.851629
$455$	8.05573	0.377658
$456$	16.1803	0.757714
$457$	28.5066	1.33348	0.666741	−	0.745290i	$-0.267689\pi$
$457$	28.5066	1.33348	0.666741	+	0.745290i	$0.267689\pi$
$458$	−15.9787	−0.746636
$459$	21.1803	0.988614
$460$	−2.00000	−0.0932505
$461$	14.1803	0.660444	0.330222	−	0.943903i	$-0.392876\pi$
$461$	14.1803	0.660444	0.330222	+	0.943903i	$0.392876\pi$
$462$	0	0
$463$	−0.472136	−0.0219420	−0.0109710	−	0.999940i	$-0.503492\pi$
$463$	−0.472136	−0.0219420	−0.0109710	+	0.999940i	$0.503492\pi$
$464$	0	0
$465$	0.381966	0.0177132
$466$	12.6525	0.586115
$467$	−7.85410	−0.363444	−0.181722	−	0.983350i	$-0.558167\pi$
$467$	−7.85410	−0.363444	−0.181722	+	0.983350i	$0.558167\pi$
$468$	17.7082	0.818562
$469$	26.9787	1.24576
$470$	−0.257354	−0.0118709
$471$	9.56231	0.440608
$472$	−5.00000	−0.230144
$473$	0	0
$474$	−3.09017	−0.141936
$475$	−35.1246	−1.61163
$476$	26.4164	1.21079
$477$	8.18034	0.374552
$478$	−11.5066	−0.526299
$479$	−6.38197	−0.291599	−0.145800	−	0.989314i	$-0.546576\pi$
$479$	−6.38197	−0.291599	−0.145800	+	0.989314i	$0.546576\pi$
$480$	2.14590	0.0979464
$481$	1.29180	0.0589008
$482$	−2.41641	−0.110064
$483$	−12.4721	−0.567502
$484$	0	0
$485$	0.360680	0.0163776
$486$	−9.88854	−0.448553
$487$	−14.0344	−0.635961	−0.317981	−	0.948097i	$-0.603005\pi$
$487$	−14.0344	−0.635961	−0.317981	+	0.948097i	$0.603005\pi$
$488$	−24.7984	−1.12257
$489$	11.0000	0.497437
$490$	−1.85410	−0.0837598
$491$	18.0000	0.812329	0.406164	−	0.913800i	$-0.366866\pi$
$491$	18.0000	0.812329	0.406164	+	0.913800i	$0.366866\pi$
$492$	−16.3262	−0.736044
$493$	0	0
$494$	24.4721	1.10105
$495$	0	0
$496$	1.85410	0.0832516
$497$	42.7426	1.91727
$498$	6.41641	0.287526
$499$	−39.3951	−1.76357	−0.881784	−	0.471654i	$-0.843657\pi$
$499$	−39.3951	−1.76357	−0.881784	+	0.471654i	$0.843657\pi$
$500$	−6.09017	−0.272361
$501$	9.38197	0.419155
$502$	9.32624	0.416250
$503$	−6.76393	−0.301589	−0.150794	−	0.988565i	$-0.548183\pi$
$503$	−6.76393	−0.301589	−0.150794	+	0.988565i	$0.548183\pi$
$504$	−17.2361	−0.767755
$505$	6.94427	0.309016
$506$	0	0
$507$	−16.9443	−0.752522
$508$	−8.56231	−0.379891
$509$	−3.81966	−0.169303	−0.0846517	−	0.996411i	$-0.526978\pi$
$509$	−3.81966	−0.169303	−0.0846517	+	0.996411i	$0.526978\pi$
$510$	1.00000	0.0442807
$511$	41.2705	1.82570
$512$	18.7082	0.826794
$513$	36.1803	1.59740
$514$	16.7295	0.737906
$515$	−3.43769	−0.151483
$516$	11.9443	0.525817
$517$	0	0
$518$	−0.562306	−0.0247063
$519$	10.9098	0.478888
$520$	4.67376	0.204958
$521$	17.4508	0.764536	0.382268	−	0.924052i	$-0.375143\pi$
$521$	17.4508	0.764536	0.382268	+	0.924052i	$0.375143\pi$
$522$	0	0
$523$	−18.2705	−0.798914	−0.399457	−	0.916752i	$-0.630801\pi$
$523$	−18.2705	−0.798914	−0.399457	+	0.916752i	$0.630801\pi$
$524$	−26.0344	−1.13732
$525$	−18.7082	−0.816493
$526$	−12.5967	−0.549244
$527$	4.23607	0.184526
$528$	0	0
$529$	−12.5279	−0.544690
$530$	0.965558	0.0419412
$531$	−4.47214	−0.194074
$532$	45.1246	1.95640
$533$	−55.2148	−2.39162
$534$	6.38197	0.276175
$535$	3.18034	0.137498
$536$	15.6525	0.676084
$537$	17.7639	0.766570
$538$	−8.49342	−0.366177
$539$	0	0
$540$	3.09017	0.132980
$541$	−39.3607	−1.69225	−0.846124	−	0.532986i	$-0.821070\pi$
$541$	−39.3607	−1.69225	−0.846124	+	0.532986i	$0.821070\pi$
$542$	13.7639	0.591212
$543$	6.09017	0.261354
$544$	23.7984	1.02035
$545$	−1.25735	−0.0538591
$546$	13.0344	0.557823
$547$	−4.38197	−0.187359	−0.0936797	−	0.995602i	$-0.529863\pi$
$547$	−4.38197	−0.187359	−0.0936797	+	0.995602i	$0.529863\pi$
$548$	−15.7082	−0.671021
$549$	−22.1803	−0.946634
$550$	0	0
$551$	0	0
$552$	−7.23607	−0.307988
$553$	−19.2705	−0.819465
$554$	8.47214	0.359947
$555$	0.0901699	0.00382750
$556$	13.6180	0.577533
$557$	−1.41641	−0.0600151	−0.0300076	−	0.999550i	$-0.509553\pi$
$557$	−1.41641	−0.0600151	−0.0300076	+	0.999550i	$0.509553\pi$
$558$	−1.23607	−0.0523269
$559$	40.3951	1.70853
$560$	2.72949	0.115342
$561$	0	0
$562$	−7.41641	−0.312842
$563$	41.1246	1.73320	0.866598	−	0.499007i	$-0.166302\pi$
$563$	41.1246	1.73320	0.866598	+	0.499007i	$0.166302\pi$
$564$	1.76393	0.0742749
$565$	−4.94427	−0.208007
$566$	−13.7295	−0.577094
$567$	−3.85410	−0.161857
$568$	24.7984	1.04052
$569$	−28.4164	−1.19128	−0.595639	−	0.803252i	$-0.703101\pi$
$569$	−28.4164	−1.19128	−0.595639	+	0.803252i	$0.703101\pi$
$570$	1.70820	0.0715488
$571$	12.2705	0.513505	0.256752	−	0.966477i	$-0.417348\pi$
$571$	12.2705	0.513505	0.256752	+	0.966477i	$0.417348\pi$
$572$	0	0
$573$	−8.18034	−0.341739
$574$	24.0344	1.00318
$575$	15.7082	0.655077
$576$	0.472136	0.0196723
$577$	−24.0344	−1.00057	−0.500283	−	0.865862i	$-0.666771\pi$
$577$	−24.0344	−1.00057	−0.500283	+	0.865862i	$0.666771\pi$
$578$	0.583592	0.0242742
$579$	2.81966	0.117181
$580$	0	0
$581$	40.0132	1.66003
$582$	0.583592	0.0241907
$583$	0	0
$584$	23.9443	0.990821
$585$	4.18034	0.172836
$586$	−0.0344419	−0.00142278
$587$	−18.9098	−0.780492	−0.390246	−	0.920711i	$-0.627610\pi$
$587$	−18.9098	−0.780492	−0.390246	+	0.920711i	$0.627610\pi$
$588$	12.7082	0.524077
$589$	7.23607	0.298157
$590$	−0.527864	−0.0217318
$591$	9.70820	0.399342
$592$	0.437694	0.0179891
$593$	−25.5066	−1.04743	−0.523715	−	0.851894i	$-0.675454\pi$
$593$	−25.5066	−1.04743	−0.523715	+	0.851894i	$0.675454\pi$
$594$	0	0
$595$	6.23607	0.255654
$596$	3.61803	0.148200
$597$	−22.8885	−0.936766
$598$	−10.9443	−0.447545
$599$	−15.6525	−0.639543	−0.319771	−	0.947495i	$-0.603606\pi$
$599$	−15.6525	−0.639543	−0.319771	+	0.947495i	$0.603606\pi$
$600$	−10.8541	−0.443117
$601$	18.0000	0.734235	0.367118	−	0.930175i	$-0.380345\pi$
$601$	18.0000	0.734235	0.367118	+	0.930175i	$0.380345\pi$
$602$	−17.5836	−0.716654
$603$	14.0000	0.570124
$604$	−27.9443	−1.13704
$605$	0	0
$606$	11.2361	0.456434
$607$	−33.1246	−1.34449	−0.672243	−	0.740330i	$-0.734669\pi$
$607$	−33.1246	−1.34449	−0.672243	+	0.740330i	$0.734669\pi$
$608$	40.6525	1.64868
$609$	0	0
$610$	−2.61803	−0.106001
$611$	5.96556	0.241341
$612$	13.7082	0.554121
$613$	28.2361	1.14044	0.570222	−	0.821491i	$-0.306857\pi$
$613$	28.2361	1.14044	0.570222	+	0.821491i	$0.306857\pi$
$614$	−15.8754	−0.640679
$615$	−3.85410	−0.155412
$616$	0	0
$617$	−33.3050	−1.34081	−0.670403	−	0.741997i	$-0.733879\pi$
$617$	−33.3050	−1.34081	−0.670403	+	0.741997i	$0.733879\pi$
$618$	−5.56231	−0.223749
$619$	−13.6180	−0.547355	−0.273677	−	0.961822i	$-0.588240\pi$
$619$	−13.6180	−0.547355	−0.273677	+	0.961822i	$0.588240\pi$
$620$	0.618034	0.0248208
$621$	−16.1803	−0.649295
$622$	−17.5836	−0.705038
$623$	39.7984	1.59449
$624$	−10.1459	−0.406161
$625$	22.8328	0.913313
$626$	−6.87539	−0.274796
$627$	0	0
$628$	15.4721	0.617405
$629$	1.00000	0.0398726
$630$	−1.81966	−0.0724970
$631$	−27.2705	−1.08562	−0.542811	−	0.839855i	$-0.682640\pi$
$631$	−27.2705	−1.08562	−0.542811	+	0.839855i	$0.682640\pi$
$632$	−11.1803	−0.444730
$633$	−18.0000	−0.715436
$634$	−10.5836	−0.420328
$635$	−2.02129	−0.0802123
$636$	−6.61803	−0.262422
$637$	42.9787	1.70288
$638$	0	0
$639$	22.1803	0.877441
$640$	4.34752	0.171851
$641$	13.9098	0.549405	0.274703	−	0.961529i	$-0.411421\pi$
$641$	13.9098	0.549405	0.274703	+	0.961529i	$0.411421\pi$
$642$	5.14590	0.203092
$643$	−28.5623	−1.12639	−0.563194	−	0.826325i	$-0.690428\pi$
$643$	−28.5623	−1.12639	−0.563194	+	0.826325i	$0.690428\pi$
$644$	−20.1803	−0.795217
$645$	2.81966	0.111024
$646$	18.9443	0.745352
$647$	34.5066	1.35659	0.678297	−	0.734788i	$-0.262718\pi$
$647$	34.5066	1.35659	0.678297	+	0.734788i	$0.262718\pi$
$648$	−2.23607	−0.0878410
$649$	0	0
$650$	−16.4164	−0.643904
$651$	3.85410	0.151054
$652$	17.7984	0.697038
$653$	19.4508	0.761171	0.380585	−	0.924746i	$-0.375723\pi$
$653$	19.4508	0.761171	0.380585	+	0.924746i	$0.375723\pi$
$654$	−2.03444	−0.0795530
$655$	−6.14590	−0.240140
$656$	−18.7082	−0.730433
$657$	21.4164	0.835534
$658$	−2.59675	−0.101232
$659$	5.12461	0.199627	0.0998133	−	0.995006i	$-0.468175\pi$
$659$	5.12461	0.199627	0.0998133	+	0.995006i	$0.468175\pi$
$660$	0	0
$661$	20.5410	0.798953	0.399477	−	0.916743i	$-0.369192\pi$
$661$	20.5410	0.798953	0.399477	+	0.916743i	$0.369192\pi$
$662$	−7.30495	−0.283915
$663$	−23.1803	−0.900250
$664$	23.2148	0.900908
$665$	10.6525	0.413085
$666$	−0.291796	−0.0113069
$667$	0	0
$668$	15.1803	0.587345
$669$	9.29180	0.359242
$670$	1.65248	0.0638407
$671$	0	0
$672$	21.6525	0.835262
$673$	3.36068	0.129545	0.0647723	−	0.997900i	$-0.479368\pi$
$673$	3.36068	0.129545	0.0647723	+	0.997900i	$0.479368\pi$
$674$	−17.8328	−0.686894
$675$	−24.2705	−0.934172
$676$	−27.4164	−1.05448
$677$	−1.74265	−0.0669753	−0.0334877	−	0.999439i	$-0.510661\pi$
$677$	−1.74265	−0.0669753	−0.0334877	+	0.999439i	$0.510661\pi$
$678$	−8.00000	−0.307238
$679$	3.63932	0.139664
$680$	3.61803	0.138745
$681$	−29.3607	−1.12510
$682$	0	0
$683$	31.3607	1.19998	0.599992	−	0.800006i	$-0.295171\pi$
$683$	31.3607	1.19998	0.599992	+	0.800006i	$0.295171\pi$
$684$	23.4164	0.895349
$685$	−3.70820	−0.141683
$686$	−2.03444	−0.0776754
$687$	25.8541	0.986396
$688$	13.6869	0.521809
$689$	−22.3820	−0.852685
$690$	−0.763932	−0.0290824
$691$	−34.1803	−1.30028	−0.650141	−	0.759814i	$-0.725290\pi$
$691$	−34.1803	−1.30028	−0.650141	+	0.759814i	$0.725290\pi$
$692$	17.6525	0.671046
$693$	0	0
$694$	3.02129	0.114686
$695$	3.21478	0.121944
$696$	0	0
$697$	−42.7426	−1.61899
$698$	−16.0557	−0.607718
$699$	−20.4721	−0.774327
$700$	−30.2705	−1.14412
$701$	31.0902	1.17426	0.587130	−	0.809493i	$-0.300258\pi$
$701$	31.0902	1.17426	0.587130	+	0.809493i	$0.300258\pi$
$702$	16.9098	0.638220
$703$	1.70820	0.0644261
$704$	0	0
$705$	0.416408	0.0156828
$706$	−18.3820	−0.691814
$707$	70.0689	2.63521
$708$	3.61803	0.135974
$709$	33.5410	1.25966	0.629830	−	0.776733i	$-0.283125\pi$
$709$	33.5410	1.25966	0.629830	+	0.776733i	$0.283125\pi$
$710$	2.61803	0.0982531
$711$	−10.0000	−0.375029
$712$	23.0902	0.865341
$713$	−3.23607	−0.121192
$714$	10.0902	0.377615
$715$	0	0
$716$	28.7426	1.07416
$717$	18.6180	0.695303
$718$	12.4377	0.464171
$719$	22.8885	0.853599	0.426799	−	0.904346i	$-0.359641\pi$
$719$	22.8885	0.853599	0.426799	+	0.904346i	$0.359641\pi$
$720$	1.41641	0.0527864
$721$	−34.6869	−1.29181
$722$	20.6180	0.767324
$723$	3.90983	0.145408
$724$	9.85410	0.366225
$725$	0	0
$726$	0	0
$727$	11.2918	0.418790	0.209395	−	0.977831i	$-0.432851\pi$
$727$	11.2918	0.418790	0.209395	+	0.977831i	$0.432851\pi$
$728$	47.1591	1.74783
$729$	13.0000	0.481481
$730$	2.52786	0.0935604
$731$	31.2705	1.15658
$732$	17.9443	0.663239
$733$	27.8328	1.02803	0.514014	−	0.857782i	$-0.328158\pi$
$733$	27.8328	1.02803	0.514014	+	0.857782i	$0.328158\pi$
$734$	4.41641	0.163013
$735$	3.00000	0.110657
$736$	−18.1803	−0.670136
$737$	0	0
$738$	12.4721	0.459106
$739$	28.4164	1.04531	0.522657	−	0.852543i	$-0.324941\pi$
$739$	28.4164	1.04531	0.522657	+	0.852543i	$0.324941\pi$
$740$	0.145898	0.00536332
$741$	−39.5967	−1.45462
$742$	9.74265	0.357664
$743$	−4.85410	−0.178080	−0.0890399	−	0.996028i	$-0.528380\pi$
$743$	−4.85410	−0.178080	−0.0890399	+	0.996028i	$0.528380\pi$
$744$	2.23607	0.0819782
$745$	0.854102	0.0312919
$746$	−10.5623	−0.386713
$747$	20.7639	0.759713
$748$	0	0
$749$	32.0902	1.17255
$750$	−2.32624	−0.0849422
$751$	40.5410	1.47936	0.739681	−	0.672957i	$-0.234976\pi$
$751$	40.5410	1.47936	0.739681	+	0.672957i	$0.234976\pi$
$752$	2.02129	0.0737087
$753$	−15.0902	−0.549916
$754$	0	0
$755$	−6.59675	−0.240080
$756$	31.1803	1.13402
$757$	19.5836	0.711778	0.355889	−	0.934528i	$-0.384178\pi$
$757$	19.5836	0.711778	0.355889	+	0.934528i	$0.384178\pi$
$758$	8.81966	0.320344
$759$	0	0
$760$	6.18034	0.224184
$761$	19.9098	0.721731	0.360865	−	0.932618i	$-0.382481\pi$
$761$	19.9098	0.721731	0.360865	+	0.932618i	$0.382481\pi$
$762$	−3.27051	−0.118478
$763$	−12.6869	−0.459297
$764$	−13.2361	−0.478864
$765$	3.23607	0.117000
$766$	−4.03444	−0.145770
$767$	12.2361	0.441819
$768$	6.56231	0.236797
$769$	−45.0000	−1.62274	−0.811371	−	0.584532i	$-0.801278\pi$
$769$	−45.0000	−1.62274	−0.811371	+	0.584532i	$0.801278\pi$
$770$	0	0
$771$	−27.0689	−0.974862
$772$	4.56231	0.164201
$773$	30.0557	1.08103	0.540515	−	0.841335i	$-0.318230\pi$
$773$	30.0557	1.08103	0.540515	+	0.841335i	$0.318230\pi$
$774$	−9.12461	−0.327977
$775$	−4.85410	−0.174364
$776$	2.11146	0.0757969
$777$	0.909830	0.0326400
$778$	−19.9230	−0.714274
$779$	−73.0132	−2.61597
$780$	−3.38197	−0.121094
$781$	0	0
$782$	−8.47214	−0.302963
$783$	0	0
$784$	14.5623	0.520082
$785$	3.65248	0.130362
$786$	−9.94427	−0.354700
$787$	17.1246	0.610426	0.305213	−	0.952284i	$-0.401272\pi$
$787$	17.1246	0.610426	0.305213	+	0.952284i	$0.401272\pi$
$788$	15.7082	0.559582
$789$	20.3820	0.725617
$790$	−1.18034	−0.0419946
$791$	−49.8885	−1.77383
$792$	0	0
$793$	60.6869	2.15506
$794$	8.03444	0.285132
$795$	−1.56231	−0.0554093
$796$	−37.0344	−1.31265
$797$	50.4853	1.78828	0.894140	−	0.447787i	$-0.147788\pi$
$797$	50.4853	1.78828	0.894140	+	0.447787i	$0.147788\pi$
$798$	17.2361	0.610150
$799$	4.61803	0.163374
$800$	−27.2705	−0.964158
$801$	20.6525	0.729719
$802$	−18.0344	−0.636818
$803$	0	0
$804$	−11.3262	−0.399446
$805$	−4.76393	−0.167907
$806$	3.38197	0.119125
$807$	13.7426	0.483764
$808$	40.6525	1.43015
$809$	24.5967	0.864776	0.432388	−	0.901688i	$-0.357671\pi$
$809$	24.5967	0.864776	0.432388	+	0.901688i	$0.357671\pi$
$810$	−0.236068	−0.00829458
$811$	5.36068	0.188239	0.0941195	−	0.995561i	$-0.469996\pi$
$811$	5.36068	0.188239	0.0941195	+	0.995561i	$0.469996\pi$
$812$	0	0
$813$	−22.2705	−0.781061
$814$	0	0
$815$	4.20163	0.147177
$816$	−7.85410	−0.274949
$817$	53.4164	1.86880
$818$	−18.6180	−0.650964
$819$	42.1803	1.47390
$820$	−6.23607	−0.217773
$821$	−45.5410	−1.58939	−0.794696	−	0.607007i	$-0.792370\pi$
$821$	−45.5410	−1.58939	−0.794696	+	0.607007i	$0.792370\pi$
$822$	−6.00000	−0.209274
$823$	−36.4508	−1.27060	−0.635298	−	0.772267i	$-0.719123\pi$
$823$	−36.4508	−1.27060	−0.635298	+	0.772267i	$0.719123\pi$
$824$	−20.1246	−0.701074
$825$	0	0
$826$	−5.32624	−0.185324
$827$	11.3475	0.394592	0.197296	−	0.980344i	$-0.436784\pi$
$827$	11.3475	0.394592	0.197296	+	0.980344i	$0.436784\pi$
$828$	−10.4721	−0.363932
$829$	30.1246	1.04627	0.523136	−	0.852250i	$-0.324762\pi$
$829$	30.1246	1.04627	0.523136	+	0.852250i	$0.324762\pi$
$830$	2.45085	0.0850702
$831$	−13.7082	−0.475532
$832$	−1.29180	−0.0447850
$833$	33.2705	1.15275
$834$	5.20163	0.180118
$835$	3.58359	0.124015
$836$	0	0
$837$	5.00000	0.172825
$838$	−8.94427	−0.308975
$839$	−28.4164	−0.981043	−0.490522	−	0.871429i	$-0.663194\pi$
$839$	−28.4164	−0.981043	−0.490522	+	0.871429i	$0.663194\pi$
$840$	3.29180	0.113578
$841$	−29.0000	−1.00000
$842$	−1.85410	−0.0638966
$843$	12.0000	0.413302
$844$	−29.1246	−1.00251
$845$	−6.47214	−0.222648
$846$	−1.34752	−0.0463288
$847$	0	0
$848$	−7.58359	−0.260422
$849$	22.2148	0.762409
$850$	−12.7082	−0.435888
$851$	−0.763932	−0.0261873
$852$	−17.9443	−0.614761
$853$	4.94427	0.169289	0.0846443	−	0.996411i	$-0.473025\pi$
$853$	4.94427	0.169289	0.0846443	+	0.996411i	$0.473025\pi$
$854$	−26.4164	−0.903951
$855$	5.52786	0.189049
$856$	18.6180	0.636351
$857$	−43.2492	−1.47737	−0.738683	−	0.674053i	$-0.764552\pi$
$857$	−43.2492	−1.47737	−0.738683	+	0.674053i	$0.764552\pi$
$858$	0	0
$859$	−33.9443	−1.15816	−0.579082	−	0.815269i	$-0.696589\pi$
$859$	−33.9443	−1.15816	−0.579082	+	0.815269i	$0.696589\pi$
$860$	4.56231	0.155573
$861$	−38.8885	−1.32532
$862$	16.1246	0.549206
$863$	16.0344	0.545819	0.272909	−	0.962040i	$-0.412014\pi$
$863$	16.0344	0.545819	0.272909	+	0.962040i	$0.412014\pi$
$864$	28.0902	0.955647
$865$	4.16718	0.141689
$866$	−5.21478	−0.177205
$867$	−0.944272	−0.0320692
$868$	6.23607	0.211666
$869$	0	0
$870$	0	0
$871$	−38.3050	−1.29791
$872$	−7.36068	−0.249264
$873$	1.88854	0.0639175
$874$	−14.4721	−0.489527
$875$	−14.5066	−0.490412
$876$	−17.3262	−0.585399
$877$	−14.0557	−0.474628	−0.237314	−	0.971433i	$-0.576267\pi$
$877$	−14.0557	−0.474628	−0.237314	+	0.971433i	$0.576267\pi$
$878$	−3.81966	−0.128907
$879$	0.0557281	0.00187966
$880$	0	0
$881$	−26.0902	−0.879000	−0.439500	−	0.898243i	$-0.644844\pi$
$881$	−26.0902	−0.879000	−0.439500	+	0.898243i	$0.644844\pi$
$882$	−9.70820	−0.326892
$883$	−6.00000	−0.201916	−0.100958	−	0.994891i	$-0.532191\pi$
$883$	−6.00000	−0.201916	−0.100958	+	0.994891i	$0.532191\pi$
$884$	−37.5066	−1.26148
$885$	0.854102	0.0287103
$886$	12.8754	0.432557
$887$	10.0902	0.338795	0.169397	−	0.985548i	$-0.445818\pi$
$887$	10.0902	0.338795	0.169397	+	0.985548i	$0.445818\pi$
$888$	0.527864	0.0177140
$889$	−20.3951	−0.684030
$890$	2.43769	0.0817117
$891$	0	0
$892$	15.0344	0.503390
$893$	7.88854	0.263980
$894$	1.38197	0.0462199
$895$	6.78522	0.226805
$896$	43.8673	1.46550
$897$	17.7082	0.591260
$898$	−12.2361	−0.408323
$899$	0	0
$900$	−15.7082	−0.523607
$901$	−17.3262	−0.577221
$902$	0	0
$903$	28.4508	0.946785
$904$	−28.9443	−0.962672
$905$	2.32624	0.0773268
$906$	−10.6738	−0.354612
$907$	−0.416408	−0.0138266	−0.00691330	−	0.999976i	$-0.502201\pi$
$907$	−0.416408	−0.0138266	−0.00691330	+	0.999976i	$0.502201\pi$
$908$	−47.5066	−1.57656
$909$	36.3607	1.20601
$910$	4.97871	0.165043
$911$	10.8197	0.358471	0.179236	−	0.983806i	$-0.442638\pi$
$911$	10.8197	0.358471	0.179236	+	0.983806i	$0.442638\pi$
$912$	−13.4164	−0.444262
$913$	0	0
$914$	17.6180	0.582753
$915$	4.23607	0.140040
$916$	41.8328	1.38220
$917$	−62.0132	−2.04785
$918$	13.0902	0.432040
$919$	57.8115	1.90703	0.953513	−	0.301351i	$-0.0974377\pi$
$919$	57.8115	1.90703	0.953513	+	0.301351i	$0.0974377\pi$
$920$	−2.76393	−0.0911241
$921$	25.6869	0.846413
$922$	8.76393	0.288625
$923$	−60.6869	−1.99753
$924$	0	0
$925$	−1.14590	−0.0376769
$926$	−0.291796	−0.00958901
$927$	−18.0000	−0.591198
$928$	0	0
$929$	−52.0344	−1.70719	−0.853597	−	0.520933i	$-0.825584\pi$
$929$	−52.0344	−1.70719	−0.853597	+	0.520933i	$0.825584\pi$
$930$	0.236068	0.00774097
$931$	56.8328	1.86262
$932$	−33.1246	−1.08503
$933$	28.4508	0.931439
$934$	−4.85410	−0.158831
$935$	0	0
$936$	24.4721	0.799897
$937$	37.8541	1.23664	0.618320	−	0.785927i	$-0.287814\pi$
$937$	37.8541	1.23664	0.618320	+	0.785927i	$0.287814\pi$
$938$	16.6738	0.544418
$939$	11.1246	0.363038
$940$	0.673762	0.0219757
$941$	−23.9098	−0.779438	−0.389719	−	0.920934i	$-0.627428\pi$
$941$	−23.9098	−0.779438	−0.389719	+	0.920934i	$0.627428\pi$
$942$	5.90983	0.192553
$943$	32.6525	1.06331
$944$	4.14590	0.134937
$945$	7.36068	0.239443
$946$	0	0
$947$	−53.4296	−1.73623	−0.868114	−	0.496365i	$-0.834668\pi$
$947$	−53.4296	−1.73623	−0.868114	+	0.496365i	$0.834668\pi$
$948$	8.09017	0.262757
$949$	−58.5967	−1.90213
$950$	−21.7082	−0.704307
$951$	17.1246	0.555304
$952$	36.5066	1.18318
$953$	−32.8673	−1.06467	−0.532337	−	0.846532i	$-0.678686\pi$
$953$	−32.8673	−1.06467	−0.532337	+	0.846532i	$0.678686\pi$
$954$	5.05573	0.163685
$955$	−3.12461	−0.101110
$956$	30.1246	0.974300
$957$	0	0
$958$	−3.94427	−0.127434
$959$	−37.4164	−1.20824
$960$	−0.0901699	−0.00291022
$961$	1.00000	0.0322581
$962$	0.798374	0.0257406
$963$	16.6525	0.536619
$964$	6.32624	0.203754
$965$	1.07701	0.0346703
$966$	−7.70820	−0.248007
$967$	−13.0000	−0.418052	−0.209026	−	0.977910i	$-0.567029\pi$
$967$	−13.0000	−0.418052	−0.209026	+	0.977910i	$0.567029\pi$
$968$	0	0
$969$	−30.6525	−0.984699
$970$	0.222912	0.00715728
$971$	−18.0000	−0.577647	−0.288824	−	0.957382i	$-0.593264\pi$
$971$	−18.0000	−0.577647	−0.288824	+	0.957382i	$0.593264\pi$
$972$	25.8885	0.830375
$973$	32.4377	1.03990
$974$	−8.67376	−0.277925
$975$	26.5623	0.850675
$976$	20.5623	0.658183
$977$	−32.7771	−1.04863	−0.524316	−	0.851524i	$-0.675679\pi$
$977$	−32.7771	−1.04863	−0.524316	+	0.851524i	$0.675679\pi$
$978$	6.79837	0.217388
$979$	0	0
$980$	4.85410	0.155059
$981$	−6.58359	−0.210198
$982$	11.1246	0.355001
$983$	−5.67376	−0.180965	−0.0904825	−	0.995898i	$-0.528841\pi$
$983$	−5.67376	−0.180965	−0.0904825	+	0.995898i	$0.528841\pi$
$984$	−22.5623	−0.719260
$985$	3.70820	0.118153
$986$	0	0
$987$	4.20163	0.133739
$988$	−64.0689	−2.03830
$989$	−23.8885	−0.759612
$990$	0	0
$991$	−26.8197	−0.851955	−0.425977	−	0.904734i	$-0.640070\pi$
$991$	−26.8197	−0.851955	−0.425977	+	0.904734i	$0.640070\pi$
$992$	5.61803	0.178373
$993$	11.8197	0.375086
$994$	26.4164	0.837878
$995$	−8.74265	−0.277161
$996$	−16.7984	−0.532277
$997$	14.7639	0.467578	0.233789	−	0.972287i	$-0.424887\pi$
$997$	14.7639	0.467578	0.233789	+	0.972287i	$0.424887\pi$
$998$	−24.3475	−0.770707
$999$	1.18034	0.0373443

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3751.2.a.a.1.2		2
11.10	odd	2		341.2.a.a.1.1	✓	2
33.32	even	2		3069.2.a.b.1.2		2
44.43	even	2		5456.2.a.r.1.2		2
55.54	odd	2		8525.2.a.d.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
341.2.a.a.1.1	✓	2	11.10	odd	2
3069.2.a.b.1.2		2	33.32	even	2
3751.2.a.a.1.2		2	1.1	even	1	trivial
5456.2.a.r.1.2		2	44.43	even	2
8525.2.a.d.1.2		2	55.54	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3751 = 11^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3751.a (trivial)

\(f(q)\)	\(=\)	\(q+0.618034 q^{2} -1.00000 q^{3} -1.61803 q^{4} -0.381966 q^{5} -0.618034 q^{6} -3.85410 q^{7} -2.23607 q^{8} -2.00000 q^{9} +O(q^{10})\)	\(q+0.618034 q^{2} -1.00000 q^{3} -1.61803 q^{4} -0.381966 q^{5} -0.618034 q^{6} -3.85410 q^{7} -2.23607 q^{8} -2.00000 q^{9} -0.236068 q^{10} +1.61803 q^{12} +5.47214 q^{13} -2.38197 q^{14} +0.381966 q^{15} +1.85410 q^{16} +4.23607 q^{17} -1.23607 q^{18} +7.23607 q^{19} +0.618034 q^{20} +3.85410 q^{21} -3.23607 q^{23} +2.23607 q^{24} -4.85410 q^{25} +3.38197 q^{26} +5.00000 q^{27} +6.23607 q^{28} +0.236068 q^{30} +1.00000 q^{31} +5.61803 q^{32} +2.61803 q^{34} +1.47214 q^{35} +3.23607 q^{36} +0.236068 q^{37} +4.47214 q^{38} -5.47214 q^{39} +0.854102 q^{40} -10.0902 q^{41} +2.38197 q^{42} +7.38197 q^{43} +0.763932 q^{45} -2.00000 q^{46} +1.09017 q^{47} -1.85410 q^{48} +7.85410 q^{49} -3.00000 q^{50} -4.23607 q^{51} -8.85410 q^{52} -4.09017 q^{53} +3.09017 q^{54} +8.61803 q^{56} -7.23607 q^{57} +2.23607 q^{59} -0.618034 q^{60} +11.0902 q^{61} +0.618034 q^{62} +7.70820 q^{63} -0.236068 q^{64} -2.09017 q^{65} -7.00000 q^{67} -6.85410 q^{68} +3.23607 q^{69} +0.909830 q^{70} -11.0902 q^{71} +4.47214 q^{72} -10.7082 q^{73} +0.145898 q^{74} +4.85410 q^{75} -11.7082 q^{76} -3.38197 q^{78} +5.00000 q^{79} -0.708204 q^{80} +1.00000 q^{81} -6.23607 q^{82} -10.3820 q^{83} -6.23607 q^{84} -1.61803 q^{85} +4.56231 q^{86} -10.3262 q^{89} +0.472136 q^{90} -21.0902 q^{91} +5.23607 q^{92} -1.00000 q^{93} +0.673762 q^{94} -2.76393 q^{95} -5.61803 q^{96} -0.944272 q^{97} +4.85410 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{2} - 2 q^{3} - q^{4} - 3 q^{5} + q^{6} - q^{7} - 4 q^{9}+O(q^{10}) \) 2 * q - q^2 - 2 * q^3 - q^4 - 3 * q^5 + q^6 - q^7 - 4 * q^9	\( 2 q - q^{2} - 2 q^{3} - q^{4} - 3 q^{5} + q^{6} - q^{7} - 4 q^{9} + 4 q^{10} + q^{12} + 2 q^{13} - 7 q^{14} + 3 q^{15} - 3 q^{16} + 4 q^{17} + 2 q^{18} + 10 q^{19} - q^{20} + q^{21} - 2 q^{23} - 3 q^{25} + 9 q^{26} + 10 q^{27} + 8 q^{28} - 4 q^{30} + 2 q^{31} + 9 q^{32} + 3 q^{34} - 6 q^{35} + 2 q^{36} - 4 q^{37} - 2 q^{39} - 5 q^{40} - 9 q^{41} + 7 q^{42} + 17 q^{43} + 6 q^{45} - 4 q^{46} - 9 q^{47} + 3 q^{48} + 9 q^{49} - 6 q^{50} - 4 q^{51} - 11 q^{52} + 3 q^{53} - 5 q^{54} + 15 q^{56} - 10 q^{57} + q^{60} + 11 q^{61} - q^{62} + 2 q^{63} + 4 q^{64} + 7 q^{65} - 14 q^{67} - 7 q^{68} + 2 q^{69} + 13 q^{70} - 11 q^{71} - 8 q^{73} + 7 q^{74} + 3 q^{75} - 10 q^{76} - 9 q^{78} + 10 q^{79} + 12 q^{80} + 2 q^{81} - 8 q^{82} - 23 q^{83} - 8 q^{84} - q^{85} - 11 q^{86} - 5 q^{89} - 8 q^{90} - 31 q^{91} + 6 q^{92} - 2 q^{93} + 17 q^{94} - 10 q^{95} - 9 q^{96} + 16 q^{97} + 3 q^{98}+O(q^{100}) \) 2 * q - q^2 - 2 * q^3 - q^4 - 3 * q^5 + q^6 - q^7 - 4 * q^9 + 4 * q^10 + q^12 + 2 * q^13 - 7 * q^14 + 3 * q^15 - 3 * q^16 + 4 * q^17 + 2 * q^18 + 10 * q^19 - q^20 + q^21 - 2 * q^23 - 3 * q^25 + 9 * q^26 + 10 * q^27 + 8 * q^28 - 4 * q^30 + 2 * q^31 + 9 * q^32 + 3 * q^34 - 6 * q^35 + 2 * q^36 - 4 * q^37 - 2 * q^39 - 5 * q^40 - 9 * q^41 + 7 * q^42 + 17 * q^43 + 6 * q^45 - 4 * q^46 - 9 * q^47 + 3 * q^48 + 9 * q^49 - 6 * q^50 - 4 * q^51 - 11 * q^52 + 3 * q^53 - 5 * q^54 + 15 * q^56 - 10 * q^57 + q^60 + 11 * q^61 - q^62 + 2 * q^63 + 4 * q^64 + 7 * q^65 - 14 * q^67 - 7 * q^68 + 2 * q^69 + 13 * q^70 - 11 * q^71 - 8 * q^73 + 7 * q^74 + 3 * q^75 - 10 * q^76 - 9 * q^78 + 10 * q^79 + 12 * q^80 + 2 * q^81 - 8 * q^82 - 23 * q^83 - 8 * q^84 - q^85 - 11 * q^86 - 5 * q^89 - 8 * q^90 - 31 * q^91 + 6 * q^92 - 2 * q^93 + 17 * q^94 - 10 * q^95 - 9 * q^96 + 16 * q^97 + 3 * q^98

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