LMFDB - Embedded newform 445.2.a.f.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [445,2,Mod(1,445)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(445, 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("445.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 445 = 5 \cdot 89 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	445.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:	$3.55334288995$
Analytic rank:	$1$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - x^{6} - 8x^{5} + 6x^{4} + 19x^{3} - 10x^{2} - 12x + 6 $ x^7 - x^6 - 8x^5 + 6x^4 + 19x^3 - 10x^2 - 12*x + 6
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$2.26266$ of defining polynomial
Character	$\chi$	$=$	445.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.11962 q^{2} -3.29058 q^{3} +2.49279 q^{4} +1.00000 q^{5} -6.97477 q^{6} -4.83304 q^{7} +1.04452 q^{8} +7.82788 q^{9} +O(q^{10})$	\(q+2.11962 q^{2} -3.29058 q^{3} +2.49279 q^{4} +1.00000 q^{5} -6.97477 q^{6} -4.83304 q^{7} +1.04452 q^{8} +7.82788 q^{9} +2.11962 q^{10} -2.21586 q^{11} -8.20270 q^{12} -2.26162 q^{13} -10.2442 q^{14} -3.29058 q^{15} -2.77159 q^{16} -3.10920 q^{17} +16.5921 q^{18} -5.93952 q^{19} +2.49279 q^{20} +15.9035 q^{21} -4.69678 q^{22} +5.79416 q^{23} -3.43708 q^{24} +1.00000 q^{25} -4.79378 q^{26} -15.8865 q^{27} -12.0477 q^{28} +2.32757 q^{29} -6.97477 q^{30} +4.47624 q^{31} -7.96375 q^{32} +7.29145 q^{33} -6.59033 q^{34} -4.83304 q^{35} +19.5133 q^{36} -8.23124 q^{37} -12.5895 q^{38} +7.44204 q^{39} +1.04452 q^{40} -0.278075 q^{41} +33.7093 q^{42} -0.176109 q^{43} -5.52366 q^{44} +7.82788 q^{45} +12.2814 q^{46} +1.91855 q^{47} +9.12011 q^{48} +16.3583 q^{49} +2.11962 q^{50} +10.2311 q^{51} -5.63775 q^{52} +8.65904 q^{53} -33.6734 q^{54} -2.21586 q^{55} -5.04821 q^{56} +19.5444 q^{57} +4.93357 q^{58} +5.39666 q^{59} -8.20270 q^{60} -9.69150 q^{61} +9.48792 q^{62} -37.8325 q^{63} -11.3370 q^{64} -2.26162 q^{65} +15.4551 q^{66} +8.06916 q^{67} -7.75058 q^{68} -19.0661 q^{69} -10.2442 q^{70} -5.00576 q^{71} +8.17640 q^{72} -9.18909 q^{73} -17.4471 q^{74} -3.29058 q^{75} -14.8060 q^{76} +10.7093 q^{77} +15.7743 q^{78} -1.78949 q^{79} -2.77159 q^{80} +28.7921 q^{81} -0.589414 q^{82} +13.8879 q^{83} +39.6440 q^{84} -3.10920 q^{85} -0.373284 q^{86} -7.65906 q^{87} -2.31451 q^{88} +1.00000 q^{89} +16.5921 q^{90} +10.9305 q^{91} +14.4436 q^{92} -14.7294 q^{93} +4.06660 q^{94} -5.93952 q^{95} +26.2053 q^{96} -5.52008 q^{97} +34.6733 q^{98} -17.3455 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - 4 q^{2} - 8 q^{3} + 8 q^{4} + 7 q^{5} - 2 q^{6} - 16 q^{7} - 12 q^{8} + 11 q^{9}+O(q^{10}) $ 7 * q - 4 * q^2 - 8 * q^3 + 8 * q^4 + 7 * q^5 - 2 * q^6 - 16 * q^7 - 12 * q^8 + 11 * q^9	\( 7 q - 4 q^{2} - 8 q^{3} + 8 q^{4} + 7 q^{5} - 2 q^{6} - 16 q^{7} - 12 q^{8} + 11 q^{9} - 4 q^{10} - 10 q^{11} - 11 q^{12} - 7 q^{13} + 3 q^{14} - 8 q^{15} + 10 q^{16} - 13 q^{17} + 4 q^{18} - 7 q^{19} + 8 q^{20} + 16 q^{21} + 2 q^{22} - 13 q^{23} + 4 q^{24} + 7 q^{25} + q^{26} - 23 q^{27} - 21 q^{28} - 4 q^{29} - 2 q^{30} + q^{31} - 13 q^{32} - 6 q^{33} + 10 q^{34} - 16 q^{35} + 20 q^{36} - 5 q^{37} - 40 q^{38} - 13 q^{39} - 12 q^{40} + 5 q^{41} + 30 q^{42} - 31 q^{43} - 21 q^{44} + 11 q^{45} + 16 q^{46} - 14 q^{47} - 7 q^{48} + 19 q^{49} - 4 q^{50} - q^{51} - 13 q^{53} - 17 q^{54} - 10 q^{55} - q^{56} + 21 q^{57} + 17 q^{58} - 14 q^{59} - 11 q^{60} + 3 q^{61} + 26 q^{62} - 54 q^{63} + 14 q^{64} - 7 q^{65} + 36 q^{66} + q^{67} - 35 q^{68} + 31 q^{69} + 3 q^{70} - 8 q^{71} + 53 q^{72} + 9 q^{73} - 35 q^{74} - 8 q^{75} + 40 q^{76} + 42 q^{77} + 46 q^{78} + 9 q^{79} + 10 q^{80} + 35 q^{81} + 29 q^{82} - 42 q^{83} + 55 q^{84} - 13 q^{85} + 35 q^{86} + 6 q^{87} + 30 q^{88} + 7 q^{89} + 4 q^{90} + 31 q^{91} + 19 q^{92} + 24 q^{93} + 37 q^{94} - 7 q^{95} + 44 q^{96} - 7 q^{97} + 9 q^{98} + 5 q^{99}+O(q^{100}) \) 7 * q - 4 * q^2 - 8 * q^3 + 8 * q^4 + 7 * q^5 - 2 * q^6 - 16 * q^7 - 12 * q^8 + 11 * q^9 - 4 * q^10 - 10 * q^11 - 11 * q^12 - 7 * q^13 + 3 * q^14 - 8 * q^15 + 10 * q^16 - 13 * q^17 + 4 * q^18 - 7 * q^19 + 8 * q^20 + 16 * q^21 + 2 * q^22 - 13 * q^23 + 4 * q^24 + 7 * q^25 + q^26 - 23 * q^27 - 21 * q^28 - 4 * q^29 - 2 * q^30 + q^31 - 13 * q^32 - 6 * q^33 + 10 * q^34 - 16 * q^35 + 20 * q^36 - 5 * q^37 - 40 * q^38 - 13 * q^39 - 12 * q^40 + 5 * q^41 + 30 * q^42 - 31 * q^43 - 21 * q^44 + 11 * q^45 + 16 * q^46 - 14 * q^47 - 7 * q^48 + 19 * q^49 - 4 * q^50 - q^51 - 13 * q^53 - 17 * q^54 - 10 * q^55 - q^56 + 21 * q^57 + 17 * q^58 - 14 * q^59 - 11 * q^60 + 3 * q^61 + 26 * q^62 - 54 * q^63 + 14 * q^64 - 7 * q^65 + 36 * q^66 + q^67 - 35 * q^68 + 31 * q^69 + 3 * q^70 - 8 * q^71 + 53 * q^72 + 9 * q^73 - 35 * q^74 - 8 * q^75 + 40 * q^76 + 42 * q^77 + 46 * q^78 + 9 * q^79 + 10 * q^80 + 35 * q^81 + 29 * q^82 - 42 * q^83 + 55 * q^84 - 13 * q^85 + 35 * q^86 + 6 * q^87 + 30 * q^88 + 7 * q^89 + 4 * q^90 + 31 * q^91 + 19 * q^92 + 24 * q^93 + 37 * q^94 - 7 * q^95 + 44 * q^96 - 7 * q^97 + 9 * q^98 + 5 * 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 $

$2$	2.11962	1.49880	0.749399	−	0.662119i	$-0.230343\pi$
$2$	2.11962	1.49880	0.749399	+	0.662119i	$0.230343\pi$
$3$	−3.29058	−1.89981	−0.949907	−	0.312532i	$-0.898823\pi$
$3$	−3.29058	−1.89981	−0.949907	+	0.312532i	$0.898823\pi$
$4$	2.49279	1.24639
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	−6.97477	−2.84744
$7$	−4.83304	−1.82672	−0.913358	−	0.407157i	$-0.866520\pi$
$7$	−4.83304	−1.82672	−0.913358	+	0.407157i	$0.866520\pi$
$8$	1.04452	0.369294
$9$	7.82788	2.60929
$10$	2.11962	0.670283
$11$	−2.21586	−0.668106	−0.334053	−	0.942554i	$-0.608417\pi$
$11$	−2.21586	−0.668106	−0.334053	+	0.942554i	$0.608417\pi$
$12$	−8.20270	−2.36792
$13$	−2.26162	−0.627261	−0.313631	−	0.949545i	$-0.601545\pi$
$13$	−2.26162	−0.627261	−0.313631	+	0.949545i	$0.601545\pi$
$14$	−10.2442	−2.73788
$15$	−3.29058	−0.849623
$16$	−2.77159	−0.692896
$17$	−3.10920	−0.754092	−0.377046	−	0.926194i	$-0.623060\pi$
$17$	−3.10920	−0.754092	−0.377046	+	0.926194i	$0.623060\pi$
$18$	16.5921	3.91080
$19$	−5.93952	−1.36262	−0.681309	−	0.731996i	$-0.738589\pi$
$19$	−5.93952	−1.36262	−0.681309	+	0.731996i	$0.738589\pi$
$20$	2.49279	0.557404
$21$	15.9035	3.47042
$22$	−4.69678	−1.00136
$23$	5.79416	1.20817	0.604083	−	0.796921i	$-0.293540\pi$
$23$	5.79416	1.20817	0.604083	+	0.796921i	$0.293540\pi$
$24$	−3.43708	−0.701590
$25$	1.00000	0.200000
$26$	−4.79378	−0.940138
$27$	−15.8865	−3.05736
$28$	−12.0477	−2.27681
$29$	2.32757	0.432220	0.216110	−	0.976369i	$-0.430663\pi$
$29$	2.32757	0.432220	0.216110	+	0.976369i	$0.430663\pi$
$30$	−6.97477	−1.27341
$31$	4.47624	0.803956	0.401978	−	0.915649i	$-0.368323\pi$
$31$	4.47624	0.803956	0.401978	+	0.915649i	$0.368323\pi$
$32$	−7.96375	−1.40781
$33$	7.29145	1.26928
$34$	−6.59033	−1.13023
$35$	−4.83304	−0.816932
$36$	19.5133	3.25221
$37$	−8.23124	−1.35321	−0.676604	−	0.736347i	$-0.736549\pi$
$37$	−8.23124	−1.35321	−0.676604	+	0.736347i	$0.736549\pi$
$38$	−12.5895	−2.04229
$39$	7.44204	1.19168
$40$	1.04452	0.165153
$41$	−0.278075	−0.0434281	−0.0217140	−	0.999764i	$-0.506912\pi$
$41$	−0.278075	−0.0434281	−0.0217140	+	0.999764i	$0.506912\pi$
$42$	33.7093	5.20146
$43$	−0.176109	−0.0268564	−0.0134282	−	0.999910i	$-0.504274\pi$
$43$	−0.176109	−0.0268564	−0.0134282	+	0.999910i	$0.504274\pi$
$44$	−5.52366	−0.832724
$45$	7.82788	1.16691
$46$	12.2814	1.81080
$47$	1.91855	0.279850	0.139925	−	0.990162i	$-0.455314\pi$
$47$	1.91855	0.279850	0.139925	+	0.990162i	$0.455314\pi$
$48$	9.12011	1.31637
$49$	16.3583	2.33689
$50$	2.11962	0.299759
$51$	10.2311	1.43264
$52$	−5.63775	−0.781815
$53$	8.65904	1.18941	0.594706	−	0.803944i	$-0.297269\pi$
$53$	8.65904	1.18941	0.594706	+	0.803944i	$0.297269\pi$
$54$	−33.6734	−4.58237
$55$	−2.21586	−0.298786
$56$	−5.04821	−0.674596
$57$	19.5444	2.58872
$58$	4.93357	0.647810
$59$	5.39666	0.702585	0.351292	−	0.936266i	$-0.385742\pi$
$59$	5.39666	0.702585	0.351292	+	0.936266i	$0.385742\pi$
$60$	−8.20270	−1.05896
$61$	−9.69150	−1.24087	−0.620434	−	0.784258i	$-0.713044\pi$
$61$	−9.69150	−1.24087	−0.620434	+	0.784258i	$0.713044\pi$
$62$	9.48792	1.20497
$63$	−37.8325	−4.76644
$64$	−11.3370	−1.41712
$65$	−2.26162	−0.280520
$66$	15.4551	1.90239
$67$	8.06916	0.985805	0.492903	−	0.870084i	$-0.335936\pi$
$67$	8.06916	0.985805	0.492903	+	0.870084i	$0.335936\pi$
$68$	−7.75058	−0.939896
$69$	−19.0661	−2.29529
$70$	−10.2442	−1.22442
$71$	−5.00576	−0.594075	−0.297037	−	0.954866i	$-0.595999\pi$
$71$	−5.00576	−0.594075	−0.297037	+	0.954866i	$0.595999\pi$
$72$	8.17640	0.963597
$73$	−9.18909	−1.07550	−0.537751	−	0.843104i	$-0.680726\pi$
$73$	−9.18909	−1.07550	−0.537751	+	0.843104i	$0.680726\pi$
$74$	−17.4471	−2.02818
$75$	−3.29058	−0.379963
$76$	−14.8060	−1.69836
$77$	10.7093	1.22044
$78$	15.7743	1.78609
$79$	−1.78949	−0.201333	−0.100666	−	0.994920i	$-0.532097\pi$
$79$	−1.78949	−0.201333	−0.100666	+	0.994920i	$0.532097\pi$
$80$	−2.77159	−0.309873
$81$	28.7921	3.19912
$82$	−0.589414	−0.0650899
$83$	13.8879	1.52439	0.762197	−	0.647345i	$-0.224121\pi$
$83$	13.8879	1.52439	0.762197	+	0.647345i	$0.224121\pi$
$84$	39.6440	4.32551
$85$	−3.10920	−0.337240
$86$	−0.373284	−0.0402523
$87$	−7.65906	−0.821137
$88$	−2.31451	−0.246728
$89$	1.00000	0.106000
$89$	1.00000	0.106000
$90$	16.5921	1.74896
$91$	10.9305	1.14583
$92$	14.4436	1.50585
$93$	−14.7294	−1.52737
$94$	4.06660	0.419438
$95$	−5.93952	−0.609382
$96$	26.2053	2.67457
$97$	−5.52008	−0.560479	−0.280240	−	0.959930i	$-0.590414\pi$
$97$	−5.52008	−0.560479	−0.280240	+	0.959930i	$0.590414\pi$
$98$	34.6733	3.50253
$99$	−17.3455	−1.74329
$100$	2.49279	0.249279
$101$	−12.8409	−1.27772	−0.638860	−	0.769323i	$-0.720594\pi$
$101$	−12.8409	−1.27772	−0.638860	+	0.769323i	$0.720594\pi$
$102$	21.6860	2.14723
$103$	−11.7697	−1.15970	−0.579852	−	0.814722i	$-0.696890\pi$
$103$	−11.7697	−1.15970	−0.579852	+	0.814722i	$0.696890\pi$
$104$	−2.36231	−0.231644
$105$	15.9035	1.55202
$106$	18.3539	1.78269
$107$	−15.4620	−1.49477	−0.747383	−	0.664393i	$-0.768690\pi$
$107$	−15.4620	−1.49477	−0.747383	+	0.664393i	$0.768690\pi$
$108$	−39.6017	−3.81068
$109$	−7.88060	−0.754824	−0.377412	−	0.926045i	$-0.623186\pi$
$109$	−7.88060	−0.754824	−0.377412	+	0.926045i	$0.623186\pi$
$110$	−4.69678	−0.447820
$111$	27.0855	2.57084
$112$	13.3952	1.26573
$113$	−2.01004	−0.189088	−0.0945442	−	0.995521i	$-0.530139\pi$
$113$	−2.01004	−0.189088	−0.0945442	+	0.995521i	$0.530139\pi$
$114$	41.4268	3.87997
$115$	5.79416	0.540308
$116$	5.80215	0.538716
$117$	−17.7037	−1.63671
$118$	11.4389	1.05303
$119$	15.0269	1.37751
$120$	−3.43708	−0.313761
$121$	−6.08997	−0.553634
$122$	−20.5423	−1.85981
$123$	0.915028	0.0825053
$124$	11.1583	1.00205
$125$	1.00000	0.0894427
$126$	−80.1904	−7.14393
$127$	−12.0158	−1.06623	−0.533113	−	0.846044i	$-0.678978\pi$
$127$	−12.0158	−1.06623	−0.533113	+	0.846044i	$0.678978\pi$
$128$	−8.10252	−0.716169
$129$	0.579500	0.0510221
$130$	−4.79378	−0.420442
$131$	−20.6066	−1.80040	−0.900202	−	0.435472i	$-0.856581\pi$
$131$	−20.6066	−1.80040	−0.900202	+	0.435472i	$0.856581\pi$
$132$	18.1760	1.58202
$133$	28.7059	2.48912
$134$	17.1036	1.47752
$135$	−15.8865	−1.36729
$136$	−3.24763	−0.278482
$137$	10.9369	0.934400	0.467200	−	0.884152i	$-0.345263\pi$
$137$	10.9369	0.934400	0.467200	+	0.884152i	$0.345263\pi$
$138$	−40.4129	−3.44018
$139$	−0.432073	−0.0366479	−0.0183240	−	0.999832i	$-0.505833\pi$
$139$	−0.432073	−0.0366479	−0.0183240	+	0.999832i	$0.505833\pi$
$140$	−12.0477	−1.01822
$141$	−6.31314	−0.531663
$142$	−10.6103	−0.890397
$143$	5.01144	0.419077
$144$	−21.6957	−1.80797
$145$	2.32757	0.193294
$146$	−19.4774	−1.61196
$147$	−53.8281	−4.43966
$148$	−20.5187	−1.68663
$149$	−13.6417	−1.11757	−0.558785	−	0.829313i	$-0.688732\pi$
$149$	−13.6417	−1.11757	−0.558785	+	0.829313i	$0.688732\pi$
$150$	−6.97477	−0.569487
$151$	22.3825	1.82146	0.910730	−	0.413002i	$-0.135520\pi$
$151$	22.3825	1.82146	0.910730	+	0.413002i	$0.135520\pi$
$152$	−6.20396	−0.503207
$153$	−24.3385	−1.96765
$154$	22.6997	1.82919
$155$	4.47624	0.359540
$156$	18.5514	1.48530
$157$	11.7366	0.936683	0.468342	−	0.883547i	$-0.344852\pi$
$157$	11.7366	0.936683	0.468342	+	0.883547i	$0.344852\pi$
$158$	−3.79303	−0.301757
$159$	−28.4932	−2.25966
$160$	−7.96375	−0.629590
$161$	−28.0034	−2.20698
$162$	61.0283	4.79484
$163$	1.49678	0.117237	0.0586184	−	0.998280i	$-0.481330\pi$
$163$	1.49678	0.117237	0.0586184	+	0.998280i	$0.481330\pi$
$164$	−0.693183	−0.0541285
$165$	7.29145	0.567638
$166$	29.4371	2.28476
$167$	−0.555849	−0.0430129	−0.0215064	−	0.999769i	$-0.506846\pi$
$167$	−0.555849	−0.0430129	−0.0215064	+	0.999769i	$0.506846\pi$
$168$	16.6115	1.28161
$169$	−7.88506	−0.606543
$170$	−6.59033	−0.505455
$171$	−46.4939	−3.55547
$172$	−0.439002	−0.0334736
$173$	14.0371	1.06722	0.533611	−	0.845730i	$-0.320835\pi$
$173$	14.0371	1.06722	0.533611	+	0.845730i	$0.320835\pi$
$174$	−16.2343	−1.23072
$175$	−4.83304	−0.365343
$176$	6.14144	0.462929
$177$	−17.7581	−1.33478
$178$	2.11962	0.158872
$179$	9.22570	0.689561	0.344780	−	0.938683i	$-0.387953\pi$
$179$	9.22570	0.689561	0.344780	+	0.938683i	$0.387953\pi$
$180$	19.5133	1.45443
$181$	−10.4704	−0.778259	−0.389129	−	0.921183i	$-0.627224\pi$
$181$	−10.4704	−0.778259	−0.389129	+	0.921183i	$0.627224\pi$
$182$	23.1685	1.71737
$183$	31.8906	2.35742
$184$	6.05213	0.446169
$185$	−8.23124	−0.605173
$186$	−31.2207	−2.28921
$187$	6.88955	0.503814
$188$	4.78255	0.348803
$189$	76.7801	5.58493
$190$	−12.5895	−0.913340
$191$	−4.73905	−0.342906	−0.171453	−	0.985192i	$-0.554846\pi$
$191$	−4.73905	−0.342906	−0.171453	+	0.985192i	$0.554846\pi$
$192$	37.3051	2.69226
$193$	1.48587	0.106955	0.0534775	−	0.998569i	$-0.482969\pi$
$193$	1.48587	0.106955	0.0534775	+	0.998569i	$0.482969\pi$
$194$	−11.7005	−0.840045
$195$	7.44204	0.532936
$196$	40.7777	2.91269
$197$	−4.51312	−0.321547	−0.160773	−	0.986991i	$-0.551399\pi$
$197$	−4.51312	−0.321547	−0.160773	+	0.986991i	$0.551399\pi$
$198$	−36.7658	−2.61283
$199$	7.28028	0.516085	0.258043	−	0.966134i	$-0.416923\pi$
$199$	7.28028	0.516085	0.258043	+	0.966134i	$0.416923\pi$
$200$	1.04452	0.0738588
$201$	−26.5522	−1.87285
$202$	−27.2179	−1.91504
$203$	−11.2493	−0.789543
$204$	25.5039	1.78563
$205$	−0.278075	−0.0194216
$206$	−24.9473	−1.73816
$207$	45.3560	3.15246
$208$	6.26828	0.434627
$209$	13.1611	0.910374
$210$	33.7093	2.32616
$211$	14.0051	0.964154	0.482077	−	0.876129i	$-0.339883\pi$
$211$	14.0051	0.964154	0.482077	+	0.876129i	$0.339883\pi$
$212$	21.5852	1.48247
$213$	16.4718	1.12863
$214$	−32.7735	−2.24035
$215$	−0.176109	−0.0120105
$216$	−16.5938	−1.12907
$217$	−21.6338	−1.46860
$218$	−16.7039	−1.13133
$219$	30.2374	2.04325
$220$	−5.52366	−0.372405
$221$	7.03184	0.473013
$222$	57.4110	3.85317
$223$	−8.76611	−0.587022	−0.293511	−	0.955956i	$-0.594824\pi$
$223$	−8.76611	−0.587022	−0.293511	+	0.955956i	$0.594824\pi$
$224$	38.4891	2.57166
$225$	7.82788	0.521859
$226$	−4.26051	−0.283405
$227$	−21.8527	−1.45042	−0.725209	−	0.688529i	$-0.758257\pi$
$227$	−21.8527	−1.45042	−0.725209	+	0.688529i	$0.758257\pi$
$228$	48.7201	3.22657
$229$	−20.0211	−1.32303	−0.661514	−	0.749932i	$-0.730086\pi$
$229$	−20.0211	−1.32303	−0.661514	+	0.749932i	$0.730086\pi$
$230$	12.2814	0.809812
$231$	−35.2398	−2.31861
$232$	2.43120	0.159616
$233$	15.3296	1.00428	0.502139	−	0.864787i	$-0.332547\pi$
$233$	15.3296	1.00428	0.502139	+	0.864787i	$0.332547\pi$
$234$	−37.5252	−2.45310
$235$	1.91855	0.125153
$236$	13.4527	0.875697
$237$	5.88844	0.382495
$238$	31.8513	2.06461
$239$	7.27188	0.470379	0.235190	−	0.971950i	$-0.424429\pi$
$239$	7.27188	0.470379	0.235190	+	0.971950i	$0.424429\pi$
$240$	9.12011	0.588701
$241$	4.02887	0.259522	0.129761	−	0.991545i	$-0.458579\pi$
$241$	4.02887	0.259522	0.129761	+	0.991545i	$0.458579\pi$
$242$	−12.9084	−0.829785
$243$	−47.0831	−3.02038
$244$	−24.1588	−1.54661
$245$	16.3583	1.04509
$246$	1.93951	0.123659
$247$	13.4329	0.854718
$248$	4.67553	0.296896
$249$	−45.6992	−2.89607
$250$	2.11962	0.134057
$251$	−6.99410	−0.441464	−0.220732	−	0.975335i	$-0.570845\pi$
$251$	−6.99410	−0.441464	−0.220732	+	0.975335i	$0.570845\pi$
$252$	−94.3083	−5.94086
$253$	−12.8390	−0.807183
$254$	−25.4688	−1.59806
$255$	10.2311	0.640694
$256$	5.49964	0.343727
$257$	−19.3582	−1.20753	−0.603767	−	0.797161i	$-0.706334\pi$
$257$	−19.3582	−1.20753	−0.603767	+	0.797161i	$0.706334\pi$
$258$	1.22832	0.0764718
$259$	39.7819	2.47193
$260$	−5.63775	−0.349638
$261$	18.2200	1.12779
$262$	−43.6781	−2.69844
$263$	−5.73446	−0.353602	−0.176801	−	0.984247i	$-0.556575\pi$
$263$	−5.73446	−0.353602	−0.176801	+	0.984247i	$0.556575\pi$
$264$	7.61608	0.468737
$265$	8.65904	0.531921
$266$	60.8456	3.73068
$267$	−3.29058	−0.201380
$268$	20.1147	1.22870
$269$	5.07725	0.309565	0.154783	−	0.987949i	$-0.450532\pi$
$269$	5.07725	0.309565	0.154783	+	0.987949i	$0.450532\pi$
$270$	−33.6734	−2.04930
$271$	14.8909	0.904557	0.452278	−	0.891877i	$-0.350611\pi$
$271$	14.8909	0.904557	0.452278	+	0.891877i	$0.350611\pi$
$272$	8.61742	0.522508
$273$	−35.9677	−2.17686
$274$	23.1820	1.40048
$275$	−2.21586	−0.133621
$276$	−47.5278	−2.86084
$277$	−30.6930	−1.84417	−0.922083	−	0.386993i	$-0.873514\pi$
$277$	−30.6930	−1.84417	−0.922083	+	0.386993i	$0.873514\pi$
$278$	−0.915829	−0.0549278
$279$	35.0395	2.09776
$280$	−5.04821	−0.301688
$281$	19.5982	1.16913	0.584564	−	0.811348i	$-0.301266\pi$
$281$	19.5982	1.16913	0.584564	+	0.811348i	$0.301266\pi$
$282$	−13.3815	−0.796855
$283$	32.1638	1.91194	0.955968	−	0.293469i	$-0.0948099\pi$
$283$	32.1638	1.91194	0.955968	+	0.293469i	$0.0948099\pi$
$284$	−12.4783	−0.740451
$285$	19.5444	1.15771
$286$	10.6223	0.628112
$287$	1.34395	0.0793308
$288$	−62.3393	−3.67338
$289$	−7.33286	−0.431345
$290$	4.93357	0.289709
$291$	18.1642	1.06481
$292$	−22.9064	−1.34050
$293$	−22.7569	−1.32947	−0.664737	−	0.747077i	$-0.731457\pi$
$293$	−22.7569	−1.32947	−0.664737	+	0.747077i	$0.731457\pi$
$294$	−114.095	−6.65416
$295$	5.39666	0.314205
$296$	−8.59771	−0.499732
$297$	35.2023	2.04264
$298$	−28.9152	−1.67501
$299$	−13.1042	−0.757836
$300$	−8.20270	−0.473583
$301$	0.851142	0.0490590
$302$	47.4423	2.73000
$303$	42.2541	2.42743
$304$	16.4619	0.944154
$305$	−9.69150	−0.554933
$306$	−51.5883	−2.94911
$307$	0.883192	0.0504064	0.0252032	−	0.999682i	$-0.491977\pi$
$307$	0.883192	0.0504064	0.0252032	+	0.999682i	$0.491977\pi$
$308$	26.6961	1.52115
$309$	38.7291	2.20322
$310$	9.48792	0.538878
$311$	−2.98323	−0.169163	−0.0845816	−	0.996417i	$-0.526955\pi$
$311$	−2.98323	−0.169163	−0.0845816	+	0.996417i	$0.526955\pi$
$312$	7.77337	0.440081
$313$	6.07859	0.343582	0.171791	−	0.985133i	$-0.445045\pi$
$313$	6.07859	0.343582	0.171791	+	0.985133i	$0.445045\pi$
$314$	24.8771	1.40390
$315$	−37.8325	−2.13162
$316$	−4.46081	−0.250940
$317$	2.02998	0.114015	0.0570075	−	0.998374i	$-0.481844\pi$
$317$	2.02998	0.114015	0.0570075	+	0.998374i	$0.481844\pi$
$318$	−60.3948	−3.38677
$319$	−5.15757	−0.288769
$320$	−11.3370	−0.633755
$321$	50.8788	2.83978
$322$	−59.3565	−3.30781
$323$	18.4672	1.02754
$324$	71.7726	3.98737
$325$	−2.26162	−0.125452
$326$	3.17260	0.175714
$327$	25.9317	1.43403
$328$	−0.290456	−0.0160377
$329$	−9.27244	−0.511206
$330$	15.4551	0.850775
$331$	−12.3999	−0.681559	−0.340779	−	0.940143i	$-0.610691\pi$
$331$	−12.3999	−0.681559	−0.340779	+	0.940143i	$0.610691\pi$
$332$	34.6196	1.90000
$333$	−64.4332	−3.53092
$334$	−1.17819	−0.0644676
$335$	8.06916	0.440866
$336$	−44.0778	−2.40464
$337$	27.1425	1.47855	0.739274	−	0.673405i	$-0.235169\pi$
$337$	27.1425	1.47855	0.739274	+	0.673405i	$0.235169\pi$
$338$	−16.7133	−0.909085
$339$	6.61418	0.359233
$340$	−7.75058	−0.420334
$341$	−9.91871	−0.537128
$342$	−98.5493	−5.32894
$343$	−45.2288	−2.44213
$344$	−0.183950	−0.00991790
$345$	−19.0661	−1.02649
$346$	29.7533	1.59955
$347$	−17.0635	−0.916019	−0.458009	−	0.888947i	$-0.651437\pi$
$347$	−17.0635	−0.916019	−0.458009	+	0.888947i	$0.651437\pi$
$348$	−19.0924	−1.02346
$349$	25.3297	1.35587	0.677933	−	0.735124i	$-0.262876\pi$
$349$	25.3297	1.35587	0.677933	+	0.735124i	$0.262876\pi$
$350$	−10.2442	−0.547576
$351$	35.9293	1.91776
$352$	17.6465	0.940564
$353$	15.3713	0.818132	0.409066	−	0.912505i	$-0.365854\pi$
$353$	15.3713	0.818132	0.409066	+	0.912505i	$0.365854\pi$
$354$	−37.6404	−2.00057
$355$	−5.00576	−0.265678
$356$	2.49279	0.132117
$357$	−49.4471	−2.61702
$358$	19.5550	1.03351
$359$	−4.81197	−0.253966	−0.126983	−	0.991905i	$-0.540529\pi$
$359$	−4.81197	−0.253966	−0.126983	+	0.991905i	$0.540529\pi$
$360$	8.17640	0.430934
$361$	16.2779	0.856730
$362$	−22.1933	−1.16645
$363$	20.0395	1.05180
$364$	27.2474	1.42815
$365$	−9.18909	−0.480979
$366$	67.5959	3.53330
$367$	23.4281	1.22294	0.611468	−	0.791269i	$-0.290579\pi$
$367$	23.4281	1.22294	0.611468	+	0.791269i	$0.290579\pi$
$368$	−16.0590	−0.837134
$369$	−2.17674	−0.113317
$370$	−17.4471	−0.907031
$371$	−41.8495	−2.17272
$372$	−36.7173	−1.90370
$373$	10.4886	0.543078	0.271539	−	0.962427i	$-0.412467\pi$
$373$	10.4886	0.543078	0.271539	+	0.962427i	$0.412467\pi$
$374$	14.6032	0.755115
$375$	−3.29058	−0.169925
$376$	2.00397	0.103347
$377$	−5.26409	−0.271115
$378$	162.745	8.37068
$379$	−20.6495	−1.06069	−0.530347	−	0.847781i	$-0.677938\pi$
$379$	−20.6495	−1.06069	−0.530347	+	0.847781i	$0.677938\pi$
$380$	−14.8060	−0.759529
$381$	39.5388	2.02563
$382$	−10.0450	−0.513946
$383$	11.8529	0.605654	0.302827	−	0.953046i	$-0.402070\pi$
$383$	11.8529	0.605654	0.302827	+	0.953046i	$0.402070\pi$
$384$	26.6620	1.36059
$385$	10.7093	0.545798
$386$	3.14947	0.160304
$387$	−1.37856	−0.0700762
$388$	−13.7604	−0.698578
$389$	−11.8172	−0.599157	−0.299578	−	0.954072i	$-0.596846\pi$
$389$	−11.8172	−0.599157	−0.299578	+	0.954072i	$0.596846\pi$
$390$	15.7743	0.798762
$391$	−18.0152	−0.911068
$392$	17.0866	0.863001
$393$	67.8075	3.42043
$394$	−9.56610	−0.481933
$395$	−1.78949	−0.0900388
$396$	−43.2386	−2.17282
$397$	36.5070	1.83223	0.916116	−	0.400912i	$-0.131307\pi$
$397$	36.5070	1.83223	0.916116	+	0.400912i	$0.131307\pi$
$398$	15.4314	0.773507
$399$	−94.4590	−4.72886
$400$	−2.77159	−0.138579
$401$	1.45221	0.0725197	0.0362599	−	0.999342i	$-0.488456\pi$
$401$	1.45221	0.0725197	0.0362599	+	0.999342i	$0.488456\pi$
$402$	−56.2805	−2.80702
$403$	−10.1236	−0.504291
$404$	−32.0097	−1.59254
$405$	28.7921	1.43069
$406$	−23.8441	−1.18336
$407$	18.2393	0.904086
$408$	10.6866	0.529064
$409$	32.3981	1.60198	0.800992	−	0.598674i	$-0.204306\pi$
$409$	32.3981	1.60198	0.800992	+	0.598674i	$0.204306\pi$
$410$	−0.589414	−0.0291091
$411$	−35.9886	−1.77519
$412$	−29.3394	−1.44545
$413$	−26.0822	−1.28342
$414$	96.1375	4.72490
$415$	13.8879	0.681730
$416$	18.0110	0.883062
$417$	1.42177	0.0696242
$418$	27.8966	1.36447
$419$	−10.4032	−0.508227	−0.254114	−	0.967174i	$-0.581784\pi$
$419$	−10.4032	−0.508227	−0.254114	+	0.967174i	$0.581784\pi$
$420$	39.6440	1.93443
$421$	1.28300	0.0625297	0.0312648	−	0.999511i	$-0.490046\pi$
$421$	1.28300	0.0625297	0.0312648	+	0.999511i	$0.490046\pi$
$422$	29.6856	1.44507
$423$	15.0182	0.730211
$424$	9.04456	0.439243
$425$	−3.10920	−0.150818
$426$	34.9140	1.69159
$427$	46.8394	2.26672
$428$	−38.5434	−1.86307
$429$	−16.4905	−0.796169
$430$	−0.373284	−0.0180014
$431$	−2.37582	−0.114439	−0.0572195	−	0.998362i	$-0.518223\pi$
$431$	−2.37582	−0.114439	−0.0572195	+	0.998362i	$0.518223\pi$
$432$	44.0308	2.11843
$433$	31.3903	1.50852	0.754262	−	0.656574i	$-0.227995\pi$
$433$	31.3903	1.50852	0.754262	+	0.656574i	$0.227995\pi$
$434$	−45.8555	−2.20113
$435$	−7.65906	−0.367224
$436$	−19.6447	−0.940808
$437$	−34.4145	−1.64627
$438$	64.0917	3.06242
$439$	−8.32745	−0.397448	−0.198724	−	0.980056i	$-0.563680\pi$
$439$	−8.32745	−0.397448	−0.198724	+	0.980056i	$0.563680\pi$
$440$	−2.31451	−0.110340
$441$	128.051	6.09764
$442$	14.9048	0.708950
$443$	−27.1908	−1.29188	−0.645938	−	0.763390i	$-0.723534\pi$
$443$	−27.1908	−1.29188	−0.645938	+	0.763390i	$0.723534\pi$
$444$	67.5184	3.20428
$445$	1.00000	0.0474045
$446$	−18.5808	−0.879827
$447$	44.8890	2.12318
$448$	54.7919	2.58867
$449$	−20.2755	−0.956861	−0.478430	−	0.878126i	$-0.658794\pi$
$449$	−20.2755	−0.956861	−0.478430	+	0.878126i	$0.658794\pi$
$450$	16.5921	0.782161
$451$	0.616176	0.0290146
$452$	−5.01059	−0.235679
$453$	−73.6512	−3.46044
$454$	−46.3195	−2.17388
$455$	10.9305	0.512430
$456$	20.4146	0.956000
$457$	−27.8204	−1.30138	−0.650691	−	0.759343i	$-0.725521\pi$
$457$	−27.8204	−1.30138	−0.650691	+	0.759343i	$0.725521\pi$
$458$	−42.4370	−1.98295
$459$	49.3944	2.30553
$460$	14.4436	0.673437
$461$	3.25129	0.151428	0.0757138	−	0.997130i	$-0.475876\pi$
$461$	3.25129	0.151428	0.0757138	+	0.997130i	$0.475876\pi$
$462$	−74.6951	−3.47513
$463$	−13.8292	−0.642699	−0.321349	−	0.946961i	$-0.604136\pi$
$463$	−13.8292	−0.642699	−0.321349	+	0.946961i	$0.604136\pi$
$464$	−6.45107	−0.299483
$465$	−14.7294	−0.683059
$466$	32.4930	1.50521
$467$	−19.5627	−0.905253	−0.452626	−	0.891700i	$-0.649513\pi$
$467$	−19.5627	−0.905253	−0.452626	+	0.891700i	$0.649513\pi$
$468$	−44.1316	−2.03998
$469$	−38.9986	−1.80079
$470$	4.06660	0.187578
$471$	−38.6202	−1.77952
$472$	5.63692	0.259461
$473$	0.390233	0.0179429
$474$	12.4812	0.573283
$475$	−5.93952	−0.272524
$476$	37.4588	1.71692
$477$	67.7820	3.10352
$478$	15.4136	0.705003
$479$	26.2070	1.19743	0.598714	−	0.800963i	$-0.295679\pi$
$479$	26.2070	1.19743	0.598714	+	0.800963i	$0.295679\pi$
$480$	26.2053	1.19610
$481$	18.6160	0.848815
$482$	8.53968	0.388972
$483$	92.1473	4.19285
$484$	−15.1810	−0.690046
$485$	−5.52008	−0.250654
$486$	−99.7982	−4.52694
$487$	28.8600	1.30777	0.653887	−	0.756592i	$-0.273137\pi$
$487$	28.8600	1.30777	0.653887	+	0.756592i	$0.273137\pi$
$488$	−10.1230	−0.458246
$489$	−4.92526	−0.222728
$490$	34.6733	1.56638
$491$	−27.5737	−1.24438	−0.622192	−	0.782865i	$-0.713758\pi$
$491$	−27.5737	−1.24438	−0.622192	+	0.782865i	$0.713758\pi$
$492$	2.28097	0.102834
$493$	−7.23690	−0.325933
$494$	28.4727	1.28105
$495$	−17.3455	−0.779621
$496$	−12.4063	−0.557058
$497$	24.1930	1.08521
$498$	−96.8648	−4.34062
$499$	32.9455	1.47484	0.737422	−	0.675432i	$-0.236043\pi$
$499$	32.9455	1.47484	0.737422	+	0.675432i	$0.236043\pi$
$500$	2.49279	0.111481
$501$	1.82906	0.0817165
$502$	−14.8248	−0.661665
$503$	−25.6894	−1.14543	−0.572717	−	0.819753i	$-0.694110\pi$
$503$	−25.6894	−1.14543	−0.572717	+	0.819753i	$0.694110\pi$
$504$	−39.5168	−1.76022
$505$	−12.8409	−0.571414
$506$	−27.2139	−1.20980
$507$	25.9464	1.15232
$508$	−29.9527	−1.32894
$509$	27.1811	1.20478	0.602390	−	0.798202i	$-0.294215\pi$
$509$	27.1811	1.20478	0.602390	+	0.798202i	$0.294215\pi$
$510$	21.6860	0.960270
$511$	44.4112	1.96464
$512$	27.8622	1.23135
$513$	94.3582	4.16602
$514$	−41.0321	−1.80985
$515$	−11.7697	−0.518635
$516$	1.44457	0.0635937
$517$	−4.25124	−0.186969
$518$	84.3225	3.70492
$519$	−46.1902	−2.02752
$520$	−2.36231	−0.103594
$521$	0.901023	0.0394745	0.0197373	−	0.999805i	$-0.493717\pi$
$521$	0.901023	0.0394745	0.0197373	+	0.999805i	$0.493717\pi$
$522$	38.6194	1.69033
$523$	0.0126235	0.000551986 0	0.000275993	−	1.00000i	$-0.499912\pi$
$523$	0.0126235	0.000551986 0	0.000275993		1.00000i	$0.499912\pi$
$524$	−51.3678	−2.24401
$525$	15.9035	0.694084
$526$	−12.1549	−0.529978
$527$	−13.9175	−0.606257
$528$	−20.2089	−0.879478
$529$	10.5723	0.459665
$530$	18.3539	0.797242
$531$	42.2444	1.83325
$532$	71.5577	3.10242
$533$	0.628902	0.0272408
$534$	−6.97477	−0.301828
$535$	−15.4620	−0.668480
$536$	8.42842	0.364052
$537$	−30.3578	−1.31004
$538$	10.7618	0.463976
$539$	−36.2476	−1.56129
$540$	−39.6017	−1.70419
$541$	9.28499	0.399193	0.199596	−	0.979878i	$-0.436037\pi$
$541$	9.28499	0.399193	0.199596	+	0.979878i	$0.436037\pi$
$542$	31.5630	1.35575
$543$	34.4536	1.47855
$544$	24.7609	1.06162
$545$	−7.88060	−0.337568
$546$	−76.2378	−3.26267
$547$	−12.6475	−0.540770	−0.270385	−	0.962752i	$-0.587151\pi$
$547$	−12.6475	−0.540770	−0.270385	+	0.962752i	$0.587151\pi$
$548$	27.2633	1.16463
$549$	−75.8639	−3.23779
$550$	−4.69678	−0.200271
$551$	−13.8247	−0.588951
$552$	−19.9150	−0.847638
$553$	8.64865	0.367778
$554$	−65.0576	−2.76403
$555$	27.0855	1.14972
$556$	−1.07707	−0.0456777
$557$	−1.89455	−0.0802747	−0.0401374	−	0.999194i	$-0.512780\pi$
$557$	−1.89455	−0.0802747	−0.0401374	+	0.999194i	$0.512780\pi$
$558$	74.2704	3.14411
$559$	0.398292	0.0168460
$560$	13.3952	0.566050
$561$	−22.6706	−0.957153
$562$	41.5406	1.75229
$563$	−39.9171	−1.68231	−0.841153	−	0.540797i	$-0.818123\pi$
$563$	−39.9171	−1.68231	−0.841153	+	0.540797i	$0.818123\pi$
$564$	−15.7373	−0.662661
$565$	−2.01004	−0.0845629
$566$	68.1749	2.86561
$567$	−139.153	−5.84389
$568$	−5.22863	−0.219388
$569$	−12.8214	−0.537499	−0.268750	−	0.963210i	$-0.586610\pi$
$569$	−12.8214	−0.537499	−0.268750	+	0.963210i	$0.586610\pi$
$570$	41.4268	1.73518
$571$	−24.0157	−1.00503	−0.502513	−	0.864570i	$-0.667591\pi$
$571$	−24.0157	−1.00503	−0.502513	+	0.864570i	$0.667591\pi$
$572$	12.4924	0.522335
$573$	15.5942	0.651457
$574$	2.84866	0.118901
$575$	5.79416	0.241633
$576$	−88.7443	−3.69768
$577$	−28.5046	−1.18666	−0.593331	−	0.804959i	$-0.702187\pi$
$577$	−28.5046	−1.18666	−0.593331	+	0.804959i	$0.702187\pi$
$578$	−15.5429	−0.646499
$579$	−4.88935	−0.203195
$580$	5.80215	0.240921
$581$	−67.1207	−2.78464
$582$	38.5013	1.59593
$583$	−19.1872	−0.794653
$584$	−9.59820	−0.397176
$585$	−17.7037	−0.731959
$586$	−48.2361	−1.99261
$587$	−2.40697	−0.0993462	−0.0496731	−	0.998766i	$-0.515818\pi$
$587$	−2.40697	−0.0993462	−0.0496731	+	0.998766i	$0.515818\pi$
$588$	−134.182	−5.53357
$589$	−26.5867	−1.09549
$590$	11.4389	0.470930
$591$	14.8508	0.610879
$592$	22.8136	0.937633
$593$	10.4715	0.430013	0.215006	−	0.976613i	$-0.431023\pi$
$593$	10.4715	0.430013	0.215006	+	0.976613i	$0.431023\pi$
$594$	74.6154	3.06151
$595$	15.0269	0.616042
$596$	−34.0058	−1.39293
$597$	−23.9563	−0.980466
$598$	−27.7759	−1.13584
$599$	9.28858	0.379521	0.189760	−	0.981830i	$-0.439229\pi$
$599$	9.28858	0.379521	0.189760	+	0.981830i	$0.439229\pi$
$600$	−3.43708	−0.140318
$601$	−18.0877	−0.737814	−0.368907	−	0.929466i	$-0.620268\pi$
$601$	−18.0877	−0.737814	−0.368907	+	0.929466i	$0.620268\pi$
$602$	1.80410	0.0735295
$603$	63.1645	2.57226
$604$	55.7948	2.27026
$605$	−6.08997	−0.247593
$606$	89.5625	3.63823
$607$	−4.08445	−0.165783	−0.0828914	−	0.996559i	$-0.526415\pi$
$607$	−4.08445	−0.165783	−0.0828914	+	0.996559i	$0.526415\pi$
$608$	47.3008	1.91830
$609$	37.0165	1.49998
$610$	−20.5423	−0.831733
$611$	−4.33904	−0.175539
$612$	−60.6706	−2.45247
$613$	20.2800	0.819101	0.409551	−	0.912287i	$-0.365685\pi$
$613$	20.2800	0.819101	0.409551	+	0.912287i	$0.365685\pi$
$614$	1.87203	0.0755490
$615$	0.915028	0.0368975
$616$	11.1861	0.450702
$617$	3.23902	0.130398	0.0651990	−	0.997872i	$-0.479232\pi$
$617$	3.23902	0.130398	0.0651990	+	0.997872i	$0.479232\pi$
$618$	82.0910	3.30218
$619$	2.62101	0.105347	0.0526737	−	0.998612i	$-0.483226\pi$
$619$	2.62101	0.105347	0.0526737	+	0.998612i	$0.483226\pi$
$620$	11.1583	0.448129
$621$	−92.0490	−3.69380
$622$	−6.32330	−0.253541
$623$	−4.83304	−0.193632
$624$	−20.6263	−0.825711
$625$	1.00000	0.0400000
$626$	12.8843	0.514960
$627$	−43.3077	−1.72954
$628$	29.2569	1.16748
$629$	25.5926	1.02044
$630$	−80.1904	−3.19486
$631$	−4.84085	−0.192711	−0.0963557	−	0.995347i	$-0.530719\pi$
$631$	−4.84085	−0.192711	−0.0963557	+	0.995347i	$0.530719\pi$
$632$	−1.86916	−0.0743511
$633$	−46.0850	−1.83171
$634$	4.30278	0.170885
$635$	−12.0158	−0.476831
$636$	−71.0276	−2.81643
$637$	−36.9962	−1.46584
$638$	−10.9321	−0.432806
$639$	−39.1845	−1.55012
$640$	−8.10252	−0.320280
$641$	−41.4377	−1.63669	−0.818344	−	0.574728i	$-0.805108\pi$
$641$	−41.4377	−1.63669	−0.818344	+	0.574728i	$0.805108\pi$
$642$	107.844	4.25625
$643$	−26.2789	−1.03634	−0.518168	−	0.855279i	$-0.673386\pi$
$643$	−26.2789	−1.03634	−0.518168	+	0.855279i	$0.673386\pi$
$644$	−69.8065	−2.75076
$645$	0.579500	0.0228178
$646$	39.1434	1.54007
$647$	32.8680	1.29218	0.646088	−	0.763263i	$-0.276404\pi$
$647$	32.8680	1.29218	0.646088	+	0.763263i	$0.276404\pi$
$648$	30.0740	1.18142
$649$	−11.9582	−0.469401
$650$	−4.79378	−0.188028
$651$	71.1877	2.79007
$652$	3.73115	0.146123
$653$	34.0840	1.33381	0.666904	−	0.745143i	$-0.267619\pi$
$653$	34.0840	1.33381	0.666904	+	0.745143i	$0.267619\pi$
$654$	54.9653	2.14931
$655$	−20.6066	−0.805165
$656$	0.770710	0.0300912
$657$	−71.9311	−2.80630
$658$	−19.6540	−0.766195
$659$	−22.3303	−0.869864	−0.434932	−	0.900463i	$-0.643228\pi$
$659$	−22.3303	−0.869864	−0.434932	+	0.900463i	$0.643228\pi$
$660$	18.1760	0.707501
$661$	−4.90059	−0.190611	−0.0953055	−	0.995448i	$-0.530383\pi$
$661$	−4.90059	−0.190611	−0.0953055	+	0.995448i	$0.530383\pi$
$662$	−26.2830	−1.02152
$663$	−23.1388	−0.898637
$664$	14.5062	0.562950
$665$	28.7059	1.11317
$666$	−136.574	−5.29213
$667$	13.4863	0.522193
$668$	−1.38561	−0.0536110
$669$	28.8455	1.11523
$670$	17.1036	0.660768
$671$	21.4750	0.829032
$672$	−126.651	−4.88568
$673$	17.8142	0.686689	0.343344	−	0.939210i	$-0.388440\pi$
$673$	17.8142	0.686689	0.343344	+	0.939210i	$0.388440\pi$
$674$	57.5318	2.21604
$675$	−15.8865	−0.611472
$676$	−19.6558	−0.755992
$677$	−2.59037	−0.0995561	−0.0497780	−	0.998760i	$-0.515851\pi$
$677$	−2.59037	−0.0995561	−0.0497780	+	0.998760i	$0.515851\pi$
$678$	14.0195	0.538417
$679$	26.6788	1.02384
$680$	−3.24763	−0.124541
$681$	71.9081	2.75552
$682$	−21.0239	−0.805046
$683$	24.9388	0.954257	0.477128	−	0.878834i	$-0.341678\pi$
$683$	24.9388	0.954257	0.477128	+	0.878834i	$0.341678\pi$
$684$	−115.899	−4.43152
$685$	10.9369	0.417876
$686$	−95.8678	−3.66025
$687$	65.8808	2.51351
$688$	0.488101	0.0186087
$689$	−19.5835	−0.746072
$690$	−40.4129	−1.53849
$691$	−29.1377	−1.10845	−0.554226	−	0.832366i	$-0.686986\pi$
$691$	−29.1377	−1.10845	−0.554226	+	0.832366i	$0.686986\pi$
$692$	34.9915	1.33018
$693$	83.8314	3.18449
$694$	−36.1682	−1.37293
$695$	−0.432073	−0.0163894
$696$	−8.00005	−0.303241
$697$	0.864593	0.0327488
$698$	53.6893	2.03217
$699$	−50.4433	−1.90794
$700$	−12.0477	−0.455362
$701$	−37.5559	−1.41847	−0.709233	−	0.704974i	$-0.750958\pi$
$701$	−37.5559	−1.41847	−0.709233	+	0.704974i	$0.750958\pi$
$702$	76.1565	2.87434
$703$	48.8896	1.84391
$704$	25.1211	0.946786
$705$	−6.31314	−0.237767
$706$	32.5813	1.22621
$707$	62.0607	2.33403
$708$	−44.2672	−1.66366
$709$	−16.0916	−0.604334	−0.302167	−	0.953255i	$-0.597710\pi$
$709$	−16.0916	−0.604334	−0.302167	+	0.953255i	$0.597710\pi$
$710$	−10.6103	−0.398198
$711$	−14.0079	−0.525337
$712$	1.04452	0.0391451
$713$	25.9360	0.971312
$714$	−104.809	−3.92238
$715$	5.01144	0.187417
$716$	22.9977	0.859464
$717$	−23.9287	−0.893633
$718$	−10.1995	−0.380644
$719$	−3.86290	−0.144062	−0.0720310	−	0.997402i	$-0.522948\pi$
$719$	−3.86290	−0.144062	−0.0720310	+	0.997402i	$0.522948\pi$
$720$	−21.6957	−0.808549
$721$	56.8834	2.11845
$722$	34.5029	1.28406
$723$	−13.2573	−0.493044
$724$	−26.1005	−0.970017
$725$	2.32757	0.0864439
$726$	42.4761	1.57644
$727$	−10.4042	−0.385871	−0.192935	−	0.981211i	$-0.561801\pi$
$727$	−10.4042	−0.385871	−0.192935	+	0.981211i	$0.561801\pi$
$728$	11.4172	0.423148
$729$	68.5541	2.53904
$730$	−19.4774	−0.720890
$731$	0.547558	0.0202522
$732$	79.4965	2.93827
$733$	−16.9908	−0.627569	−0.313784	−	0.949494i	$-0.601597\pi$
$733$	−16.9908	−0.627569	−0.313784	+	0.949494i	$0.601597\pi$
$734$	49.6587	1.83293
$735$	−53.8281	−1.98548
$736$	−46.1432	−1.70086
$737$	−17.8801	−0.658623
$738$	−4.61387	−0.169839
$739$	10.1630	0.373854	0.186927	−	0.982374i	$-0.440147\pi$
$739$	10.1630	0.373854	0.186927	+	0.982374i	$0.440147\pi$
$740$	−20.5187	−0.754283
$741$	−44.2021	−1.62381
$742$	−88.7050	−3.25646
$743$	−25.6666	−0.941617	−0.470809	−	0.882235i	$-0.656038\pi$
$743$	−25.6666	−0.941617	−0.470809	+	0.882235i	$0.656038\pi$
$744$	−15.3852	−0.564048
$745$	−13.6417	−0.499793
$746$	22.2318	0.813964
$747$	108.713	3.97759
$748$	17.1742	0.627950
$749$	74.7284	2.73051
$750$	−6.97477	−0.254682
$751$	−1.88134	−0.0686510	−0.0343255	−	0.999411i	$-0.510928\pi$
$751$	−1.88134	−0.0686510	−0.0343255	+	0.999411i	$0.510928\pi$
$752$	−5.31744	−0.193907
$753$	23.0146	0.838699
$754$	−11.1579	−0.406346
$755$	22.3825	0.814582
$756$	191.397	6.96102
$757$	11.6427	0.423161	0.211580	−	0.977361i	$-0.432139\pi$
$757$	11.6427	0.423161	0.211580	+	0.977361i	$0.432139\pi$
$758$	−43.7691	−1.58977
$759$	42.2478	1.53350
$760$	−6.20396	−0.225041
$761$	20.9113	0.758035	0.379017	−	0.925390i	$-0.376262\pi$
$761$	20.9113	0.758035	0.379017	+	0.925390i	$0.376262\pi$
$762$	83.8071	3.03601
$763$	38.0872	1.37885
$764$	−11.8134	−0.427395
$765$	−24.3385	−0.879959
$766$	25.1236	0.907753
$767$	−12.2052	−0.440704
$768$	−18.0970	−0.653018
$769$	−21.0699	−0.759800	−0.379900	−	0.925028i	$-0.624042\pi$
$769$	−21.0699	−0.759800	−0.379900	+	0.925028i	$0.624042\pi$
$770$	22.6997	0.818040
$771$	63.6997	2.29409
$772$	3.70395	0.133308
$773$	4.26541	0.153416	0.0767080	−	0.997054i	$-0.475559\pi$
$773$	4.26541	0.153416	0.0767080	+	0.997054i	$0.475559\pi$
$774$	−2.92203	−0.105030
$775$	4.47624	0.160791
$776$	−5.76584	−0.206982
$777$	−130.905	−4.69620
$778$	−25.0480	−0.898015
$779$	1.65163	0.0591759
$780$	18.5514	0.664248
$781$	11.0921	0.396905
$782$	−38.1854	−1.36551
$783$	−36.9770	−1.32145
$784$	−45.3383	−1.61923
$785$	11.7366	0.418898
$786$	143.726	5.12654
$787$	46.9571	1.67384	0.836920	−	0.547325i	$-0.184354\pi$
$787$	46.9571	1.67384	0.836920	+	0.547325i	$0.184354\pi$
$788$	−11.2503	−0.400774
$789$	18.8697	0.671778
$790$	−3.79303	−0.134950
$791$	9.71458	0.345411
$792$	−18.1177	−0.643786
$793$	21.9185	0.778349
$794$	77.3809	2.74615
$795$	−28.4932	−1.01055
$796$	18.1482	0.643245
$797$	−36.9401	−1.30849	−0.654243	−	0.756284i	$-0.727013\pi$
$797$	−36.9401	−1.30849	−0.654243	+	0.756284i	$0.727013\pi$
$798$	−200.217	−7.08761
$799$	−5.96517	−0.211033
$800$	−7.96375	−0.281561
$801$	7.82788	0.276585
$802$	3.07813	0.108692
$803$	20.3617	0.718549
$804$	−66.1890	−2.33431
$805$	−28.0034	−0.986990
$806$	−21.4581	−0.755829
$807$	−16.7071	−0.588116
$808$	−13.4126	−0.471855
$809$	17.3676	0.610613	0.305306	−	0.952254i	$-0.401241\pi$
$809$	17.3676	0.610613	0.305306	+	0.952254i	$0.401241\pi$
$810$	61.0283	2.14432
$811$	−44.5363	−1.56388	−0.781941	−	0.623352i	$-0.785770\pi$
$811$	−44.5363	−1.56388	−0.781941	+	0.623352i	$0.785770\pi$
$812$	−28.0420	−0.984081
$813$	−48.9996	−1.71849
$814$	38.6603	1.35504
$815$	1.49678	0.0524299
$816$	−28.3563	−0.992668
$817$	1.04600	0.0365950
$818$	68.6717	2.40105
$819$	85.5628	2.98980
$820$	−0.693183	−0.0242070
$821$	−17.7581	−0.619761	−0.309880	−	0.950776i	$-0.600289\pi$
$821$	−17.7581	−0.619761	−0.309880	+	0.950776i	$0.600289\pi$
$822$	−76.2821	−2.66064
$823$	−27.7787	−0.968304	−0.484152	−	0.874984i	$-0.660872\pi$
$823$	−27.7787	−0.968304	−0.484152	+	0.874984i	$0.660872\pi$
$824$	−12.2937	−0.428272
$825$	7.29145	0.253856
$826$	−55.2844	−1.92359
$827$	−28.8725	−1.00400	−0.501998	−	0.864869i	$-0.667401\pi$
$827$	−28.8725	−1.00400	−0.501998	+	0.864869i	$0.667401\pi$
$828$	113.063	3.92921
$829$	41.4440	1.43941	0.719704	−	0.694281i	$-0.244277\pi$
$829$	41.4440	1.43941	0.719704	+	0.694281i	$0.244277\pi$
$830$	29.4371	1.02178
$831$	100.998	3.50357
$832$	25.6399	0.888904
$833$	−50.8611	−1.76223
$834$	3.01361	0.104353
$835$	−0.555849	−0.0192360
$836$	32.8079	1.13468
$837$	−71.1118	−2.45798
$838$	−22.0507	−0.761730
$839$	−15.0790	−0.520583	−0.260292	−	0.965530i	$-0.583819\pi$
$839$	−15.0790	−0.520583	−0.260292	+	0.965530i	$0.583819\pi$
$840$	16.6115	0.573152
$841$	−23.5824	−0.813186
$842$	2.71948	0.0937193
$843$	−64.4892	−2.22113
$844$	34.9118	1.20172
$845$	−7.88506	−0.271254
$846$	31.8329	1.09444
$847$	29.4331	1.01133
$848$	−23.9993	−0.824139
$849$	−105.837	−3.63233
$850$	−6.59033	−0.226046
$851$	−47.6931	−1.63490
$852$	41.0608	1.40672
$853$	36.1724	1.23852	0.619260	−	0.785186i	$-0.287433\pi$
$853$	36.1724	1.23852	0.619260	+	0.785186i	$0.287433\pi$
$854$	99.2816	3.39735
$855$	−46.4939	−1.59006
$856$	−16.1504	−0.552009
$857$	14.1138	0.482117	0.241058	−	0.970511i	$-0.422505\pi$
$857$	14.1138	0.482117	0.241058	+	0.970511i	$0.422505\pi$
$858$	−34.9536	−1.19330
$859$	−3.04904	−0.104032	−0.0520159	−	0.998646i	$-0.516565\pi$
$859$	−3.04904	−0.104032	−0.0520159	+	0.998646i	$0.516565\pi$
$860$	−0.439002	−0.0149699
$861$	−4.42237	−0.150714
$862$	−5.03582	−0.171521
$863$	−44.3700	−1.51037	−0.755187	−	0.655510i	$-0.772454\pi$
$863$	−44.3700	−1.51037	−0.755187	+	0.655510i	$0.772454\pi$
$864$	126.516	4.30417
$865$	14.0371	0.477276
$866$	66.5356	2.26097
$867$	24.1293	0.819475
$868$	−53.9285	−1.83045
$869$	3.96525	0.134512
$870$	−16.2343	−0.550394
$871$	−18.2494	−0.618358
$872$	−8.23146	−0.278752
$873$	−43.2105	−1.46246
$874$	−72.9457	−2.46742
$875$	−4.83304	−0.163386
$876$	75.3754	2.54670
$877$	−39.0474	−1.31854	−0.659268	−	0.751908i	$-0.729134\pi$
$877$	−39.0474	−1.31854	−0.659268	+	0.751908i	$0.729134\pi$
$878$	−17.6510	−0.595693
$879$	74.8834	2.52575
$880$	6.14144	0.207028
$881$	8.96712	0.302110	0.151055	−	0.988525i	$-0.451733\pi$
$881$	8.96712	0.302110	0.151055	+	0.988525i	$0.451733\pi$
$882$	271.418	9.13913
$883$	6.30062	0.212033	0.106016	−	0.994364i	$-0.466190\pi$
$883$	6.30062	0.212033	0.106016	+	0.994364i	$0.466190\pi$
$884$	17.5289	0.589560
$885$	−17.7581	−0.596932
$886$	−57.6343	−1.93626
$887$	51.6559	1.73443	0.867217	−	0.497931i	$-0.165907\pi$
$887$	51.6559	1.73443	0.867217	+	0.497931i	$0.165907\pi$
$888$	28.2914	0.949397
$889$	58.0726	1.94769
$890$	2.11962	0.0710498
$891$	−63.7993	−2.13736
$892$	−21.8520	−0.731660
$893$	−11.3953	−0.381329
$894$	95.1476	3.18221
$895$	9.22570	0.308381
$896$	39.1598	1.30824
$897$	43.1204	1.43975
$898$	−42.9764	−1.43414
$899$	10.4188	0.347486
$900$	19.5133	0.650442
$901$	−26.9227	−0.896926
$902$	1.30606	0.0434870
$903$	−2.80075	−0.0932030
$904$	−2.09953	−0.0698292
$905$	−10.4704	−0.348048
$906$	−156.113	−5.18649
$907$	−30.5492	−1.01437	−0.507185	−	0.861837i	$-0.669314\pi$
$907$	−30.5492	−1.01437	−0.507185	+	0.861837i	$0.669314\pi$
$908$	−54.4742	−1.80779
$909$	−100.517	−3.33395
$910$	23.1685	0.768029
$911$	13.9379	0.461784	0.230892	−	0.972979i	$-0.425836\pi$
$911$	13.9379	0.461784	0.230892	+	0.972979i	$0.425836\pi$
$912$	−54.1691	−1.79372
$913$	−30.7736	−1.01846
$914$	−58.9686	−1.95051
$915$	31.8906	1.05427
$916$	−49.9082	−1.64901
$917$	99.5923	3.28883
$918$	104.697	3.45553
$919$	21.5909	0.712217	0.356108	−	0.934445i	$-0.384103\pi$
$919$	21.5909	0.712217	0.356108	+	0.934445i	$0.384103\pi$
$920$	6.05213	0.199533
$921$	−2.90621	−0.0957628
$922$	6.89150	0.226959
$923$	11.3211	0.372640
$924$	−87.8454	−2.88990
$925$	−8.23124	−0.270641
$926$	−29.3127	−0.963275
$927$	−92.1319	−3.02601
$928$	−18.5362	−0.608481
$929$	−34.8126	−1.14216	−0.571082	−	0.820893i	$-0.693476\pi$
$929$	−34.8126	−1.14216	−0.571082	+	0.820893i	$0.693476\pi$
$930$	−31.2207	−1.02377
$931$	−97.1601	−3.18429
$932$	38.2135	1.25172
$933$	9.81653	0.321379
$934$	−41.4654	−1.35679
$935$	6.88955	0.225312
$936$	−18.4919	−0.604427
$937$	29.3529	0.958918	0.479459	−	0.877564i	$-0.340833\pi$
$937$	29.3529	0.958918	0.479459	+	0.877564i	$0.340833\pi$
$938$	−82.6622	−2.69901
$939$	−20.0021	−0.652743
$940$	4.78255	0.155989
$941$	39.9726	1.30307	0.651534	−	0.758619i	$-0.274126\pi$
$941$	39.9726	1.30307	0.651534	+	0.758619i	$0.274126\pi$
$942$	−81.8601	−2.66715
$943$	−1.61121	−0.0524683
$944$	−14.9573	−0.486819
$945$	76.7801	2.49766
$946$	0.827145	0.0268928
$947$	57.0642	1.85434	0.927168	−	0.374646i	$-0.122236\pi$
$947$	57.0642	1.85434	0.927168	+	0.374646i	$0.122236\pi$
$948$	14.6786	0.476739
$949$	20.7822	0.674620
$950$	−12.5895	−0.408458
$951$	−6.67980	−0.216607
$952$	15.6959	0.508707
$953$	47.9488	1.55321	0.776607	−	0.629985i	$-0.216939\pi$
$953$	47.9488	1.55321	0.776607	+	0.629985i	$0.216939\pi$
$954$	143.672	4.65155
$955$	−4.73905	−0.153352
$956$	18.1273	0.586278
$957$	16.9714	0.548607
$958$	55.5488	1.79470
$959$	−52.8583	−1.70688
$960$	37.3051	1.20402
$961$	−10.9633	−0.353655
$962$	39.4587	1.27220
$963$	−121.035	−3.90029
$964$	10.0431	0.323467
$965$	1.48587	0.0478317
$966$	195.317	6.28423
$967$	−22.9873	−0.739222	−0.369611	−	0.929187i	$-0.620509\pi$
$967$	−22.9873	−0.739222	−0.369611	+	0.929187i	$0.620509\pi$
$968$	−6.36111	−0.204454
$969$	−60.7676	−1.95214
$970$	−11.7005	−0.375679
$971$	4.86938	0.156266	0.0781330	−	0.996943i	$-0.475104\pi$
$971$	4.86938	0.156266	0.0781330	+	0.996943i	$0.475104\pi$
$972$	−117.368	−3.76458
$973$	2.08822	0.0669454
$974$	61.1723	1.96009
$975$	7.44204	0.238336
$976$	26.8608	0.859794
$977$	5.28623	0.169121	0.0845607	−	0.996418i	$-0.473051\pi$
$977$	5.28623	0.169121	0.0845607	+	0.996418i	$0.473051\pi$
$978$	−10.4397	−0.333824
$979$	−2.21586	−0.0708191
$980$	40.7777	1.30259
$981$	−61.6884	−1.96956
$982$	−58.4458	−1.86508
$983$	21.4830	0.685202	0.342601	−	0.939481i	$-0.388692\pi$
$983$	21.4830	0.685202	0.342601	+	0.939481i	$0.388692\pi$
$984$	0.955767	0.0304687
$985$	−4.51312	−0.143800
$986$	−15.3395	−0.488508
$987$	30.5117	0.971197
$988$	33.4855	1.06532
$989$	−1.02040	−0.0324470
$990$	−36.7658	−1.16849
$991$	−39.9796	−1.26999	−0.634997	−	0.772514i	$-0.718999\pi$
$991$	−39.9796	−1.26999	−0.634997	+	0.772514i	$0.718999\pi$
$992$	−35.6476	−1.13181
$993$	40.8027	1.29484
$994$	51.2800	1.62650
$995$	7.28028	0.230800
$996$	−113.918	−3.60964
$997$	−40.8157	−1.29265	−0.646323	−	0.763064i	$-0.723694\pi$
$997$	−40.8157	−1.29265	−0.646323	+	0.763064i	$0.723694\pi$
$998$	69.8320	2.21049
$999$	130.766	4.13724

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	445.2.a.f.1.7	✓	7
3.2	odd	2		4005.2.a.o.1.1		7
4.3	odd	2		7120.2.a.bj.1.7		7
5.4	even	2		2225.2.a.k.1.1		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
445.2.a.f.1.7	✓	7	1.1	even	1	trivial
2225.2.a.k.1.1		7	5.4	even	2
4005.2.a.o.1.1		7	3.2	odd	2
7120.2.a.bj.1.7		7	4.3	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:	\( N \)	\(=\)	\( 445 = 5 \cdot 89 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	445.a (trivial)

\(f(q)\)	\(=\)	\(q+2.11962 q^{2} -3.29058 q^{3} +2.49279 q^{4} +1.00000 q^{5} -6.97477 q^{6} -4.83304 q^{7} +1.04452 q^{8} +7.82788 q^{9} +O(q^{10})\)	\(q+2.11962 q^{2} -3.29058 q^{3} +2.49279 q^{4} +1.00000 q^{5} -6.97477 q^{6} -4.83304 q^{7} +1.04452 q^{8} +7.82788 q^{9} +2.11962 q^{10} -2.21586 q^{11} -8.20270 q^{12} -2.26162 q^{13} -10.2442 q^{14} -3.29058 q^{15} -2.77159 q^{16} -3.10920 q^{17} +16.5921 q^{18} -5.93952 q^{19} +2.49279 q^{20} +15.9035 q^{21} -4.69678 q^{22} +5.79416 q^{23} -3.43708 q^{24} +1.00000 q^{25} -4.79378 q^{26} -15.8865 q^{27} -12.0477 q^{28} +2.32757 q^{29} -6.97477 q^{30} +4.47624 q^{31} -7.96375 q^{32} +7.29145 q^{33} -6.59033 q^{34} -4.83304 q^{35} +19.5133 q^{36} -8.23124 q^{37} -12.5895 q^{38} +7.44204 q^{39} +1.04452 q^{40} -0.278075 q^{41} +33.7093 q^{42} -0.176109 q^{43} -5.52366 q^{44} +7.82788 q^{45} +12.2814 q^{46} +1.91855 q^{47} +9.12011 q^{48} +16.3583 q^{49} +2.11962 q^{50} +10.2311 q^{51} -5.63775 q^{52} +8.65904 q^{53} -33.6734 q^{54} -2.21586 q^{55} -5.04821 q^{56} +19.5444 q^{57} +4.93357 q^{58} +5.39666 q^{59} -8.20270 q^{60} -9.69150 q^{61} +9.48792 q^{62} -37.8325 q^{63} -11.3370 q^{64} -2.26162 q^{65} +15.4551 q^{66} +8.06916 q^{67} -7.75058 q^{68} -19.0661 q^{69} -10.2442 q^{70} -5.00576 q^{71} +8.17640 q^{72} -9.18909 q^{73} -17.4471 q^{74} -3.29058 q^{75} -14.8060 q^{76} +10.7093 q^{77} +15.7743 q^{78} -1.78949 q^{79} -2.77159 q^{80} +28.7921 q^{81} -0.589414 q^{82} +13.8879 q^{83} +39.6440 q^{84} -3.10920 q^{85} -0.373284 q^{86} -7.65906 q^{87} -2.31451 q^{88} +1.00000 q^{89} +16.5921 q^{90} +10.9305 q^{91} +14.4436 q^{92} -14.7294 q^{93} +4.06660 q^{94} -5.93952 q^{95} +26.2053 q^{96} -5.52008 q^{97} +34.6733 q^{98} -17.3455 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 4 q^{2} - 8 q^{3} + 8 q^{4} + 7 q^{5} - 2 q^{6} - 16 q^{7} - 12 q^{8} + 11 q^{9}+O(q^{10}) \) 7 * q - 4 * q^2 - 8 * q^3 + 8 * q^4 + 7 * q^5 - 2 * q^6 - 16 * q^7 - 12 * q^8 + 11 * q^9	\( 7 q - 4 q^{2} - 8 q^{3} + 8 q^{4} + 7 q^{5} - 2 q^{6} - 16 q^{7} - 12 q^{8} + 11 q^{9} - 4 q^{10} - 10 q^{11} - 11 q^{12} - 7 q^{13} + 3 q^{14} - 8 q^{15} + 10 q^{16} - 13 q^{17} + 4 q^{18} - 7 q^{19} + 8 q^{20} + 16 q^{21} + 2 q^{22} - 13 q^{23} + 4 q^{24} + 7 q^{25} + q^{26} - 23 q^{27} - 21 q^{28} - 4 q^{29} - 2 q^{30} + q^{31} - 13 q^{32} - 6 q^{33} + 10 q^{34} - 16 q^{35} + 20 q^{36} - 5 q^{37} - 40 q^{38} - 13 q^{39} - 12 q^{40} + 5 q^{41} + 30 q^{42} - 31 q^{43} - 21 q^{44} + 11 q^{45} + 16 q^{46} - 14 q^{47} - 7 q^{48} + 19 q^{49} - 4 q^{50} - q^{51} - 13 q^{53} - 17 q^{54} - 10 q^{55} - q^{56} + 21 q^{57} + 17 q^{58} - 14 q^{59} - 11 q^{60} + 3 q^{61} + 26 q^{62} - 54 q^{63} + 14 q^{64} - 7 q^{65} + 36 q^{66} + q^{67} - 35 q^{68} + 31 q^{69} + 3 q^{70} - 8 q^{71} + 53 q^{72} + 9 q^{73} - 35 q^{74} - 8 q^{75} + 40 q^{76} + 42 q^{77} + 46 q^{78} + 9 q^{79} + 10 q^{80} + 35 q^{81} + 29 q^{82} - 42 q^{83} + 55 q^{84} - 13 q^{85} + 35 q^{86} + 6 q^{87} + 30 q^{88} + 7 q^{89} + 4 q^{90} + 31 q^{91} + 19 q^{92} + 24 q^{93} + 37 q^{94} - 7 q^{95} + 44 q^{96} - 7 q^{97} + 9 q^{98} + 5 q^{99}+O(q^{100}) \) 7 * q - 4 * q^2 - 8 * q^3 + 8 * q^4 + 7 * q^5 - 2 * q^6 - 16 * q^7 - 12 * q^8 + 11 * q^9 - 4 * q^10 - 10 * q^11 - 11 * q^12 - 7 * q^13 + 3 * q^14 - 8 * q^15 + 10 * q^16 - 13 * q^17 + 4 * q^18 - 7 * q^19 + 8 * q^20 + 16 * q^21 + 2 * q^22 - 13 * q^23 + 4 * q^24 + 7 * q^25 + q^26 - 23 * q^27 - 21 * q^28 - 4 * q^29 - 2 * q^30 + q^31 - 13 * q^32 - 6 * q^33 + 10 * q^34 - 16 * q^35 + 20 * q^36 - 5 * q^37 - 40 * q^38 - 13 * q^39 - 12 * q^40 + 5 * q^41 + 30 * q^42 - 31 * q^43 - 21 * q^44 + 11 * q^45 + 16 * q^46 - 14 * q^47 - 7 * q^48 + 19 * q^49 - 4 * q^50 - q^51 - 13 * q^53 - 17 * q^54 - 10 * q^55 - q^56 + 21 * q^57 + 17 * q^58 - 14 * q^59 - 11 * q^60 + 3 * q^61 + 26 * q^62 - 54 * q^63 + 14 * q^64 - 7 * q^65 + 36 * q^66 + q^67 - 35 * q^68 + 31 * q^69 + 3 * q^70 - 8 * q^71 + 53 * q^72 + 9 * q^73 - 35 * q^74 - 8 * q^75 + 40 * q^76 + 42 * q^77 + 46 * q^78 + 9 * q^79 + 10 * q^80 + 35 * q^81 + 29 * q^82 - 42 * q^83 + 55 * q^84 - 13 * q^85 + 35 * q^86 + 6 * q^87 + 30 * q^88 + 7 * q^89 + 4 * q^90 + 31 * q^91 + 19 * q^92 + 24 * q^93 + 37 * q^94 - 7 * q^95 + 44 * q^96 - 7 * q^97 + 9 * q^98 + 5 * q^99

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