LMFDB - Embedded newform 2005.2.a.f.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2005,2,Mod(1,2005)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2005, 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("2005.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2005 = 5 \cdot 401 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2005.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$16.0100056053$
Analytic rank:	$0$
Dimension:	$37$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Character	$\chi$	$=$	2005.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.45894 q^{2} +0.456001 q^{3} +4.04640 q^{4} -1.00000 q^{5} -1.12128 q^{6} +2.81754 q^{7} -5.03197 q^{8} -2.79206 q^{9} +O(q^{10})$	\(q-2.45894 q^{2} +0.456001 q^{3} +4.04640 q^{4} -1.00000 q^{5} -1.12128 q^{6} +2.81754 q^{7} -5.03197 q^{8} -2.79206 q^{9} +2.45894 q^{10} +6.16829 q^{11} +1.84516 q^{12} -4.63807 q^{13} -6.92816 q^{14} -0.456001 q^{15} +4.28053 q^{16} -3.36554 q^{17} +6.86552 q^{18} -1.84724 q^{19} -4.04640 q^{20} +1.28480 q^{21} -15.1675 q^{22} +5.76035 q^{23} -2.29458 q^{24} +1.00000 q^{25} +11.4047 q^{26} -2.64119 q^{27} +11.4009 q^{28} +7.03310 q^{29} +1.12128 q^{30} +1.16232 q^{31} -0.461635 q^{32} +2.81275 q^{33} +8.27566 q^{34} -2.81754 q^{35} -11.2978 q^{36} +0.0653049 q^{37} +4.54226 q^{38} -2.11497 q^{39} +5.03197 q^{40} +1.37585 q^{41} -3.15925 q^{42} -5.29780 q^{43} +24.9593 q^{44} +2.79206 q^{45} -14.1644 q^{46} +0.357747 q^{47} +1.95193 q^{48} +0.938510 q^{49} -2.45894 q^{50} -1.53469 q^{51} -18.7675 q^{52} +4.97683 q^{53} +6.49453 q^{54} -6.16829 q^{55} -14.1778 q^{56} -0.842344 q^{57} -17.2940 q^{58} +2.59693 q^{59} -1.84516 q^{60} +4.56539 q^{61} -2.85807 q^{62} -7.86674 q^{63} -7.42593 q^{64} +4.63807 q^{65} -6.91638 q^{66} +4.65806 q^{67} -13.6183 q^{68} +2.62673 q^{69} +6.92816 q^{70} +5.69482 q^{71} +14.0496 q^{72} +7.61994 q^{73} -0.160581 q^{74} +0.456001 q^{75} -7.47467 q^{76} +17.3794 q^{77} +5.20058 q^{78} +9.48038 q^{79} -4.28053 q^{80} +7.17180 q^{81} -3.38313 q^{82} +5.32643 q^{83} +5.19881 q^{84} +3.36554 q^{85} +13.0270 q^{86} +3.20710 q^{87} -31.0386 q^{88} +0.226950 q^{89} -6.86552 q^{90} -13.0679 q^{91} +23.3087 q^{92} +0.530018 q^{93} -0.879678 q^{94} +1.84724 q^{95} -0.210506 q^{96} -9.53815 q^{97} -2.30774 q^{98} -17.2222 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 37 q + 7 q^{2} + 3 q^{3} + 43 q^{4} - 37 q^{5} + 8 q^{6} - 16 q^{7} + 21 q^{8} + 54 q^{9}+O(q^{10}) $ 37 * q + 7 * q^2 + 3 * q^3 + 43 * q^4 - 37 * q^5 + 8 * q^6 - 16 * q^7 + 21 * q^8 + 54 * q^9	\( 37 q + 7 q^{2} + 3 q^{3} + 43 q^{4} - 37 q^{5} + 8 q^{6} - 16 q^{7} + 21 q^{8} + 54 q^{9} - 7 q^{10} + 42 q^{11} - 13 q^{13} + 14 q^{14} - 3 q^{15} + 63 q^{16} + 18 q^{17} + 22 q^{18} + 22 q^{19} - 43 q^{20} + 16 q^{21} - 10 q^{22} + 23 q^{23} + 23 q^{24} + 37 q^{25} + 21 q^{26} + 3 q^{27} - 18 q^{28} + 33 q^{29} - 8 q^{30} + 11 q^{31} + 54 q^{32} + 2 q^{33} + 8 q^{34} + 16 q^{35} + 91 q^{36} - 11 q^{37} + 29 q^{38} + 25 q^{39} - 21 q^{40} + 24 q^{41} + 4 q^{42} + 25 q^{43} + 84 q^{44} - 54 q^{45} + 31 q^{46} + 7 q^{47} + 4 q^{48} + 45 q^{49} + 7 q^{50} + 94 q^{51} - 43 q^{52} + 49 q^{53} + 38 q^{54} - 42 q^{55} + 46 q^{56} + 6 q^{57} + 15 q^{58} + 69 q^{59} + 9 q^{61} + 17 q^{62} - 38 q^{63} + 107 q^{64} + 13 q^{65} + 74 q^{66} + 13 q^{67} + 86 q^{68} - 14 q^{70} + 51 q^{71} + 81 q^{72} - 47 q^{73} + 79 q^{74} + 3 q^{75} + 59 q^{76} + 2 q^{77} + 20 q^{78} + 67 q^{79} - 63 q^{80} + 125 q^{81} - 24 q^{82} + 80 q^{83} + 50 q^{84} - 18 q^{85} + 69 q^{86} - 32 q^{87} - 12 q^{88} + 34 q^{89} - 22 q^{90} + 39 q^{91} + 85 q^{92} + q^{93} + 12 q^{94} - 22 q^{95} + 77 q^{96} - 14 q^{97} + 40 q^{98} + 133 q^{99}+O(q^{100}) \) 37 * q + 7 * q^2 + 3 * q^3 + 43 * q^4 - 37 * q^5 + 8 * q^6 - 16 * q^7 + 21 * q^8 + 54 * q^9 - 7 * q^10 + 42 * q^11 - 13 * q^13 + 14 * q^14 - 3 * q^15 + 63 * q^16 + 18 * q^17 + 22 * q^18 + 22 * q^19 - 43 * q^20 + 16 * q^21 - 10 * q^22 + 23 * q^23 + 23 * q^24 + 37 * q^25 + 21 * q^26 + 3 * q^27 - 18 * q^28 + 33 * q^29 - 8 * q^30 + 11 * q^31 + 54 * q^32 + 2 * q^33 + 8 * q^34 + 16 * q^35 + 91 * q^36 - 11 * q^37 + 29 * q^38 + 25 * q^39 - 21 * q^40 + 24 * q^41 + 4 * q^42 + 25 * q^43 + 84 * q^44 - 54 * q^45 + 31 * q^46 + 7 * q^47 + 4 * q^48 + 45 * q^49 + 7 * q^50 + 94 * q^51 - 43 * q^52 + 49 * q^53 + 38 * q^54 - 42 * q^55 + 46 * q^56 + 6 * q^57 + 15 * q^58 + 69 * q^59 + 9 * q^61 + 17 * q^62 - 38 * q^63 + 107 * q^64 + 13 * q^65 + 74 * q^66 + 13 * q^67 + 86 * q^68 - 14 * q^70 + 51 * q^71 + 81 * q^72 - 47 * q^73 + 79 * q^74 + 3 * q^75 + 59 * q^76 + 2 * q^77 + 20 * q^78 + 67 * q^79 - 63 * q^80 + 125 * q^81 - 24 * q^82 + 80 * q^83 + 50 * q^84 - 18 * q^85 + 69 * q^86 - 32 * q^87 - 12 * q^88 + 34 * q^89 - 22 * q^90 + 39 * q^91 + 85 * q^92 + q^93 + 12 * q^94 - 22 * q^95 + 77 * q^96 - 14 * q^97 + 40 * q^98 + 133 * 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 $

$2$	−2.45894	−1.73873	−0.869367	−	0.494166i	$-0.835473\pi$
$2$	−2.45894	−1.73873	−0.869367	+	0.494166i	$0.835473\pi$
$3$	0.456001	0.263272	0.131636	−	0.991298i	$-0.457977\pi$
$3$	0.456001	0.263272	0.131636	+	0.991298i	$0.457977\pi$
$4$	4.04640	2.02320
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	−1.12128	−0.457761
$7$	2.81754	1.06493	0.532464	−	0.846452i	$-0.321266\pi$
$7$	2.81754	1.06493	0.532464	+	0.846452i	$0.321266\pi$
$8$	−5.03197	−1.77907
$9$	−2.79206	−0.930688
$10$	2.45894	0.777586
$11$	6.16829	1.85981	0.929904	−	0.367802i	$-0.119889\pi$
$11$	6.16829	1.85981	0.929904	+	0.367802i	$0.119889\pi$
$12$	1.84516	0.532652
$13$	−4.63807	−1.28637	−0.643185	−	0.765711i	$-0.722387\pi$
$13$	−4.63807	−1.28637	−0.643185	+	0.765711i	$0.722387\pi$
$14$	−6.92816	−1.85163
$15$	−0.456001	−0.117739
$16$	4.28053	1.07013
$17$	−3.36554	−0.816263	−0.408131	−	0.912923i	$-0.633819\pi$
$17$	−3.36554	−0.816263	−0.408131	+	0.912923i	$0.633819\pi$
$18$	6.86552	1.61822
$19$	−1.84724	−0.423786	−0.211893	−	0.977293i	$-0.567963\pi$
$19$	−1.84724	−0.423786	−0.211893	+	0.977293i	$0.567963\pi$
$20$	−4.04640	−0.904802
$21$	1.28480	0.280366
$22$	−15.1675	−3.23371
$23$	5.76035	1.20112	0.600558	−	0.799581i	$-0.294945\pi$
$23$	5.76035	1.20112	0.600558	+	0.799581i	$0.294945\pi$
$24$	−2.29458	−0.468380
$25$	1.00000	0.200000
$26$	11.4047	2.23666
$27$	−2.64119	−0.508297
$28$	11.4009	2.15456
$29$	7.03310	1.30601	0.653007	−	0.757352i	$-0.273507\pi$
$29$	7.03310	1.30601	0.653007	+	0.757352i	$0.273507\pi$
$30$	1.12128	0.204717
$31$	1.16232	0.208758	0.104379	−	0.994538i	$-0.466714\pi$
$31$	1.16232	0.208758	0.104379	+	0.994538i	$0.466714\pi$
$32$	−0.461635	−0.0816063
$33$	2.81275	0.489636
$34$	8.27566	1.41926
$35$	−2.81754	−0.476251
$36$	−11.2978	−1.88297
$37$	0.0653049	0.0107361	0.00536803	−	0.999986i	$-0.498291\pi$
$37$	0.0653049	0.0107361	0.00536803	+	0.999986i	$0.498291\pi$
$38$	4.54226	0.736852
$39$	−2.11497	−0.338666
$40$	5.03197	0.795624
$41$	1.37585	0.214871	0.107436	−	0.994212i	$-0.465736\pi$
$41$	1.37585	0.214871	0.107436	+	0.994212i	$0.465736\pi$
$42$	−3.15925	−0.487483
$43$	−5.29780	−0.807907	−0.403954	−	0.914780i	$-0.632364\pi$
$43$	−5.29780	−0.807907	−0.403954	+	0.914780i	$0.632364\pi$
$44$	24.9593	3.76276
$45$	2.79206	0.416216
$46$	−14.1644	−2.08842
$47$	0.357747	0.0521827	0.0260913	−	0.999660i	$-0.491694\pi$
$47$	0.357747	0.0521827	0.0260913	+	0.999660i	$0.491694\pi$
$48$	1.95193	0.281736
$49$	0.938510	0.134073
$50$	−2.45894	−0.347747
$51$	−1.53469	−0.214899
$52$	−18.7675	−2.60258
$53$	4.97683	0.683621	0.341810	−	0.939769i	$-0.388960\pi$
$53$	4.97683	0.683621	0.341810	+	0.939769i	$0.388960\pi$
$54$	6.49453	0.883793
$55$	−6.16829	−0.831732
$56$	−14.1778	−1.89458
$57$	−0.842344	−0.111571
$58$	−17.2940	−2.27081
$59$	2.59693	0.338092	0.169046	−	0.985608i	$-0.445931\pi$
$59$	2.59693	0.338092	0.169046	+	0.985608i	$0.445931\pi$
$60$	−1.84516	−0.238209
$61$	4.56539	0.584539	0.292269	−	0.956336i	$-0.405590\pi$
$61$	4.56539	0.584539	0.292269	+	0.956336i	$0.405590\pi$
$62$	−2.85807	−0.362975
$63$	−7.86674	−0.991116
$64$	−7.42593	−0.928241
$65$	4.63807	0.575282
$66$	−6.91638	−0.851347
$67$	4.65806	0.569073	0.284536	−	0.958665i	$-0.408160\pi$
$67$	4.65806	0.569073	0.284536	+	0.958665i	$0.408160\pi$
$68$	−13.6183	−1.65146
$69$	2.62673	0.316221
$70$	6.92816	0.828073
$71$	5.69482	0.675851	0.337925	−	0.941173i	$-0.390275\pi$
$71$	5.69482	0.675851	0.337925	+	0.941173i	$0.390275\pi$
$72$	14.0496	1.65576
$73$	7.61994	0.891846	0.445923	−	0.895071i	$-0.352875\pi$
$73$	7.61994	0.891846	0.445923	+	0.895071i	$0.352875\pi$
$74$	−0.160581	−0.0186672
$75$	0.456001	0.0526545
$76$	−7.47467	−0.857403
$77$	17.3794	1.98056
$78$	5.20058	0.588850
$79$	9.48038	1.06663	0.533313	−	0.845918i	$-0.320947\pi$
$79$	9.48038	1.06663	0.533313	+	0.845918i	$0.320947\pi$
$80$	−4.28053	−0.478578
$81$	7.17180	0.796867
$82$	−3.38313	−0.373604
$83$	5.32643	0.584652	0.292326	−	0.956319i	$-0.405571\pi$
$83$	5.32643	0.584652	0.292326	+	0.956319i	$0.405571\pi$
$84$	5.19881	0.567237
$85$	3.36554	0.365044
$86$	13.0270	1.40474
$87$	3.20710	0.343837
$88$	−31.0386	−3.30873
$89$	0.226950	0.0240567	0.0120283	−	0.999928i	$-0.496171\pi$
$89$	0.226950	0.0240567	0.0120283	+	0.999928i	$0.496171\pi$
$90$	−6.86552	−0.723689
$91$	−13.0679	−1.36989
$92$	23.3087	2.43010
$93$	0.530018	0.0549603
$94$	−0.879678	−0.0907319
$95$	1.84724	0.189523
$96$	−0.210506	−0.0214847
$97$	−9.53815	−0.968453	−0.484226	−	0.874943i	$-0.660899\pi$
$97$	−9.53815	−0.968453	−0.484226	+	0.874943i	$0.660899\pi$
$98$	−2.30774	−0.233117
$99$	−17.2222	−1.73090
$100$	4.04640	0.404640
$101$	4.70548	0.468213	0.234106	−	0.972211i	$-0.424784\pi$
$101$	4.70548	0.468213	0.234106	+	0.972211i	$0.424784\pi$
$102$	3.77371	0.373653
$103$	−17.8924	−1.76299	−0.881497	−	0.472189i	$-0.843464\pi$
$103$	−17.8924	−1.76299	−0.881497	+	0.472189i	$0.843464\pi$
$104$	23.3386	2.28854
$105$	−1.28480	−0.125384
$106$	−12.2377	−1.18864
$107$	−18.5841	−1.79659	−0.898297	−	0.439389i	$-0.855195\pi$
$107$	−18.5841	−1.79659	−0.898297	+	0.439389i	$0.855195\pi$
$108$	−10.6873	−1.02839
$109$	−8.12841	−0.778560	−0.389280	−	0.921119i	$-0.627276\pi$
$109$	−8.12841	−0.778560	−0.389280	+	0.921119i	$0.627276\pi$
$110$	15.1675	1.44616
$111$	0.0297791	0.00282651
$112$	12.0605	1.13961
$113$	−13.7528	−1.29375	−0.646876	−	0.762596i	$-0.723925\pi$
$113$	−13.7528	−1.29375	−0.646876	+	0.762596i	$0.723925\pi$
$114$	2.07127	0.193993
$115$	−5.76035	−0.537155
$116$	28.4587	2.64233
$117$	12.9498	1.19721
$118$	−6.38571	−0.587852
$119$	−9.48252	−0.869262
$120$	2.29458	0.209466
$121$	27.0478	2.45889
$122$	−11.2260	−1.01636
$123$	0.627388	0.0565696
$124$	4.70320	0.422360
$125$	−1.00000	−0.0894427
$126$	19.3439	1.72329
$127$	−4.54697	−0.403479	−0.201739	−	0.979439i	$-0.564659\pi$
$127$	−4.54697	−0.403479	−0.201739	+	0.979439i	$0.564659\pi$
$128$	19.1832	1.69557
$129$	−2.41580	−0.212700
$130$	−11.4047	−1.00026
$131$	10.0571	0.878692	0.439346	−	0.898318i	$-0.355210\pi$
$131$	10.0571	0.878692	0.439346	+	0.898318i	$0.355210\pi$
$132$	11.3815	0.990631
$133$	−5.20467	−0.451302
$134$	−11.4539	−0.989467
$135$	2.64119	0.227317
$136$	16.9353	1.45219
$137$	6.31720	0.539715	0.269857	−	0.962900i	$-0.413023\pi$
$137$	6.31720	0.539715	0.269857	+	0.962900i	$0.413023\pi$
$138$	−6.45897	−0.549824
$139$	0.371988	0.0315516	0.0157758	−	0.999876i	$-0.494978\pi$
$139$	0.371988	0.0315516	0.0157758	+	0.999876i	$0.494978\pi$
$140$	−11.4009	−0.963549
$141$	0.163133	0.0137383
$142$	−14.0032	−1.17512
$143$	−28.6090	−2.39240
$144$	−11.9515	−0.995959
$145$	−7.03310	−0.584067
$146$	−18.7370	−1.55068
$147$	0.427962	0.0352977
$148$	0.264250	0.0217212
$149$	17.6484	1.44581	0.722906	−	0.690947i	$-0.242806\pi$
$149$	17.6484	1.44581	0.722906	+	0.690947i	$0.242806\pi$
$150$	−1.12128	−0.0915522
$151$	23.1035	1.88014	0.940068	−	0.340986i	$-0.110761\pi$
$151$	23.1035	1.88014	0.940068	+	0.340986i	$0.110761\pi$
$152$	9.29526	0.753945
$153$	9.39679	0.759686
$154$	−42.7349	−3.44367
$155$	−1.16232	−0.0933596
$156$	−8.55799	−0.685188
$157$	7.03561	0.561503	0.280751	−	0.959781i	$-0.409416\pi$
$157$	7.03561	0.561503	0.280751	+	0.959781i	$0.409416\pi$
$158$	−23.3117	−1.85458
$159$	2.26944	0.179978
$160$	0.461635	0.0364954
$161$	16.2300	1.27910
$162$	−17.6351	−1.38554
$163$	8.88489	0.695918	0.347959	−	0.937510i	$-0.386875\pi$
$163$	8.88489	0.695918	0.347959	+	0.937510i	$0.386875\pi$
$164$	5.56722	0.434727
$165$	−2.81275	−0.218972
$166$	−13.0974	−1.01655
$167$	16.4838	1.27555	0.637777	−	0.770221i	$-0.279854\pi$
$167$	16.4838	1.27555	0.637777	+	0.770221i	$0.279854\pi$
$168$	−6.46507	−0.498791
$169$	8.51170	0.654747
$170$	−8.27566	−0.634714
$171$	5.15761	0.394412
$172$	−21.4370	−1.63456
$173$	14.7564	1.12191	0.560955	−	0.827847i	$-0.310434\pi$
$173$	14.7564	1.12191	0.560955	+	0.827847i	$0.310434\pi$
$174$	−7.88608	−0.597842
$175$	2.81754	0.212986
$176$	26.4035	1.99024
$177$	1.18420	0.0890103
$178$	−0.558057	−0.0418282
$179$	−9.84350	−0.735737	−0.367869	−	0.929878i	$-0.619912\pi$
$179$	−9.84350	−0.735737	−0.367869	+	0.929878i	$0.619912\pi$
$180$	11.2978	0.842088
$181$	−4.80283	−0.356991	−0.178496	−	0.983941i	$-0.557123\pi$
$181$	−4.80283	−0.356991	−0.178496	+	0.983941i	$0.557123\pi$
$182$	32.1333	2.38188
$183$	2.08183	0.153893
$184$	−28.9859	−2.13687
$185$	−0.0653049	−0.00480131
$186$	−1.30328	−0.0955614
$187$	−20.7596	−1.51809
$188$	1.44758	0.105576
$189$	−7.44164	−0.541300
$190$	−4.54226	−0.329530
$191$	18.5470	1.34201	0.671007	−	0.741451i	$-0.265862\pi$
$191$	18.5470	1.34201	0.671007	+	0.741451i	$0.265862\pi$
$192$	−3.38623	−0.244380
$193$	11.6925	0.841643	0.420822	−	0.907143i	$-0.361742\pi$
$193$	11.6925	0.841643	0.420822	+	0.907143i	$0.361742\pi$
$194$	23.4538	1.68388
$195$	2.11497	0.151456
$196$	3.79758	0.271256
$197$	0.723682	0.0515602	0.0257801	−	0.999668i	$-0.491793\pi$
$197$	0.723682	0.0515602	0.0257801	+	0.999668i	$0.491793\pi$
$198$	42.3485	3.00958
$199$	−1.00768	−0.0714329	−0.0357164	−	0.999362i	$-0.511371\pi$
$199$	−1.00768	−0.0714329	−0.0357164	+	0.999362i	$0.511371\pi$
$200$	−5.03197	−0.355814
$201$	2.12408	0.149821
$202$	−11.5705	−0.814098
$203$	19.8160	1.39081
$204$	−6.20996	−0.434784
$205$	−1.37585	−0.0960933
$206$	43.9965	3.06538
$207$	−16.0833	−1.11786
$208$	−19.8534	−1.37659
$209$	−11.3943	−0.788161
$210$	3.15925	0.218009
$211$	−7.71494	−0.531118	−0.265559	−	0.964095i	$-0.585557\pi$
$211$	−7.71494	−0.531118	−0.265559	+	0.964095i	$0.585557\pi$
$212$	20.1382	1.38310
$213$	2.59684	0.177933
$214$	45.6972	3.12380
$215$	5.29780	0.361307
$216$	13.2904	0.904295
$217$	3.27487	0.222313
$218$	19.9873	1.35371
$219$	3.47470	0.234799
$220$	−24.9593	−1.68276
$221$	15.6096	1.05002
$222$	−0.0732251	−0.00491455
$223$	14.5923	0.977173	0.488586	−	0.872516i	$-0.337513\pi$
$223$	14.5923	0.977173	0.488586	+	0.872516i	$0.337513\pi$
$224$	−1.30067	−0.0869049
$225$	−2.79206	−0.186138
$226$	33.8172	2.24949
$227$	−2.86614	−0.190233	−0.0951164	−	0.995466i	$-0.530322\pi$
$227$	−2.86614	−0.190233	−0.0951164	+	0.995466i	$0.530322\pi$
$228$	−3.40846	−0.225731
$229$	20.1181	1.32944	0.664719	−	0.747093i	$-0.268551\pi$
$229$	20.1181	1.32944	0.664719	+	0.747093i	$0.268551\pi$
$230$	14.1644	0.933971
$231$	7.92501	0.521428
$232$	−35.3904	−2.32349
$233$	7.54295	0.494155	0.247078	−	0.968996i	$-0.420530\pi$
$233$	7.54295	0.494155	0.247078	+	0.968996i	$0.420530\pi$
$234$	−31.8428	−2.08163
$235$	−0.357747	−0.0233368
$236$	10.5082	0.684027
$237$	4.32306	0.280813
$238$	23.3170	1.51142
$239$	−24.5308	−1.58676	−0.793381	−	0.608725i	$-0.791681\pi$
$239$	−24.5308	−1.58676	−0.793381	+	0.608725i	$0.791681\pi$
$240$	−1.95193	−0.125996
$241$	23.3103	1.50155	0.750776	−	0.660557i	$-0.229680\pi$
$241$	23.3103	1.50155	0.750776	+	0.660557i	$0.229680\pi$
$242$	−66.5089	−4.27535
$243$	11.1939	0.718090
$244$	18.4734	1.18264
$245$	−0.938510	−0.0599592
$246$	−1.54271	−0.0983596
$247$	8.56763	0.545145
$248$	−5.84875	−0.371396
$249$	2.42886	0.153923
$250$	2.45894	0.155517
$251$	30.5876	1.93067	0.965335	−	0.261013i	$-0.0840566\pi$
$251$	30.5876	1.93067	0.965335	+	0.261013i	$0.0840566\pi$
$252$	−31.8319	−2.00522
$253$	35.5315	2.23385
$254$	11.1807	0.701542
$255$	1.53469	0.0961060
$256$	−32.3185	−2.01991
$257$	17.6407	1.10040	0.550199	−	0.835033i	$-0.314552\pi$
$257$	17.6407	1.10040	0.550199	+	0.835033i	$0.314552\pi$
$258$	5.94032	0.369828
$259$	0.183999	0.0114331
$260$	18.7675	1.16391
$261$	−19.6369	−1.21549
$262$	−24.7298	−1.52781
$263$	−27.7422	−1.71066	−0.855329	−	0.518085i	$-0.826645\pi$
$263$	−27.7422	−1.71066	−0.855329	+	0.518085i	$0.826645\pi$
$264$	−14.1537	−0.871097
$265$	−4.97683	−0.305724
$266$	12.7980	0.784694
$267$	0.103490	0.00633346
$268$	18.8484	1.15135
$269$	−5.87349	−0.358113	−0.179057	−	0.983839i	$-0.557304\pi$
$269$	−5.87349	−0.358113	−0.179057	+	0.983839i	$0.557304\pi$
$270$	−6.49453	−0.395244
$271$	−20.1044	−1.22125	−0.610627	−	0.791918i	$-0.709083\pi$
$271$	−20.1044	−1.22125	−0.610627	+	0.791918i	$0.709083\pi$
$272$	−14.4063	−0.873509
$273$	−5.95899	−0.360655
$274$	−15.5336	−0.938421
$275$	6.16829	0.371962
$276$	10.6288	0.639777
$277$	6.79908	0.408517	0.204259	−	0.978917i	$-0.434522\pi$
$277$	6.79908	0.408517	0.204259	+	0.978917i	$0.434522\pi$
$278$	−0.914698	−0.0548599
$279$	−3.24526	−0.194289
$280$	14.1778	0.847283
$281$	26.0067	1.55143	0.775714	−	0.631085i	$-0.217390\pi$
$281$	26.0067	1.55143	0.775714	+	0.631085i	$0.217390\pi$
$282$	−0.401134	−0.0238872
$283$	−15.1996	−0.903523	−0.451761	−	0.892139i	$-0.649204\pi$
$283$	−15.1996	−0.903523	−0.451761	+	0.892139i	$0.649204\pi$
$284$	23.0435	1.36738
$285$	0.842344	0.0498961
$286$	70.3478	4.15975
$287$	3.87650	0.228822
$288$	1.28891	0.0759500
$289$	−5.67315	−0.333715
$290$	17.2940	1.01554
$291$	−4.34941	−0.254967
$292$	30.8333	1.80438
$293$	−15.4835	−0.904553	−0.452277	−	0.891878i	$-0.649388\pi$
$293$	−15.4835	−0.904553	−0.452277	+	0.891878i	$0.649388\pi$
$294$	−1.05233	−0.0613733
$295$	−2.59693	−0.151199
$296$	−0.328612	−0.0191002
$297$	−16.2916	−0.945335
$298$	−43.3963	−2.51388
$299$	−26.7169	−1.54508
$300$	1.84516	0.106530
$301$	−14.9267	−0.860363
$302$	−56.8102	−3.26906
$303$	2.14570	0.123268
$304$	−7.90717	−0.453507
$305$	−4.56539	−0.261414
$306$	−23.1062	−1.32089
$307$	1.89839	0.108347	0.0541734	−	0.998532i	$-0.482748\pi$
$307$	1.89839	0.108347	0.0541734	+	0.998532i	$0.482748\pi$
$308$	70.3238	4.00707
$309$	−8.15897	−0.464148
$310$	2.85807	0.162328
$311$	8.15925	0.462669	0.231334	−	0.972874i	$-0.425691\pi$
$311$	8.15925	0.462669	0.231334	+	0.972874i	$0.425691\pi$
$312$	10.6424	0.602510
$313$	16.9657	0.958958	0.479479	−	0.877553i	$-0.340826\pi$
$313$	16.9657	0.958958	0.479479	+	0.877553i	$0.340826\pi$
$314$	−17.3002	−0.976304
$315$	7.86674	0.443240
$316$	38.3614	2.15800
$317$	20.1249	1.13033	0.565164	−	0.824978i	$-0.308813\pi$
$317$	20.1249	1.13033	0.565164	+	0.824978i	$0.308813\pi$
$318$	−5.58043	−0.312935
$319$	43.3822	2.42894
$320$	7.42593	0.415122
$321$	−8.47437	−0.472993
$322$	−39.9086	−2.22402
$323$	6.21696	0.345921
$324$	29.0200	1.61222
$325$	−4.63807	−0.257274
$326$	−21.8474	−1.21002
$327$	−3.70656	−0.204973
$328$	−6.92322	−0.382271
$329$	1.00796	0.0555708
$330$	6.91638	0.380734
$331$	19.3966	1.06613	0.533067	−	0.846073i	$-0.321039\pi$
$331$	19.3966	1.06613	0.533067	+	0.846073i	$0.321039\pi$
$332$	21.5529	1.18287
$333$	−0.182335	−0.00999192
$334$	−40.5327	−2.21785
$335$	−4.65806	−0.254497
$336$	5.49962	0.300029
$337$	5.05224	0.275213	0.137607	−	0.990487i	$-0.456059\pi$
$337$	5.05224	0.275213	0.137607	+	0.990487i	$0.456059\pi$
$338$	−20.9298	−1.13843
$339$	−6.27128	−0.340609
$340$	13.6183	0.738556
$341$	7.16951	0.388251
$342$	−12.6823	−0.685779
$343$	−17.0785	−0.922151
$344$	26.6584	1.43732
$345$	−2.62673	−0.141418
$346$	−36.2852	−1.95070
$347$	20.8308	1.11826	0.559128	−	0.829081i	$-0.311136\pi$
$347$	20.8308	1.11826	0.559128	+	0.829081i	$0.311136\pi$
$348$	12.9772	0.695651
$349$	35.0063	1.87384	0.936921	−	0.349541i	$-0.113662\pi$
$349$	35.0063	1.87384	0.936921	+	0.349541i	$0.113662\pi$
$350$	−6.92816	−0.370326
$351$	12.2500	0.653857
$352$	−2.84750	−0.151772
$353$	−9.97464	−0.530896	−0.265448	−	0.964125i	$-0.585520\pi$
$353$	−9.97464	−0.530896	−0.265448	+	0.964125i	$0.585520\pi$
$354$	−2.91189	−0.154765
$355$	−5.69482	−0.302250
$356$	0.918330	0.0486714
$357$	−4.32404	−0.228853
$358$	24.2046	1.27925
$359$	29.1400	1.53795	0.768975	−	0.639279i	$-0.220767\pi$
$359$	29.1400	1.53795	0.768975	+	0.639279i	$0.220767\pi$
$360$	−14.0496	−0.740478
$361$	−15.5877	−0.820405
$362$	11.8099	0.620713
$363$	12.3338	0.647357
$364$	−52.8780	−2.77156
$365$	−7.61994	−0.398846
$366$	−5.11909	−0.267579
$367$	20.6809	1.07953	0.539767	−	0.841814i	$-0.318512\pi$
$367$	20.6809	1.07953	0.539767	+	0.841814i	$0.318512\pi$
$368$	24.6574	1.28535
$369$	−3.84145	−0.199978
$370$	0.160581	0.00834821
$371$	14.0224	0.728007
$372$	2.14466	0.111196
$373$	0.148414	0.00768458	0.00384229	−	0.999993i	$-0.498777\pi$
$373$	0.148414	0.00768458	0.00384229	+	0.999993i	$0.498777\pi$
$374$	51.0467	2.63956
$375$	−0.456001	−0.0235478
$376$	−1.80017	−0.0928367
$377$	−32.6200	−1.68002
$378$	18.2986	0.941177
$379$	−9.78841	−0.502797	−0.251398	−	0.967884i	$-0.580890\pi$
$379$	−9.78841	−0.502797	−0.251398	+	0.967884i	$0.580890\pi$
$380$	7.47467	0.383442
$381$	−2.07342	−0.106225
$382$	−45.6060	−2.33341
$383$	−2.65834	−0.135835	−0.0679174	−	0.997691i	$-0.521635\pi$
$383$	−2.65834	−0.135835	−0.0679174	+	0.997691i	$0.521635\pi$
$384$	8.74756	0.446397
$385$	−17.3794	−0.885735
$386$	−28.7511	−1.46339
$387$	14.7918	0.751909
$388$	−38.5951	−1.95937
$389$	−34.5161	−1.75004	−0.875019	−	0.484089i	$-0.839151\pi$
$389$	−34.5161	−1.75004	−0.875019	+	0.484089i	$0.839151\pi$
$390$	−5.20058	−0.263342
$391$	−19.3867	−0.980427
$392$	−4.72256	−0.238525
$393$	4.58604	0.231335
$394$	−1.77949	−0.0896495
$395$	−9.48038	−0.477010
$396$	−69.6880	−3.50196
$397$	−38.1342	−1.91390	−0.956950	−	0.290252i	$-0.906261\pi$
$397$	−38.1342	−1.91390	−0.956950	+	0.290252i	$0.906261\pi$
$398$	2.47784	0.124203
$399$	−2.37333	−0.118815
$400$	4.28053	0.214026
$401$	1.00000	0.0499376
$401$	1.00000	0.0499376
$402$	−5.22300	−0.260499
$403$	−5.39091	−0.268540
$404$	19.0402	0.947287
$405$	−7.17180	−0.356370
$406$	−48.7264	−2.41825
$407$	0.402819	0.0199670
$408$	7.72251	0.382321
$409$	−30.4865	−1.50746	−0.753730	−	0.657184i	$-0.771747\pi$
$409$	−30.4865	−1.50746	−0.753730	+	0.657184i	$0.771747\pi$
$410$	3.38313	0.167081
$411$	2.88065	0.142092
$412$	−72.3999	−3.56689
$413$	7.31695	0.360044
$414$	39.5478	1.94367
$415$	−5.32643	−0.261464
$416$	2.14110	0.104976
$417$	0.169627	0.00830668
$418$	28.0180	1.37040
$419$	17.5763	0.858659	0.429330	−	0.903148i	$-0.358750\pi$
$419$	17.5763	0.858659	0.429330	+	0.903148i	$0.358750\pi$
$420$	−5.19881	−0.253676
$421$	−8.18789	−0.399053	−0.199527	−	0.979892i	$-0.563940\pi$
$421$	−8.18789	−0.399053	−0.199527	+	0.979892i	$0.563940\pi$
$422$	18.9706	0.923474
$423$	−0.998851	−0.0485658
$424$	−25.0433	−1.21621
$425$	−3.36554	−0.163253
$426$	−6.38549	−0.309378
$427$	12.8632	0.622492
$428$	−75.1987	−3.63486
$429$	−13.0457	−0.629853
$430$	−13.0270	−0.628217
$431$	−40.3517	−1.94367	−0.971836	−	0.235658i	$-0.924276\pi$
$431$	−40.3517	−1.94367	−0.971836	+	0.235658i	$0.924276\pi$
$432$	−11.3057	−0.543945
$433$	34.9697	1.68054	0.840268	−	0.542172i	$-0.182398\pi$
$433$	34.9697	1.68054	0.840268	+	0.542172i	$0.182398\pi$
$434$	−8.05272	−0.386543
$435$	−3.20710	−0.153769
$436$	−32.8907	−1.57518
$437$	−10.6408	−0.509016
$438$	−8.54409	−0.408252
$439$	−25.8777	−1.23508	−0.617538	−	0.786541i	$-0.711870\pi$
$439$	−25.8777	−1.23508	−0.617538	+	0.786541i	$0.711870\pi$
$440$	31.0386	1.47971
$441$	−2.62038	−0.124780
$442$	−38.3831	−1.82570
$443$	−20.3139	−0.965143	−0.482572	−	0.875856i	$-0.660297\pi$
$443$	−20.3139	−0.965143	−0.482572	+	0.875856i	$0.660297\pi$
$444$	0.120498	0.00571859
$445$	−0.226950	−0.0107585
$446$	−35.8816	−1.69904
$447$	8.04768	0.380642
$448$	−20.9228	−0.988510
$449$	−14.6954	−0.693520	−0.346760	−	0.937954i	$-0.612718\pi$
$449$	−14.6954	−0.693520	−0.346760	+	0.937954i	$0.612718\pi$
$450$	6.86552	0.323644
$451$	8.48662	0.399619
$452$	−55.6491	−2.61751
$453$	10.5352	0.494988
$454$	7.04768	0.330764
$455$	13.0679	0.612634
$456$	4.23865	0.198493
$457$	29.2627	1.36885	0.684426	−	0.729082i	$-0.260053\pi$
$457$	29.2627	1.36885	0.684426	+	0.729082i	$0.260053\pi$
$458$	−49.4691	−2.31154
$459$	8.88902	0.414904
$460$	−23.3087	−1.08677
$461$	−0.384661	−0.0179154	−0.00895771	−	0.999960i	$-0.502851\pi$
$461$	−0.384661	−0.0179154	−0.00895771	+	0.999960i	$0.502851\pi$
$462$	−19.4872	−0.906624
$463$	−0.225457	−0.0104779	−0.00523894	−	0.999986i	$-0.501668\pi$
$463$	−0.225457	−0.0104779	−0.00523894	+	0.999986i	$0.501668\pi$
$464$	30.1054	1.39761
$465$	−0.530018	−0.0245790
$466$	−18.5477	−0.859205
$467$	13.7734	0.637356	0.318678	−	0.947863i	$-0.396761\pi$
$467$	13.7734	0.637356	0.318678	+	0.947863i	$0.396761\pi$
$468$	52.4000	2.42219
$469$	13.1243	0.606022
$470$	0.879678	0.0405765
$471$	3.20825	0.147828
$472$	−13.0677	−0.601489
$473$	−32.6784	−1.50255
$474$	−10.6302	−0.488259
$475$	−1.84724	−0.0847572
$476$	−38.3701	−1.75869
$477$	−13.8956	−0.636237
$478$	60.3197	2.75896
$479$	−19.5784	−0.894559	−0.447279	−	0.894394i	$-0.647607\pi$
$479$	−19.5784	−0.894559	−0.447279	+	0.894394i	$0.647607\pi$
$480$	0.210506	0.00960824
$481$	−0.302889	−0.0138105
$482$	−57.3188	−2.61080
$483$	7.40090	0.336753
$484$	109.446	4.97482
$485$	9.53815	0.433105
$486$	−27.5252	−1.24857
$487$	22.8198	1.03407	0.517033	−	0.855965i	$-0.327036\pi$
$487$	22.8198	1.03407	0.517033	+	0.855965i	$0.327036\pi$
$488$	−22.9729	−1.03994
$489$	4.05152	0.183216
$490$	2.30774	0.104253
$491$	11.3996	0.514455	0.257228	−	0.966351i	$-0.417191\pi$
$491$	11.3996	0.514455	0.257228	+	0.966351i	$0.417191\pi$
$492$	2.53866	0.114452
$493$	−23.6702	−1.06605
$494$	−21.0673	−0.947863
$495$	17.2222	0.774082
$496$	4.97533	0.223399
$497$	16.0454	0.719733
$498$	−5.97243	−0.267631
$499$	11.9052	0.532952	0.266476	−	0.963842i	$-0.414141\pi$
$499$	11.9052	0.532952	0.266476	+	0.963842i	$0.414141\pi$
$500$	−4.04640	−0.180960
$501$	7.51663	0.335818
$502$	−75.2131	−3.35692
$503$	−2.39580	−0.106823	−0.0534116	−	0.998573i	$-0.517010\pi$
$503$	−2.39580	−0.106823	−0.0534116	+	0.998573i	$0.517010\pi$
$504$	39.5852	1.76326
$505$	−4.70548	−0.209391
$506$	−87.3699	−3.88407
$507$	3.88135	0.172377
$508$	−18.3989	−0.816317
$509$	−16.4023	−0.727020	−0.363510	−	0.931590i	$-0.618422\pi$
$509$	−16.4023	−0.727020	−0.363510	+	0.931590i	$0.618422\pi$
$510$	−3.77371	−0.167103
$511$	21.4695	0.949753
$512$	41.1029	1.81651
$513$	4.87891	0.215409
$514$	−43.3775	−1.91330
$515$	17.8924	0.788435
$516$	−9.77530	−0.430333
$517$	2.20668	0.0970498
$518$	−0.452443	−0.0198792
$519$	6.72894	0.295368
$520$	−23.3386	−1.02347
$521$	−29.4302	−1.28936	−0.644680	−	0.764452i	$-0.723010\pi$
$521$	−29.4302	−1.28936	−0.644680	+	0.764452i	$0.723010\pi$
$522$	48.2859	2.11342
$523$	−1.57545	−0.0688895	−0.0344447	−	0.999407i	$-0.510966\pi$
$523$	−1.57545	−0.0688895	−0.0344447	+	0.999407i	$0.510966\pi$
$524$	40.6950	1.77777
$525$	1.28480	0.0560733
$526$	68.2165	2.97438
$527$	−3.91182	−0.170402
$528$	12.0400	0.523976
$529$	10.1816	0.442680
$530$	12.2377	0.531574
$531$	−7.25080	−0.314658
$532$	−21.0601	−0.913073
$533$	−6.38128	−0.276404
$534$	−0.254475	−0.0110122
$535$	18.5841	0.803461
$536$	−23.4392	−1.01242
$537$	−4.48865	−0.193699
$538$	14.4426	0.622664
$539$	5.78900	0.249350
$540$	10.6873	0.459908
$541$	−13.6959	−0.588833	−0.294417	−	0.955677i	$-0.595125\pi$
$541$	−13.6959	−0.588833	−0.294417	+	0.955677i	$0.595125\pi$
$542$	49.4355	2.12344
$543$	−2.19009	−0.0939859
$544$	1.55365	0.0666122
$545$	8.12841	0.348183
$546$	14.6528	0.627083
$547$	4.20345	0.179727	0.0898633	−	0.995954i	$-0.471357\pi$
$547$	4.20345	0.179727	0.0898633	+	0.995954i	$0.471357\pi$
$548$	25.5619	1.09195
$549$	−12.7469	−0.544023
$550$	−15.1675	−0.646743
$551$	−12.9918	−0.553471
$552$	−13.2176	−0.562579
$553$	26.7113	1.13588
$554$	−16.7185	−0.710303
$555$	−0.0297791	−0.00126405
$556$	1.50521	0.0638352
$557$	20.6715	0.875881	0.437941	−	0.899004i	$-0.355708\pi$
$557$	20.6715	0.875881	0.437941	+	0.899004i	$0.355708\pi$
$558$	7.97992	0.337817
$559$	24.5716	1.03927
$560$	−12.0605	−0.509651
$561$	−9.46640	−0.399672
$562$	−63.9489	−2.69752
$563$	26.5422	1.11862	0.559311	−	0.828958i	$-0.311066\pi$
$563$	26.5422	1.11862	0.559311	+	0.828958i	$0.311066\pi$
$564$	0.660100	0.0277952
$565$	13.7528	0.578583
$566$	37.3749	1.57099
$567$	20.2068	0.848607
$568$	−28.6562	−1.20239
$569$	−35.8843	−1.50435	−0.752173	−	0.658965i	$-0.770994\pi$
$569$	−35.8843	−1.50435	−0.752173	+	0.658965i	$0.770994\pi$
$570$	−2.07127	−0.0867562
$571$	30.7934	1.28866	0.644331	−	0.764747i	$-0.277136\pi$
$571$	30.7934	1.28866	0.644331	+	0.764747i	$0.277136\pi$
$572$	−115.763	−4.84030
$573$	8.45746	0.353315
$574$	−9.53208	−0.397861
$575$	5.76035	0.240223
$576$	20.7337	0.863902
$577$	−22.9702	−0.956264	−0.478132	−	0.878288i	$-0.658686\pi$
$577$	−22.9702	−0.956264	−0.478132	+	0.878288i	$0.658686\pi$
$578$	13.9500	0.580242
$579$	5.33178	0.221581
$580$	−28.4587	−1.18168
$581$	15.0074	0.622613
$582$	10.6949	0.443320
$583$	30.6985	1.27140
$584$	−38.3433	−1.58666
$585$	−12.9498	−0.535408
$586$	38.0729	1.57278
$587$	−34.4739	−1.42289	−0.711446	−	0.702741i	$-0.751959\pi$
$587$	−34.4739	−1.42289	−0.711446	+	0.702741i	$0.751959\pi$
$588$	1.73170	0.0714142
$589$	−2.14708	−0.0884689
$590$	6.38571	0.262895
$591$	0.330000	0.0135744
$592$	0.279540	0.0114890
$593$	−26.6788	−1.09557	−0.547783	−	0.836620i	$-0.684528\pi$
$593$	−26.6788	−1.09557	−0.547783	+	0.836620i	$0.684528\pi$
$594$	40.0601	1.64369
$595$	9.48252	0.388746
$596$	71.4123	2.92516
$597$	−0.459505	−0.0188063
$598$	65.6954	2.68648
$599$	9.98077	0.407803	0.203902	−	0.978991i	$-0.434638\pi$
$599$	9.98077	0.407803	0.203902	+	0.978991i	$0.434638\pi$
$600$	−2.29458	−0.0936760
$601$	−37.7449	−1.53964	−0.769822	−	0.638258i	$-0.779655\pi$
$601$	−37.7449	−1.53964	−0.769822	+	0.638258i	$0.779655\pi$
$602$	36.7040	1.49594
$603$	−13.0056	−0.529629
$604$	93.4860	3.80389
$605$	−27.0478	−1.09965
$606$	−5.27616	−0.214330
$607$	−33.3187	−1.35236	−0.676182	−	0.736734i	$-0.736367\pi$
$607$	−33.3187	−1.35236	−0.676182	+	0.736734i	$0.736367\pi$
$608$	0.852751	0.0345836
$609$	9.03613	0.366162
$610$	11.2260	0.454529
$611$	−1.65925	−0.0671262
$612$	38.0231	1.53699
$613$	26.2658	1.06087	0.530433	−	0.847727i	$-0.322029\pi$
$613$	26.2658	1.06087	0.530433	+	0.847727i	$0.322029\pi$
$614$	−4.66803	−0.188386
$615$	−0.627388	−0.0252987
$616$	−87.4525	−3.52356
$617$	40.1876	1.61789	0.808946	−	0.587883i	$-0.200038\pi$
$617$	40.1876	1.61789	0.808946	+	0.587883i	$0.200038\pi$
$618$	20.0624	0.807030
$619$	−23.3648	−0.939110	−0.469555	−	0.882903i	$-0.655586\pi$
$619$	−23.3648	−0.939110	−0.469555	+	0.882903i	$0.655586\pi$
$620$	−4.70320	−0.188885
$621$	−15.2142	−0.610523
$622$	−20.0631	−0.804458
$623$	0.639440	0.0256186
$624$	−9.05317	−0.362417
$625$	1.00000	0.0400000
$626$	−41.7177	−1.66737
$627$	−5.19582	−0.207501
$628$	28.4689	1.13603
$629$	−0.219786	−0.00876345
$630$	−19.3439	−0.770678
$631$	−19.0048	−0.756568	−0.378284	−	0.925690i	$-0.623486\pi$
$631$	−19.0048	−0.756568	−0.378284	+	0.925690i	$0.623486\pi$
$632$	−47.7050	−1.89760
$633$	−3.51802	−0.139829
$634$	−49.4860	−1.96534
$635$	4.54697	0.180441
$636$	9.18306	0.364132
$637$	−4.35288	−0.172467
$638$	−106.674	−4.22328
$639$	−15.9003	−0.629006
$640$	−19.1832	−0.758282
$641$	−0.0250248	−0.000988421 0	−0.000494210	−	1.00000i	$-0.500157\pi$
$641$	−0.0250248	−0.000988421 0	−0.000494210		1.00000i	$0.500157\pi$
$642$	20.8380	0.822410
$643$	−38.6240	−1.52318	−0.761591	−	0.648058i	$-0.775581\pi$
$643$	−38.6240	−1.52318	−0.761591	+	0.648058i	$0.775581\pi$
$644$	65.6730	2.58788
$645$	2.41580	0.0951222
$646$	−15.2871	−0.601465
$647$	−9.48422	−0.372863	−0.186431	−	0.982468i	$-0.559692\pi$
$647$	−9.48422	−0.372863	−0.186431	+	0.982468i	$0.559692\pi$
$648$	−36.0883	−1.41768
$649$	16.0186	0.628786
$650$	11.4047	0.447331
$651$	1.49335	0.0585288
$652$	35.9518	1.40798
$653$	−39.3233	−1.53884	−0.769420	−	0.638743i	$-0.779455\pi$
$653$	−39.3233	−1.53884	−0.769420	+	0.638743i	$0.779455\pi$
$654$	9.11422	0.356394
$655$	−10.0571	−0.392963
$656$	5.88935	0.229941
$657$	−21.2753	−0.830030
$658$	−2.47852	−0.0966230
$659$	17.1495	0.668049	0.334025	−	0.942564i	$-0.391593\pi$
$659$	17.1495	0.668049	0.334025	+	0.942564i	$0.391593\pi$
$660$	−11.3815	−0.443024
$661$	−9.25909	−0.360137	−0.180068	−	0.983654i	$-0.557632\pi$
$661$	−9.25909	−0.360137	−0.180068	+	0.983654i	$0.557632\pi$
$662$	−47.6951	−1.85372
$663$	7.11800	0.276440
$664$	−26.8025	−1.04014
$665$	5.20467	0.201828
$666$	0.448352	0.0173733
$667$	40.5131	1.56867
$668$	66.6999	2.58070
$669$	6.65411	0.257263
$670$	11.4539	0.442503
$671$	28.1607	1.08713
$672$	−0.593109	−0.0228797
$673$	27.2288	1.04959	0.524796	−	0.851228i	$-0.324141\pi$
$673$	27.2288	1.04959	0.524796	+	0.851228i	$0.324141\pi$
$674$	−12.4232	−0.478523
$675$	−2.64119	−0.101659
$676$	34.4417	1.32468
$677$	−3.02935	−0.116427	−0.0582137	−	0.998304i	$-0.518540\pi$
$677$	−3.02935	−0.116427	−0.0582137	+	0.998304i	$0.518540\pi$
$678$	15.4207	0.592229
$679$	−26.8741	−1.03133
$680$	−16.9353	−0.649439
$681$	−1.30697	−0.0500830
$682$	−17.6294	−0.675065
$683$	27.5906	1.05572	0.527862	−	0.849330i	$-0.322994\pi$
$683$	27.5906	1.05572	0.527862	+	0.849330i	$0.322994\pi$
$684$	20.8697	0.797975
$685$	−6.31720	−0.241368
$686$	41.9950	1.60338
$687$	9.17386	0.350004
$688$	−22.6774	−0.864568
$689$	−23.0829	−0.879389
$690$	6.45897	0.245889
$691$	−44.1854	−1.68089	−0.840446	−	0.541896i	$-0.817707\pi$
$691$	−44.1854	−1.68089	−0.840446	+	0.541896i	$0.817707\pi$
$692$	59.7103	2.26984
$693$	−48.5243	−1.84329
$694$	−51.2218	−1.94435
$695$	−0.371988	−0.0141103
$696$	−16.1380	−0.611711
$697$	−4.63046	−0.175391
$698$	−86.0784	−3.25811
$699$	3.43959	0.130097
$700$	11.4009	0.430912
$701$	−12.8695	−0.486076	−0.243038	−	0.970017i	$-0.578144\pi$
$701$	−12.8695	−0.486076	−0.243038	+	0.970017i	$0.578144\pi$
$702$	−30.1221	−1.13688
$703$	−0.120634	−0.00454979
$704$	−45.8052	−1.72635
$705$	−0.163133	−0.00614394
$706$	24.5271	0.923088
$707$	13.2579	0.498613
$708$	4.79176	0.180085
$709$	−28.7255	−1.07881	−0.539405	−	0.842047i	$-0.681351\pi$
$709$	−28.7255	−1.07881	−0.539405	+	0.842047i	$0.681351\pi$
$710$	14.0032	0.525532
$711$	−26.4698	−0.992695
$712$	−1.14201	−0.0427985
$713$	6.69536	0.250743
$714$	10.6326	0.397914
$715$	28.6090	1.06991
$716$	−39.8307	−1.48854
$717$	−11.1861	−0.417751
$718$	−71.6535	−2.67409
$719$	23.9432	0.892932	0.446466	−	0.894801i	$-0.352682\pi$
$719$	23.9432	0.892932	0.446466	+	0.894801i	$0.352682\pi$
$720$	11.9515	0.445406
$721$	−50.4126	−1.87746
$722$	38.3293	1.42647
$723$	10.6295	0.395317
$724$	−19.4341	−0.722264
$725$	7.03310	0.261203
$726$	−30.3281	−1.12558
$727$	30.3808	1.12676	0.563380	−	0.826198i	$-0.309501\pi$
$727$	30.3808	1.12676	0.563380	+	0.826198i	$0.309501\pi$
$728$	65.7574	2.43713
$729$	−16.4110	−0.607814
$730$	18.7370	0.693487
$731$	17.8299	0.659465
$732$	8.42389	0.311356
$733$	47.7424	1.76341	0.881704	−	0.471804i	$-0.156397\pi$
$733$	47.7424	1.76341	0.881704	+	0.471804i	$0.156397\pi$
$734$	−50.8531	−1.87702
$735$	−0.427962	−0.0157856
$736$	−2.65918	−0.0980186
$737$	28.7323	1.05837
$738$	9.44590	0.347709
$739$	10.9997	0.404630	0.202315	−	0.979320i	$-0.435154\pi$
$739$	10.9997	0.404630	0.202315	+	0.979320i	$0.435154\pi$
$740$	−0.264250	−0.00971401
$741$	3.90685	0.143522
$742$	−34.4803	−1.26581
$743$	30.6082	1.12290	0.561452	−	0.827509i	$-0.310243\pi$
$743$	30.6082	1.12290	0.561452	+	0.827509i	$0.310243\pi$
$744$	−2.66704	−0.0977783
$745$	−17.6484	−0.646586
$746$	−0.364941	−0.0133614
$747$	−14.8717	−0.544129
$748$	−84.0016	−3.07140
$749$	−52.3614	−1.91324
$750$	1.12128	0.0409434
$751$	−16.8213	−0.613818	−0.306909	−	0.951739i	$-0.599295\pi$
$751$	−16.8213	−0.613818	−0.306909	+	0.951739i	$0.599295\pi$
$752$	1.53134	0.0558424
$753$	13.9480	0.508292
$754$	80.2108	2.92110
$755$	−23.1035	−0.840823
$756$	−30.1118	−1.09516
$757$	−12.0207	−0.436901	−0.218450	−	0.975848i	$-0.570100\pi$
$757$	−12.0207	−0.436901	−0.218450	+	0.975848i	$0.570100\pi$
$758$	24.0691	0.874230
$759$	16.2024	0.588110
$760$	−9.29526	−0.337174
$761$	−38.7121	−1.40331	−0.701657	−	0.712515i	$-0.747556\pi$
$761$	−38.7121	−1.40331	−0.701657	+	0.712515i	$0.747556\pi$
$762$	5.09843	0.184697
$763$	−22.9021	−0.829111
$764$	75.0486	2.71516
$765$	−9.39679	−0.339742
$766$	6.53670	0.236181
$767$	−12.0448	−0.434911
$768$	−14.7373	−0.531786
$769$	28.9073	1.04243	0.521213	−	0.853427i	$-0.325480\pi$
$769$	28.9073	1.04243	0.521213	+	0.853427i	$0.325480\pi$
$770$	42.7349	1.54006
$771$	8.04419	0.289705
$772$	47.3124	1.70281
$773$	28.7729	1.03489	0.517445	−	0.855717i	$-0.326883\pi$
$773$	28.7729	1.03489	0.517445	+	0.855717i	$0.326883\pi$
$774$	−36.3722	−1.30737
$775$	1.16232	0.0417517
$776$	47.9957	1.72295
$777$	0.0839037	0.00301003
$778$	84.8732	3.04285
$779$	−2.54152	−0.0910594
$780$	8.55799	0.306425
$781$	35.1273	1.25695
$782$	47.6707	1.70470
$783$	−18.5757	−0.663843
$784$	4.01732	0.143476
$785$	−7.03561	−0.251112
$786$	−11.2768	−0.402231
$787$	−47.0917	−1.67864	−0.839319	−	0.543639i	$-0.817046\pi$
$787$	−47.0917	−1.67864	−0.839319	+	0.543639i	$0.817046\pi$
$788$	2.92830	0.104317
$789$	−12.6505	−0.450369
$790$	23.3117	0.829393
$791$	−38.7489	−1.37775
$792$	86.6618	3.07939
$793$	−21.1746	−0.751933
$794$	93.7698	3.32777
$795$	−2.26944	−0.0804888
$796$	−4.07749	−0.144523
$797$	−42.2856	−1.49783	−0.748917	−	0.662664i	$-0.769426\pi$
$797$	−42.2856	−1.49783	−0.748917	+	0.662664i	$0.769426\pi$
$798$	5.83589	0.206588
$799$	−1.20401	−0.0425948
$800$	−0.461635	−0.0163213
$801$	−0.633659	−0.0223893
$802$	−2.45894	−0.0868283
$803$	47.0020	1.65866
$804$	8.59488	0.303118
$805$	−16.2300	−0.572032
$806$	13.2559	0.466921
$807$	−2.67832	−0.0942813
$808$	−23.6778	−0.832983
$809$	32.1708	1.13106	0.565532	−	0.824726i	$-0.308671\pi$
$809$	32.1708	1.13106	0.565532	+	0.824726i	$0.308671\pi$
$810$	17.6351	0.619633
$811$	17.4710	0.613490	0.306745	−	0.951792i	$-0.400760\pi$
$811$	17.4710	0.613490	0.306745	+	0.951792i	$0.400760\pi$
$812$	80.1835	2.81389
$813$	−9.16762	−0.321523
$814$	−0.990510	−0.0347173
$815$	−8.88489	−0.311224
$816$	−6.56928	−0.229971
$817$	9.78631	0.342380
$818$	74.9645	2.62107
$819$	36.4865	1.27494
$820$	−5.56722	−0.194416
$821$	41.2543	1.43978	0.719892	−	0.694086i	$-0.244191\pi$
$821$	41.2543	1.43978	0.719892	+	0.694086i	$0.244191\pi$
$822$	−7.08335	−0.247060
$823$	12.9615	0.451811	0.225906	−	0.974149i	$-0.427466\pi$
$823$	12.9615	0.451811	0.225906	+	0.974149i	$0.427466\pi$
$824$	90.0342	3.13649
$825$	2.81275	0.0979272
$826$	−17.9920	−0.626020
$827$	−7.68160	−0.267116	−0.133558	−	0.991041i	$-0.542640\pi$
$827$	−7.68160	−0.267116	−0.133558	+	0.991041i	$0.542640\pi$
$828$	−65.0792	−2.26166
$829$	−48.0965	−1.67046	−0.835231	−	0.549899i	$-0.814666\pi$
$829$	−48.0965	−1.67046	−0.835231	+	0.549899i	$0.814666\pi$
$830$	13.0974	0.454617
$831$	3.10039	0.107551
$832$	34.4420	1.19406
$833$	−3.15859	−0.109439
$834$	−0.417103	−0.0144431
$835$	−16.4838	−0.570445
$836$	−46.1059	−1.59461
$837$	−3.06990	−0.106111
$838$	−43.2192	−1.49298
$839$	−25.5547	−0.882248	−0.441124	−	0.897446i	$-0.645420\pi$
$839$	−25.5547	−0.882248	−0.441124	+	0.897446i	$0.645420\pi$
$840$	6.46507	0.223066
$841$	20.4645	0.705673
$842$	20.1335	0.693848
$843$	11.8591	0.408448
$844$	−31.2177	−1.07456
$845$	−8.51170	−0.292812
$846$	2.45612	0.0844430
$847$	76.2081	2.61854
$848$	21.3035	0.731565
$849$	−6.93104	−0.237873
$850$	8.27566	0.283853
$851$	0.376179	0.0128953
$852$	10.5079	0.359993
$853$	−44.9468	−1.53895	−0.769474	−	0.638678i	$-0.779482\pi$
$853$	−44.9468	−1.53895	−0.769474	+	0.638678i	$0.779482\pi$
$854$	−31.6298	−1.08235
$855$	−5.15761	−0.176387
$856$	93.5147	3.19627
$857$	−2.60742	−0.0890678	−0.0445339	−	0.999008i	$-0.514180\pi$
$857$	−2.60742	−0.0890678	−0.0445339	+	0.999008i	$0.514180\pi$
$858$	32.0787	1.09515
$859$	0.974960	0.0332652	0.0166326	−	0.999862i	$-0.494705\pi$
$859$	0.974960	0.0332652	0.0166326	+	0.999862i	$0.494705\pi$
$860$	21.4370	0.730996
$861$	1.76769	0.0602426
$862$	99.2225	3.37953
$863$	44.2053	1.50476	0.752382	−	0.658727i	$-0.228905\pi$
$863$	44.2053	1.50476	0.752382	+	0.658727i	$0.228905\pi$
$864$	1.21926	0.0414802
$865$	−14.7564	−0.501733
$866$	−85.9884	−2.92200
$867$	−2.58697	−0.0878579
$868$	13.2514	0.449783
$869$	58.4777	1.98372
$870$	7.88608	0.267363
$871$	−21.6044	−0.732038
$872$	40.9019	1.38511
$873$	26.6311	0.901327
$874$	26.1650	0.885044
$875$	−2.81754	−0.0952501
$876$	14.0600	0.475044
$877$	−10.8667	−0.366941	−0.183470	−	0.983025i	$-0.558733\pi$
$877$	−10.8667	−0.366941	−0.183470	+	0.983025i	$0.558733\pi$
$878$	63.6318	2.14747
$879$	−7.06048	−0.238144
$880$	−26.4035	−0.890063
$881$	35.6999	1.20276	0.601379	−	0.798964i	$-0.294618\pi$
$881$	35.6999	1.20276	0.601379	+	0.798964i	$0.294618\pi$
$882$	6.44336	0.216959
$883$	32.9933	1.11031	0.555156	−	0.831746i	$-0.312659\pi$
$883$	32.9933	1.11031	0.555156	+	0.831746i	$0.312659\pi$
$884$	63.1626	2.12439
$885$	−1.18420	−0.0398066
$886$	49.9508	1.67813
$887$	−48.8530	−1.64032	−0.820162	−	0.572131i	$-0.806117\pi$
$887$	−48.8530	−1.64032	−0.820162	+	0.572131i	$0.806117\pi$
$888$	−0.149848	−0.00502856
$889$	−12.8113	−0.429676
$890$	0.558057	0.0187061
$891$	44.2377	1.48202
$892$	59.0462	1.97701
$893$	−0.660844	−0.0221143
$894$	−19.7888	−0.661836
$895$	9.84350	0.329032
$896$	54.0493	1.80566
$897$	−12.1829	−0.406777
$898$	36.1352	1.20585
$899$	8.17470	0.272641
$900$	−11.2978	−0.376593
$901$	−16.7497	−0.558014
$902$	−20.8681	−0.694832
$903$	−6.80661	−0.226510
$904$	69.2035	2.30167
$905$	4.80283	0.159651
$906$	−25.9055	−0.860653
$907$	−7.35283	−0.244147	−0.122073	−	0.992521i	$-0.538954\pi$
$907$	−7.35283	−0.244147	−0.122073	+	0.992521i	$0.538954\pi$
$908$	−11.5976	−0.384878
$909$	−13.1380	−0.435760
$910$	−32.1333	−1.06521
$911$	−2.42939	−0.0804894	−0.0402447	−	0.999190i	$-0.512814\pi$
$911$	−2.42939	−0.0804894	−0.0402447	+	0.999190i	$0.512814\pi$
$912$	−3.60568	−0.119396
$913$	32.8550	1.08734
$914$	−71.9554	−2.38007
$915$	−2.08183	−0.0688230
$916$	81.4056	2.68972
$917$	28.3362	0.935744
$918$	−21.8576	−0.721408
$919$	−4.05142	−0.133644	−0.0668220	−	0.997765i	$-0.521286\pi$
$919$	−4.05142	−0.133644	−0.0668220	+	0.997765i	$0.521286\pi$
$920$	28.9859	0.955637
$921$	0.865667	0.0285247
$922$	0.945858	0.0311502
$923$	−26.4130	−0.869394
$924$	32.0677	1.05495
$925$	0.0653049	0.00214721
$926$	0.554386	0.0182182
$927$	49.9568	1.64080
$928$	−3.24673	−0.106579
$929$	−29.4153	−0.965085	−0.482543	−	0.875872i	$-0.660287\pi$
$929$	−29.4153	−0.965085	−0.482543	+	0.875872i	$0.660287\pi$
$930$	1.30328	0.0427364
$931$	−1.73365	−0.0568182
$932$	30.5218	0.999774
$933$	3.72063	0.121808
$934$	−33.8680	−1.10819
$935$	20.7596	0.678912
$936$	−65.1629	−2.12992
$937$	−16.9371	−0.553310	−0.276655	−	0.960969i	$-0.589226\pi$
$937$	−16.9371	−0.553310	−0.276655	+	0.960969i	$0.589226\pi$
$938$	−32.2718	−1.05371
$939$	7.73638	0.252467
$940$	−1.44758	−0.0472150
$941$	−10.3890	−0.338670	−0.169335	−	0.985559i	$-0.554162\pi$
$941$	−10.3890	−0.338670	−0.169335	+	0.985559i	$0.554162\pi$
$942$	−7.88889	−0.257034
$943$	7.92536	0.258085
$944$	11.1162	0.361803
$945$	7.44164	0.242077
$946$	80.3542	2.61254
$947$	42.1470	1.36959	0.684797	−	0.728734i	$-0.259891\pi$
$947$	42.1470	1.36959	0.684797	+	0.728734i	$0.259891\pi$
$948$	17.4928	0.568141
$949$	−35.3418	−1.14724
$950$	4.54226	0.147370
$951$	9.17699	0.297584
$952$	47.7158	1.54648
$953$	34.8875	1.13012	0.565058	−	0.825051i	$-0.308854\pi$
$953$	34.8875	1.13012	0.565058	+	0.825051i	$0.308854\pi$
$954$	34.1686	1.10625
$955$	−18.5470	−0.600167
$956$	−99.2612	−3.21034
$957$	19.7823	0.639472
$958$	48.1421	1.55540
$959$	17.7989	0.574758
$960$	3.38623	0.109290
$961$	−29.6490	−0.956420
$962$	0.744786	0.0240129
$963$	51.8880	1.67207
$964$	94.3229	3.03794
$965$	−11.6925	−0.376394
$966$	−18.1984	−0.585523
$967$	−9.85574	−0.316939	−0.158470	−	0.987364i	$-0.550656\pi$
$967$	−9.85574	−0.316939	−0.158470	+	0.987364i	$0.550656\pi$
$968$	−136.104	−4.37453
$969$	2.83494	0.0910714
$970$	−23.4538	−0.753055
$971$	16.1754	0.519094	0.259547	−	0.965730i	$-0.416427\pi$
$971$	16.1754	0.519094	0.259547	+	0.965730i	$0.416427\pi$
$972$	45.2950	1.45284
$973$	1.04809	0.0336002
$974$	−56.1127	−1.79797
$975$	−2.11497	−0.0677331
$976$	19.5423	0.625534
$977$	−20.0587	−0.641733	−0.320867	−	0.947124i	$-0.603974\pi$
$977$	−20.0587	−0.641733	−0.320867	+	0.947124i	$0.603974\pi$
$978$	−9.96246	−0.318564
$979$	1.39989	0.0447408
$980$	−3.79758	−0.121309
$981$	22.6950	0.724596
$982$	−28.0309	−0.894501
$983$	−40.8580	−1.30317	−0.651585	−	0.758576i	$-0.725895\pi$
$983$	−40.8580	−1.30317	−0.651585	+	0.758576i	$0.725895\pi$
$984$	−3.15700	−0.100641
$985$	−0.723682	−0.0230584
$986$	58.2036	1.85358
$987$	0.459633	0.0146303
$988$	34.6680	1.10294
$989$	−30.5172	−0.970390
$990$	−42.3485	−1.34592
$991$	−24.5627	−0.780260	−0.390130	−	0.920760i	$-0.627570\pi$
$991$	−24.5627	−0.780260	−0.390130	+	0.920760i	$0.627570\pi$
$992$	−0.536566	−0.0170360
$993$	8.84488	0.280684
$994$	−39.4546	−1.25142
$995$	1.00768	0.0319457
$996$	9.82813	0.311416
$997$	39.2017	1.24153	0.620766	−	0.783996i	$-0.286822\pi$
$997$	39.2017	1.24153	0.620766	+	0.783996i	$0.286822\pi$
$998$	−29.2743	−0.926661
$999$	−0.172483	−0.00545711

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2005.2.a.f.1.3	✓	37

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2005.2.a.f.1.3	✓	37	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2005 = 5 \cdot 401 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2005.a (trivial)

\(f(q)\)	\(=\)	\(q-2.45894 q^{2} +0.456001 q^{3} +4.04640 q^{4} -1.00000 q^{5} -1.12128 q^{6} +2.81754 q^{7} -5.03197 q^{8} -2.79206 q^{9} +O(q^{10})\)	\(q-2.45894 q^{2} +0.456001 q^{3} +4.04640 q^{4} -1.00000 q^{5} -1.12128 q^{6} +2.81754 q^{7} -5.03197 q^{8} -2.79206 q^{9} +2.45894 q^{10} +6.16829 q^{11} +1.84516 q^{12} -4.63807 q^{13} -6.92816 q^{14} -0.456001 q^{15} +4.28053 q^{16} -3.36554 q^{17} +6.86552 q^{18} -1.84724 q^{19} -4.04640 q^{20} +1.28480 q^{21} -15.1675 q^{22} +5.76035 q^{23} -2.29458 q^{24} +1.00000 q^{25} +11.4047 q^{26} -2.64119 q^{27} +11.4009 q^{28} +7.03310 q^{29} +1.12128 q^{30} +1.16232 q^{31} -0.461635 q^{32} +2.81275 q^{33} +8.27566 q^{34} -2.81754 q^{35} -11.2978 q^{36} +0.0653049 q^{37} +4.54226 q^{38} -2.11497 q^{39} +5.03197 q^{40} +1.37585 q^{41} -3.15925 q^{42} -5.29780 q^{43} +24.9593 q^{44} +2.79206 q^{45} -14.1644 q^{46} +0.357747 q^{47} +1.95193 q^{48} +0.938510 q^{49} -2.45894 q^{50} -1.53469 q^{51} -18.7675 q^{52} +4.97683 q^{53} +6.49453 q^{54} -6.16829 q^{55} -14.1778 q^{56} -0.842344 q^{57} -17.2940 q^{58} +2.59693 q^{59} -1.84516 q^{60} +4.56539 q^{61} -2.85807 q^{62} -7.86674 q^{63} -7.42593 q^{64} +4.63807 q^{65} -6.91638 q^{66} +4.65806 q^{67} -13.6183 q^{68} +2.62673 q^{69} +6.92816 q^{70} +5.69482 q^{71} +14.0496 q^{72} +7.61994 q^{73} -0.160581 q^{74} +0.456001 q^{75} -7.47467 q^{76} +17.3794 q^{77} +5.20058 q^{78} +9.48038 q^{79} -4.28053 q^{80} +7.17180 q^{81} -3.38313 q^{82} +5.32643 q^{83} +5.19881 q^{84} +3.36554 q^{85} +13.0270 q^{86} +3.20710 q^{87} -31.0386 q^{88} +0.226950 q^{89} -6.86552 q^{90} -13.0679 q^{91} +23.3087 q^{92} +0.530018 q^{93} -0.879678 q^{94} +1.84724 q^{95} -0.210506 q^{96} -9.53815 q^{97} -2.30774 q^{98} -17.2222 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 37 q + 7 q^{2} + 3 q^{3} + 43 q^{4} - 37 q^{5} + 8 q^{6} - 16 q^{7} + 21 q^{8} + 54 q^{9}+O(q^{10}) \) 37 * q + 7 * q^2 + 3 * q^3 + 43 * q^4 - 37 * q^5 + 8 * q^6 - 16 * q^7 + 21 * q^8 + 54 * q^9	\( 37 q + 7 q^{2} + 3 q^{3} + 43 q^{4} - 37 q^{5} + 8 q^{6} - 16 q^{7} + 21 q^{8} + 54 q^{9} - 7 q^{10} + 42 q^{11} - 13 q^{13} + 14 q^{14} - 3 q^{15} + 63 q^{16} + 18 q^{17} + 22 q^{18} + 22 q^{19} - 43 q^{20} + 16 q^{21} - 10 q^{22} + 23 q^{23} + 23 q^{24} + 37 q^{25} + 21 q^{26} + 3 q^{27} - 18 q^{28} + 33 q^{29} - 8 q^{30} + 11 q^{31} + 54 q^{32} + 2 q^{33} + 8 q^{34} + 16 q^{35} + 91 q^{36} - 11 q^{37} + 29 q^{38} + 25 q^{39} - 21 q^{40} + 24 q^{41} + 4 q^{42} + 25 q^{43} + 84 q^{44} - 54 q^{45} + 31 q^{46} + 7 q^{47} + 4 q^{48} + 45 q^{49} + 7 q^{50} + 94 q^{51} - 43 q^{52} + 49 q^{53} + 38 q^{54} - 42 q^{55} + 46 q^{56} + 6 q^{57} + 15 q^{58} + 69 q^{59} + 9 q^{61} + 17 q^{62} - 38 q^{63} + 107 q^{64} + 13 q^{65} + 74 q^{66} + 13 q^{67} + 86 q^{68} - 14 q^{70} + 51 q^{71} + 81 q^{72} - 47 q^{73} + 79 q^{74} + 3 q^{75} + 59 q^{76} + 2 q^{77} + 20 q^{78} + 67 q^{79} - 63 q^{80} + 125 q^{81} - 24 q^{82} + 80 q^{83} + 50 q^{84} - 18 q^{85} + 69 q^{86} - 32 q^{87} - 12 q^{88} + 34 q^{89} - 22 q^{90} + 39 q^{91} + 85 q^{92} + q^{93} + 12 q^{94} - 22 q^{95} + 77 q^{96} - 14 q^{97} + 40 q^{98} + 133 q^{99}+O(q^{100}) \) 37 * q + 7 * q^2 + 3 * q^3 + 43 * q^4 - 37 * q^5 + 8 * q^6 - 16 * q^7 + 21 * q^8 + 54 * q^9 - 7 * q^10 + 42 * q^11 - 13 * q^13 + 14 * q^14 - 3 * q^15 + 63 * q^16 + 18 * q^17 + 22 * q^18 + 22 * q^19 - 43 * q^20 + 16 * q^21 - 10 * q^22 + 23 * q^23 + 23 * q^24 + 37 * q^25 + 21 * q^26 + 3 * q^27 - 18 * q^28 + 33 * q^29 - 8 * q^30 + 11 * q^31 + 54 * q^32 + 2 * q^33 + 8 * q^34 + 16 * q^35 + 91 * q^36 - 11 * q^37 + 29 * q^38 + 25 * q^39 - 21 * q^40 + 24 * q^41 + 4 * q^42 + 25 * q^43 + 84 * q^44 - 54 * q^45 + 31 * q^46 + 7 * q^47 + 4 * q^48 + 45 * q^49 + 7 * q^50 + 94 * q^51 - 43 * q^52 + 49 * q^53 + 38 * q^54 - 42 * q^55 + 46 * q^56 + 6 * q^57 + 15 * q^58 + 69 * q^59 + 9 * q^61 + 17 * q^62 - 38 * q^63 + 107 * q^64 + 13 * q^65 + 74 * q^66 + 13 * q^67 + 86 * q^68 - 14 * q^70 + 51 * q^71 + 81 * q^72 - 47 * q^73 + 79 * q^74 + 3 * q^75 + 59 * q^76 + 2 * q^77 + 20 * q^78 + 67 * q^79 - 63 * q^80 + 125 * q^81 - 24 * q^82 + 80 * q^83 + 50 * q^84 - 18 * q^85 + 69 * q^86 - 32 * q^87 - 12 * q^88 + 34 * q^89 - 22 * q^90 + 39 * q^91 + 85 * q^92 + q^93 + 12 * q^94 - 22 * q^95 + 77 * q^96 - 14 * q^97 + 40 * q^98 + 133 * q^99

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