LMFDB - Embedded newform 1997.4.a.a.1.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1997,4,Mod(1,1997)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1997, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1997.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1997 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1997.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:	$117.826814281$
Analytic rank:	$1$
Dimension:	$239$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
Character	$\chi$	$=$	1997.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.19350 q^{2} +0.362833 q^{3} +18.9725 q^{4} +7.86550 q^{5} -1.88438 q^{6} +18.4887 q^{7} -56.9857 q^{8} -26.8684 q^{9} +O(q^{10})$	\(q-5.19350 q^{2} +0.362833 q^{3} +18.9725 q^{4} +7.86550 q^{5} -1.88438 q^{6} +18.4887 q^{7} -56.9857 q^{8} -26.8684 q^{9} -40.8495 q^{10} +25.0970 q^{11} +6.88385 q^{12} -0.620782 q^{13} -96.0211 q^{14} +2.85387 q^{15} +144.175 q^{16} -69.6979 q^{17} +139.541 q^{18} -24.6676 q^{19} +149.228 q^{20} +6.70832 q^{21} -130.341 q^{22} +84.4351 q^{23} -20.6763 q^{24} -63.1339 q^{25} +3.22404 q^{26} -19.5452 q^{27} +350.777 q^{28} -35.0886 q^{29} -14.8216 q^{30} -281.442 q^{31} -292.891 q^{32} +9.10602 q^{33} +361.977 q^{34} +145.423 q^{35} -509.760 q^{36} +354.822 q^{37} +128.111 q^{38} -0.225241 q^{39} -448.221 q^{40} -8.59883 q^{41} -34.8397 q^{42} +528.843 q^{43} +476.152 q^{44} -211.333 q^{45} -438.514 q^{46} +259.921 q^{47} +52.3117 q^{48} -1.16799 q^{49} +327.886 q^{50} -25.2887 q^{51} -11.7778 q^{52} +419.325 q^{53} +101.508 q^{54} +197.400 q^{55} -1053.59 q^{56} -8.95022 q^{57} +182.233 q^{58} -762.943 q^{59} +54.1449 q^{60} -70.6025 q^{61} +1461.67 q^{62} -496.761 q^{63} +367.725 q^{64} -4.88276 q^{65} -47.2922 q^{66} -890.884 q^{67} -1322.34 q^{68} +30.6359 q^{69} -755.254 q^{70} +865.528 q^{71} +1531.11 q^{72} -565.140 q^{73} -1842.77 q^{74} -22.9071 q^{75} -468.005 q^{76} +464.011 q^{77} +1.16979 q^{78} -767.532 q^{79} +1134.01 q^{80} +718.354 q^{81} +44.6580 q^{82} -1157.63 q^{83} +127.273 q^{84} -548.209 q^{85} -2746.55 q^{86} -12.7313 q^{87} -1430.17 q^{88} -606.989 q^{89} +1097.56 q^{90} -11.4775 q^{91} +1601.94 q^{92} -102.117 q^{93} -1349.90 q^{94} -194.023 q^{95} -106.270 q^{96} -520.270 q^{97} +6.06594 q^{98} -674.315 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 239 q - 16 q^{2} - 106 q^{3} + 872 q^{4} - 85 q^{5} - 111 q^{6} - 352 q^{7} - 210 q^{8} + 1961 q^{9}+O(q^{10}) $ 239 * q - 16 * q^2 - 106 * q^3 + 872 * q^4 - 85 * q^5 - 111 * q^6 - 352 * q^7 - 210 * q^8 + 1961 * q^9	\( 239 q - 16 q^{2} - 106 q^{3} + 872 q^{4} - 85 q^{5} - 111 q^{6} - 352 q^{7} - 210 q^{8} + 1961 q^{9} - 273 q^{10} - 294 q^{11} - 864 q^{12} - 797 q^{13} - 220 q^{14} - 580 q^{15} + 2816 q^{16} - 439 q^{17} - 536 q^{18} - 1704 q^{19} - 933 q^{20} - 596 q^{21} - 1046 q^{22} - 829 q^{23} - 1237 q^{24} + 4364 q^{25} - 818 q^{26} - 3670 q^{27} - 3690 q^{28} - 316 q^{29} - 888 q^{30} - 2595 q^{31} - 1881 q^{32} - 2066 q^{33} - 2605 q^{34} - 2450 q^{35} + 5863 q^{36} - 1912 q^{37} - 1709 q^{38} - 914 q^{39} - 3582 q^{40} - 1064 q^{41} - 3228 q^{42} - 5184 q^{43} - 2656 q^{44} - 3967 q^{45} - 2521 q^{46} - 4909 q^{47} - 7461 q^{48} + 7193 q^{49} - 1906 q^{50} - 3240 q^{51} - 9614 q^{52} - 2722 q^{53} - 3754 q^{54} - 6018 q^{55} - 2347 q^{56} - 2032 q^{57} - 6709 q^{58} - 6318 q^{59} - 5821 q^{60} - 2990 q^{61} - 2117 q^{62} - 8738 q^{63} + 6866 q^{64} - 1738 q^{65} - 3080 q^{66} - 14729 q^{67} - 3897 q^{68} - 2080 q^{69} - 7445 q^{70} - 3240 q^{71} - 8263 q^{72} - 8828 q^{73} - 3103 q^{74} - 12716 q^{75} - 14843 q^{76} - 3818 q^{77} - 8029 q^{78} - 4794 q^{79} - 10336 q^{80} + 11899 q^{81} - 13447 q^{82} - 11434 q^{83} - 7957 q^{84} - 8188 q^{85} - 5196 q^{86} - 11266 q^{87} - 11861 q^{88} - 4845 q^{89} - 7759 q^{90} - 12734 q^{91} - 8644 q^{92} - 10130 q^{93} - 6909 q^{94} - 3686 q^{95} - 11958 q^{96} - 16108 q^{97} - 6845 q^{98} - 12372 q^{99}+O(q^{100}) \) 239 * q - 16 * q^2 - 106 * q^3 + 872 * q^4 - 85 * q^5 - 111 * q^6 - 352 * q^7 - 210 * q^8 + 1961 * q^9 - 273 * q^10 - 294 * q^11 - 864 * q^12 - 797 * q^13 - 220 * q^14 - 580 * q^15 + 2816 * q^16 - 439 * q^17 - 536 * q^18 - 1704 * q^19 - 933 * q^20 - 596 * q^21 - 1046 * q^22 - 829 * q^23 - 1237 * q^24 + 4364 * q^25 - 818 * q^26 - 3670 * q^27 - 3690 * q^28 - 316 * q^29 - 888 * q^30 - 2595 * q^31 - 1881 * q^32 - 2066 * q^33 - 2605 * q^34 - 2450 * q^35 + 5863 * q^36 - 1912 * q^37 - 1709 * q^38 - 914 * q^39 - 3582 * q^40 - 1064 * q^41 - 3228 * q^42 - 5184 * q^43 - 2656 * q^44 - 3967 * q^45 - 2521 * q^46 - 4909 * q^47 - 7461 * q^48 + 7193 * q^49 - 1906 * q^50 - 3240 * q^51 - 9614 * q^52 - 2722 * q^53 - 3754 * q^54 - 6018 * q^55 - 2347 * q^56 - 2032 * q^57 - 6709 * q^58 - 6318 * q^59 - 5821 * q^60 - 2990 * q^61 - 2117 * q^62 - 8738 * q^63 + 6866 * q^64 - 1738 * q^65 - 3080 * q^66 - 14729 * q^67 - 3897 * q^68 - 2080 * q^69 - 7445 * q^70 - 3240 * q^71 - 8263 * q^72 - 8828 * q^73 - 3103 * q^74 - 12716 * q^75 - 14843 * q^76 - 3818 * q^77 - 8029 * q^78 - 4794 * q^79 - 10336 * q^80 + 11899 * q^81 - 13447 * q^82 - 11434 * q^83 - 7957 * q^84 - 8188 * q^85 - 5196 * q^86 - 11266 * q^87 - 11861 * q^88 - 4845 * q^89 - 7759 * q^90 - 12734 * q^91 - 8644 * q^92 - 10130 * q^93 - 6909 * q^94 - 3686 * q^95 - 11958 * q^96 - 16108 * q^97 - 6845 * q^98 - 12372 * 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 $

$2$	−5.19350	−1.83618	−0.918091	−	0.396371i	$-0.870270\pi$
$2$	−5.19350	−1.83618	−0.918091	+	0.396371i	$0.870270\pi$
$3$	0.362833	0.0698273	0.0349137	−	0.999390i	$-0.488884\pi$
$3$	0.362833	0.0698273	0.0349137	+	0.999390i	$0.488884\pi$
$4$	18.9725	2.37156
$5$	7.86550	0.703512	0.351756	−	0.936092i	$-0.385585\pi$
$5$	7.86550	0.703512	0.351756	+	0.936092i	$0.385585\pi$
$6$	−1.88438	−0.128216
$7$	18.4887	0.998296	0.499148	−	0.866517i	$-0.333646\pi$
$7$	18.4887	0.998296	0.499148	+	0.866517i	$0.333646\pi$
$8$	−56.9857	−2.51844
$9$	−26.8684	−0.995124
$10$	−40.8495	−1.29178
$11$	25.0970	0.687911	0.343956	−	0.938986i	$-0.388233\pi$
$11$	25.0970	0.687911	0.343956	+	0.938986i	$0.388233\pi$
$12$	6.88385	0.165600
$13$	−0.620782	−0.0132442	−0.00662208	−	0.999978i	$-0.502108\pi$
$13$	−0.620782	−0.0132442	−0.00662208	+	0.999978i	$0.502108\pi$
$14$	−96.0211	−1.83305
$15$	2.85387	0.0491243
$16$	144.175	2.25274
$17$	−69.6979	−0.994366	−0.497183	−	0.867646i	$-0.665632\pi$
$17$	−69.6979	−0.994366	−0.497183	+	0.867646i	$0.665632\pi$
$18$	139.541	1.82723
$19$	−24.6676	−0.297849	−0.148924	−	0.988849i	$-0.547581\pi$
$19$	−24.6676	−0.297849	−0.148924	+	0.988849i	$0.547581\pi$
$20$	149.228	1.66842
$21$	6.70832	0.0697083
$22$	−130.341	−1.26313
$23$	84.4351	0.765475	0.382738	−	0.923857i	$-0.374981\pi$
$23$	84.4351	0.765475	0.382738	+	0.923857i	$0.374981\pi$
$24$	−20.6763	−0.175856
$25$	−63.1339	−0.505071
$26$	3.22404	0.0243187
$27$	−19.5452	−0.139314
$28$	350.777	2.36752
$29$	−35.0886	−0.224682	−0.112341	−	0.993670i	$-0.535835\pi$
$29$	−35.0886	−0.224682	−0.112341	+	0.993670i	$0.535835\pi$
$30$	−14.8216	−0.0902012
$31$	−281.442	−1.63060	−0.815298	−	0.579041i	$-0.803427\pi$
$31$	−281.442	−1.63060	−0.815298	+	0.579041i	$0.803427\pi$
$32$	−292.891	−1.61801
$33$	9.10602	0.0480350
$34$	361.977	1.82584
$35$	145.423	0.702313
$36$	−509.760	−2.36000
$37$	354.822	1.57655	0.788275	−	0.615323i	$-0.210974\pi$
$37$	354.822	1.57655	0.788275	+	0.615323i	$0.210974\pi$
$38$	128.111	0.546905
$39$	−0.225241	−0.000924804 0
$40$	−448.221	−1.77175
$41$	−8.59883	−0.0327539	−0.0163770	−	0.999866i	$-0.505213\pi$
$41$	−8.59883	−0.0327539	−0.0163770	+	0.999866i	$0.505213\pi$
$42$	−34.8397	−0.127997
$43$	528.843	1.87553	0.937765	−	0.347270i	$-0.112891\pi$
$43$	528.843	1.87553	0.937765	+	0.347270i	$0.112891\pi$
$44$	476.152	1.63142
$45$	−211.333	−0.700082
$46$	−438.514	−1.40555
$47$	259.921	0.806667	0.403334	−	0.915053i	$-0.367851\pi$
$47$	259.921	0.806667	0.403334	+	0.915053i	$0.367851\pi$
$48$	52.3117	0.157303
$49$	−1.16799	−0.00340521
$50$	327.886	0.927402
$51$	−25.2887	−0.0694339
$52$	−11.7778	−0.0314093
$53$	419.325	1.08677	0.543384	−	0.839484i	$-0.317143\pi$
$53$	419.325	1.08677	0.543384	+	0.839484i	$0.317143\pi$
$54$	101.508	0.255806
$55$	197.400	0.483954
$56$	−1053.59	−2.51414
$57$	−8.95022	−0.0207980
$58$	182.233	0.412557
$59$	−762.943	−1.68350	−0.841752	−	0.539865i	$-0.818475\pi$
$59$	−762.943	−1.68350	−0.841752	+	0.539865i	$0.818475\pi$
$60$	54.1449	0.116501
$61$	−70.6025	−0.148192	−0.0740961	−	0.997251i	$-0.523607\pi$
$61$	−70.6025	−0.148192	−0.0740961	+	0.997251i	$0.523607\pi$
$62$	1461.67	2.99407
$63$	−496.761	−0.993428
$64$	367.725	0.718213
$65$	−4.88276	−0.00931742
$66$	−47.2922	−0.0882010
$67$	−890.884	−1.62446	−0.812230	−	0.583337i	$-0.801747\pi$
$67$	−890.884	−1.62446	−0.812230	+	0.583337i	$0.801747\pi$
$68$	−1322.34	−2.35820
$69$	30.6359	0.0534511
$70$	−755.254	−1.28957
$71$	865.528	1.44675	0.723375	−	0.690455i	$-0.242590\pi$
$71$	865.528	1.44675	0.723375	+	0.690455i	$0.242590\pi$
$72$	1531.11	2.50616
$73$	−565.140	−0.906091	−0.453045	−	0.891487i	$-0.649662\pi$
$73$	−565.140	−0.906091	−0.453045	+	0.891487i	$0.649662\pi$
$74$	−1842.77	−2.89483
$75$	−22.9071	−0.0352678
$76$	−468.005	−0.706367
$77$	464.011	0.686739
$78$	1.16979	0.00169811
$79$	−767.532	−1.09309	−0.546545	−	0.837430i	$-0.684057\pi$
$79$	−767.532	−1.09309	−0.546545	+	0.837430i	$0.684057\pi$
$80$	1134.01	1.58483
$81$	718.354	0.985396
$82$	44.6580	0.0601421
$83$	−1157.63	−1.53092	−0.765462	−	0.643481i	$-0.777490\pi$
$83$	−1157.63	−1.53092	−0.765462	+	0.643481i	$0.777490\pi$
$84$	127.273	0.165318
$85$	−548.209	−0.699549
$86$	−2746.55	−3.44381
$87$	−12.7313	−0.0156890
$88$	−1430.17	−1.73246
$89$	−606.989	−0.722929	−0.361464	−	0.932386i	$-0.617723\pi$
$89$	−606.989	−0.722929	−0.361464	+	0.932386i	$0.617723\pi$
$90$	1097.56	1.28548
$91$	−11.4775	−0.0132216
$92$	1601.94	1.81537
$93$	−102.117	−0.113860
$94$	−1349.90	−1.48119
$95$	−194.023	−0.209540
$96$	−106.270	−0.112981
$97$	−520.270	−0.544592	−0.272296	−	0.962214i	$-0.587783\pi$
$97$	−520.270	−0.544592	−0.272296	+	0.962214i	$0.587783\pi$
$98$	6.06594	0.00625258
$99$	−674.315	−0.684557
$100$	−1197.81	−1.19781
$101$	−649.736	−0.640111	−0.320055	−	0.947399i	$-0.603701\pi$
$101$	−649.736	−0.640111	−0.320055	+	0.947399i	$0.603701\pi$
$102$	131.337	0.127493
$103$	845.928	0.809240	0.404620	−	0.914485i	$-0.367404\pi$
$103$	845.928	0.809240	0.404620	+	0.914485i	$0.367404\pi$
$104$	35.3757	0.0333546
$105$	52.7643	0.0490406
$106$	−2177.77	−1.99550
$107$	429.148	0.387732	0.193866	−	0.981028i	$-0.437897\pi$
$107$	429.148	0.387732	0.193866	+	0.981028i	$0.437897\pi$
$108$	−370.822	−0.330392
$109$	−516.889	−0.454211	−0.227106	−	0.973870i	$-0.572926\pi$
$109$	−516.889	−0.454211	−0.227106	+	0.973870i	$0.572926\pi$
$110$	−1025.20	−0.888627
$111$	128.741	0.110086
$112$	2665.62	2.24890
$113$	1129.97	0.940695	0.470348	−	0.882481i	$-0.344129\pi$
$113$	1129.97	0.940695	0.470348	+	0.882481i	$0.344129\pi$
$114$	46.4830	0.0381889
$115$	664.124	0.538521
$116$	−665.718	−0.532848
$117$	16.6794	0.0131796
$118$	3962.35	3.09122
$119$	−1288.62	−0.992672
$120$	−162.630	−0.123716
$121$	−701.142	−0.526778
$122$	366.675	0.272108
$123$	−3.11994	−0.00228712
$124$	−5339.66	−3.86706
$125$	−1479.77	−1.05884
$126$	2579.93	1.82411
$127$	−1929.46	−1.34813	−0.674064	−	0.738673i	$-0.735453\pi$
$127$	−1929.46	−1.34813	−0.674064	+	0.738673i	$0.735453\pi$
$128$	433.344	0.299239
$129$	191.882	0.130963
$130$	25.3587	0.0171085
$131$	1639.50	1.09347	0.546733	−	0.837307i	$-0.315871\pi$
$131$	1639.50	1.09347	0.546733	+	0.837307i	$0.315871\pi$
$132$	172.764	0.113918
$133$	−456.071	−0.297341
$134$	4626.81	2.98280
$135$	−153.733	−0.0980091
$136$	3971.78	2.50425
$137$	−1847.44	−1.15210	−0.576050	−	0.817414i	$-0.695407\pi$
$137$	−1847.44	−1.15210	−0.576050	+	0.817414i	$0.695407\pi$
$138$	−159.107	−0.0981458
$139$	2748.18	1.67696	0.838481	−	0.544931i	$-0.183444\pi$
$139$	2748.18	1.67696	0.838481	+	0.544931i	$0.183444\pi$
$140$	2759.03	1.66558
$141$	94.3080	0.0563274
$142$	−4495.13	−2.65650
$143$	−15.5798	−0.00911081
$144$	−3873.76	−2.24176
$145$	−275.989	−0.158067
$146$	2935.06	1.66375
$147$	−0.423784	−0.000237776 0
$148$	6731.86	3.73889
$149$	1180.40	0.649007	0.324504	−	0.945884i	$-0.394803\pi$
$149$	1180.40	0.649007	0.324504	+	0.945884i	$0.394803\pi$
$150$	118.968	0.0647580
$151$	376.049	0.202665	0.101333	−	0.994853i	$-0.467689\pi$
$151$	376.049	0.202665	0.101333	+	0.994853i	$0.467689\pi$
$152$	1405.70	0.750113
$153$	1872.67	0.989518
$154$	−2409.84	−1.26098
$155$	−2213.68	−1.14714
$156$	−4.27337	−0.00219323
$157$	−3143.93	−1.59817	−0.799086	−	0.601217i	$-0.794683\pi$
$157$	−3143.93	−1.59817	−0.799086	+	0.601217i	$0.794683\pi$
$158$	3986.18	2.00711
$159$	152.145	0.0758861
$160$	−2303.73	−1.13829
$161$	1561.09	0.764171
$162$	−3730.77	−1.80937
$163$	−1774.45	−0.852674	−0.426337	−	0.904564i	$-0.640196\pi$
$163$	−1774.45	−0.852674	−0.426337	+	0.904564i	$0.640196\pi$
$164$	−163.141	−0.0776779
$165$	71.6234	0.0337932
$166$	6012.17	2.81106
$167$	862.626	0.399713	0.199856	−	0.979825i	$-0.435952\pi$
$167$	862.626	0.399713	0.199856	+	0.979825i	$0.435952\pi$
$168$	−382.278	−0.175556
$169$	−2196.61	−0.999825
$170$	2847.13	1.28450
$171$	662.777	0.296397
$172$	10033.5	4.44794
$173$	−3453.35	−1.51765	−0.758825	−	0.651294i	$-0.774226\pi$
$173$	−3453.35	−1.51765	−0.758825	+	0.651294i	$0.774226\pi$
$174$	66.1201	0.0288078
$175$	−1167.26	−0.504210
$176$	3618.37	1.54969
$177$	−276.821	−0.117554
$178$	3152.40	1.32743
$179$	1736.24	0.724985	0.362493	−	0.931987i	$-0.381926\pi$
$179$	1736.24	0.724985	0.362493	+	0.931987i	$0.381926\pi$
$180$	−4009.51	−1.66029
$181$	1701.19	0.698612	0.349306	−	0.937009i	$-0.386417\pi$
$181$	1701.19	0.698612	0.349306	+	0.937009i	$0.386417\pi$
$182$	59.6082	0.0242772
$183$	−25.6170	−0.0103479
$184$	−4811.59	−1.92780
$185$	2790.85	1.10912
$186$	530.343	0.209068
$187$	−1749.21	−0.684036
$188$	4931.35	1.91306
$189$	−361.366	−0.139077
$190$	1007.66	0.384754
$191$	632.649	0.239669	0.119835	−	0.992794i	$-0.461764\pi$
$191$	632.649	0.239669	0.119835	+	0.992794i	$0.461764\pi$
$192$	133.423	0.0501508
$193$	−2393.85	−0.892814	−0.446407	−	0.894830i	$-0.647297\pi$
$193$	−2393.85	−0.892814	−0.446407	+	0.894830i	$0.647297\pi$
$194$	2702.03	0.999970
$195$	−1.77163	−0.000650610 0
$196$	−22.1596	−0.00807566
$197$	162.969	0.0589393	0.0294696	−	0.999566i	$-0.490618\pi$
$197$	162.969	0.0589393	0.0294696	+	0.999566i	$0.490618\pi$
$198$	3502.06	1.25697
$199$	4097.51	1.45962	0.729811	−	0.683649i	$-0.239608\pi$
$199$	4097.51	1.45962	0.729811	+	0.683649i	$0.239608\pi$
$200$	3597.73	1.27199
$201$	−323.243	−0.113432
$202$	3374.41	1.17536
$203$	−648.742	−0.224299
$204$	−479.790	−0.164667
$205$	−67.6341	−0.0230428
$206$	−4393.33	−1.48591
$207$	−2268.63	−0.761743
$208$	−89.5016	−0.0298357
$209$	−619.082	−0.204894
$210$	−274.031	−0.0900475
$211$	1542.05	0.503124	0.251562	−	0.967841i	$-0.419056\pi$
$211$	1542.05	0.503124	0.251562	+	0.967841i	$0.419056\pi$
$212$	7955.64	2.57734
$213$	314.042	0.101023
$214$	−2228.78	−0.711946
$215$	4159.62	1.31946
$216$	1113.80	0.350854
$217$	−5203.50	−1.62782
$218$	2684.47	0.834014
$219$	−205.052	−0.0632699
$220$	3745.18	1.14773
$221$	43.2672	0.0131695
$222$	−668.618	−0.202138
$223$	3967.13	1.19129	0.595647	−	0.803246i	$-0.296896\pi$
$223$	3967.13	1.19129	0.595647	+	0.803246i	$0.296896\pi$
$224$	−5415.17	−1.61525
$225$	1696.30	0.502608
$226$	−5868.50	−1.72729
$227$	2367.21	0.692146	0.346073	−	0.938208i	$-0.387515\pi$
$227$	2367.21	0.692146	0.346073	+	0.938208i	$0.387515\pi$
$228$	−169.808	−0.0493237
$229$	−4416.44	−1.27444	−0.637220	−	0.770682i	$-0.719916\pi$
$229$	−4416.44	−1.27444	−0.637220	+	0.770682i	$0.719916\pi$
$230$	−3449.13	−0.988822
$231$	168.358	0.0479531
$232$	1999.55	0.565848
$233$	2721.40	0.765172	0.382586	−	0.923920i	$-0.375034\pi$
$233$	2721.40	0.765172	0.382586	+	0.923920i	$0.375034\pi$
$234$	−86.6245	−0.0242001
$235$	2044.41	0.567500
$236$	−14474.9	−3.99253
$237$	−278.486	−0.0763275
$238$	6692.47	1.82273
$239$	−3455.28	−0.935161	−0.467581	−	0.883950i	$-0.654874\pi$
$239$	−3455.28	−0.935161	−0.467581	+	0.883950i	$0.654874\pi$
$240$	411.458	0.110664
$241$	−305.559	−0.0816714	−0.0408357	−	0.999166i	$-0.513002\pi$
$241$	−305.559	−0.0816714	−0.0408357	+	0.999166i	$0.513002\pi$
$242$	3641.38	0.967260
$243$	788.364	0.208122
$244$	−1339.51	−0.351447
$245$	−9.18680	−0.00239560
$246$	16.2034	0.00419956
$247$	15.3132	0.00394476
$248$	16038.2	4.10655
$249$	−420.028	−0.106900
$250$	7685.18	1.94421
$251$	−4285.15	−1.07759	−0.538797	−	0.842436i	$-0.681121\pi$
$251$	−4285.15	−1.07759	−0.538797	+	0.842436i	$0.681121\pi$
$252$	−9424.79	−2.35598
$253$	2119.07	0.526579
$254$	10020.7	2.47541
$255$	−198.909	−0.0488476
$256$	−5192.37	−1.26767
$257$	−5718.16	−1.38790	−0.693948	−	0.720026i	$-0.744130\pi$
$257$	−5718.16	−1.38790	−0.693948	+	0.720026i	$0.744130\pi$
$258$	−996.539	−0.240472
$259$	6560.20	1.57386
$260$	−92.6382	−0.0220968
$261$	942.773	0.223587
$262$	−8514.76	−2.00780
$263$	5711.01	1.33900	0.669498	−	0.742814i	$-0.266509\pi$
$263$	5711.01	1.33900	0.669498	+	0.742814i	$0.266509\pi$
$264$	−518.913	−0.120973
$265$	3298.20	0.764554
$266$	2368.61	0.545973
$267$	−220.236	−0.0504802
$268$	−16902.3	−3.85251
$269$	6814.52	1.54457	0.772284	−	0.635278i	$-0.219114\pi$
$269$	6814.52	1.54457	0.772284	+	0.635278i	$0.219114\pi$
$270$	798.413	0.179963
$271$	−6587.75	−1.47667	−0.738334	−	0.674435i	$-0.764387\pi$
$271$	−6587.75	−1.47667	−0.738334	+	0.674435i	$0.764387\pi$
$272$	−10048.7	−2.24005
$273$	−4.16440	−0.000923228 0
$274$	9594.71	2.11546
$275$	−1584.47	−0.347444
$276$	581.238	0.126762
$277$	−7070.00	−1.53356	−0.766779	−	0.641912i	$-0.778142\pi$
$277$	−7070.00	−1.53356	−0.766779	+	0.641912i	$0.778142\pi$
$278$	−14272.7	−3.07921
$279$	7561.88	1.62265
$280$	−8287.02	−1.76873
$281$	602.699	0.127950	0.0639751	−	0.997951i	$-0.479622\pi$
$281$	602.699	0.127950	0.0639751	+	0.997951i	$0.479622\pi$
$282$	−489.789	−0.103427
$283$	2132.52	0.447933	0.223967	−	0.974597i	$-0.428099\pi$
$283$	2132.52	0.447933	0.223967	+	0.974597i	$0.428099\pi$
$284$	16421.2	3.43106
$285$	−70.3980	−0.0146316
$286$	80.9136	0.0167291
$287$	−158.981	−0.0326981
$288$	7869.49	1.61012
$289$	−55.1992	−0.0112353
$290$	1433.35	0.290239
$291$	−188.771	−0.0380274
$292$	−10722.1	−2.14885
$293$	−5632.77	−1.12310	−0.561552	−	0.827441i	$-0.689796\pi$
$293$	−5632.77	−1.12310	−0.561552	+	0.827441i	$0.689796\pi$
$294$	2.20093	0.000436601 0
$295$	−6000.93	−1.18436
$296$	−20219.8	−3.97044
$297$	−490.526	−0.0958358
$298$	−6130.41	−1.19170
$299$	−52.4158	−0.0101381
$300$	−434.604	−0.0836396
$301$	9777.62	1.87233
$302$	−1953.01	−0.372130
$303$	−235.746	−0.0446972
$304$	−3556.46	−0.670977
$305$	−555.324	−0.104255
$306$	−9725.71	−1.81693
$307$	6258.43	1.16348	0.581738	−	0.813376i	$-0.302373\pi$
$307$	6258.43	1.16348	0.581738	+	0.813376i	$0.302373\pi$
$308$	8803.44	1.62864
$309$	306.931	0.0565071
$310$	11496.8	2.10636
$311$	−2397.82	−0.437196	−0.218598	−	0.975815i	$-0.570148\pi$
$311$	−2397.82	−0.437196	−0.218598	+	0.975815i	$0.570148\pi$
$312$	12.8355	0.00232906
$313$	−7388.17	−1.33420	−0.667100	−	0.744969i	$-0.732464\pi$
$313$	−7388.17	−1.33420	−0.667100	+	0.744969i	$0.732464\pi$
$314$	16328.0	2.93453
$315$	−3907.27	−0.698889
$316$	−14562.0	−2.59233
$317$	6422.42	1.13792	0.568958	−	0.822367i	$-0.307347\pi$
$317$	6422.42	1.13792	0.568958	+	0.822367i	$0.307347\pi$
$318$	−790.166	−0.139341
$319$	−880.618	−0.154562
$320$	2892.34	0.505271
$321$	155.709	0.0270743
$322$	−8107.55	−1.40316
$323$	1719.28	0.296171
$324$	13629.0	2.33693
$325$	39.1924	0.00668924
$326$	9215.63	1.56566
$327$	−187.545	−0.0317163
$328$	490.010	0.0824886
$329$	4805.60	0.805293
$330$	−371.977	−0.0620504
$331$	6899.55	1.14572	0.572860	−	0.819653i	$-0.305834\pi$
$331$	6899.55	1.14572	0.572860	+	0.819653i	$0.305834\pi$
$332$	−21963.2	−3.63068
$333$	−9533.48	−1.56886
$334$	−4480.05	−0.733945
$335$	−7007.25	−1.14283
$336$	967.175	0.157035
$337$	−5298.77	−0.856506	−0.428253	−	0.903659i	$-0.640871\pi$
$337$	−5298.77	−0.856506	−0.428253	+	0.903659i	$0.640871\pi$
$338$	11408.1	1.83586
$339$	409.990	0.0656862
$340$	−10400.9	−1.65902
$341$	−7063.35	−1.12171
$342$	−3442.14	−0.544238
$343$	−6363.22	−1.00170
$344$	−30136.5	−4.72340
$345$	240.966	0.0376035
$346$	17935.0	2.78668
$347$	−5325.77	−0.823926	−0.411963	−	0.911201i	$-0.635157\pi$
$347$	−5325.77	−0.823926	−0.411963	+	0.911201i	$0.635157\pi$
$348$	−241.545	−0.0372073
$349$	−12531.7	−1.92207	−0.961037	−	0.276419i	$-0.910852\pi$
$349$	−12531.7	−1.92207	−0.961037	+	0.276419i	$0.910852\pi$
$350$	6062.19	0.925822
$351$	12.1333	0.00184510
$352$	−7350.67	−1.11305
$353$	3885.76	0.585887	0.292944	−	0.956130i	$-0.405365\pi$
$353$	3885.76	0.585887	0.292944	+	0.956130i	$0.405365\pi$
$354$	1437.67	0.215851
$355$	6807.81	1.01781
$356$	−11516.1	−1.71447
$357$	−467.556	−0.0693156
$358$	−9017.15	−1.33120
$359$	11811.8	1.73650	0.868252	−	0.496123i	$-0.165243\pi$
$359$	11811.8	1.73650	0.868252	+	0.496123i	$0.165243\pi$
$360$	12043.0	1.76311
$361$	−6250.51	−0.911286
$362$	−8835.16	−1.28278
$363$	−254.398	−0.0367835
$364$	−217.756	−0.0313558
$365$	−4445.11	−0.637446
$366$	133.042	0.0190006
$367$	−12961.6	−1.84357	−0.921784	−	0.387704i	$-0.873268\pi$
$367$	−12961.6	−1.84357	−0.921784	+	0.387704i	$0.873268\pi$
$368$	12173.5	1.72442
$369$	231.036	0.0325942
$370$	−14494.3	−2.03655
$371$	7752.78	1.08492
$372$	−1937.41	−0.270026
$373$	10705.5	1.48608	0.743041	−	0.669246i	$-0.233383\pi$
$373$	10705.5	1.48608	0.743041	+	0.669246i	$0.233383\pi$
$374$	9084.52	1.25601
$375$	−536.909	−0.0739356
$376$	−14811.8	−2.03154
$377$	21.7824	0.00297573
$378$	1876.76	0.255370
$379$	6359.29	0.861886	0.430943	−	0.902379i	$-0.358181\pi$
$379$	6359.29	0.861886	0.430943	+	0.902379i	$0.358181\pi$
$380$	−3681.10	−0.496938
$381$	−700.073	−0.0941361
$382$	−3285.66	−0.440077
$383$	424.797	0.0566739	0.0283370	−	0.999598i	$-0.490979\pi$
$383$	424.797	0.0566739	0.0283370	+	0.999598i	$0.490979\pi$
$384$	157.232	0.0208950
$385$	3649.68	0.483129
$386$	12432.5	1.63937
$387$	−14209.1	−1.86639
$388$	−9870.82	−1.29153
$389$	−11860.9	−1.54595	−0.772974	−	0.634438i	$-0.781232\pi$
$389$	−11860.9	−1.54595	−0.772974	+	0.634438i	$0.781232\pi$
$390$	9.20097	0.00119464
$391$	−5884.95	−0.761163
$392$	66.5585	0.00857579
$393$	594.866	0.0763537
$394$	−846.378	−0.108223
$395$	−6037.03	−0.769002
$396$	−12793.4	−1.62347
$397$	−363.943	−0.0460096	−0.0230048	−	0.999735i	$-0.507323\pi$
$397$	−363.943	−0.0460096	−0.0230048	+	0.999735i	$0.507323\pi$
$398$	−21280.4	−2.68013
$399$	−165.478	−0.0207625
$400$	−9102.36	−1.13779
$401$	−44.3454	−0.00552245	−0.00276123	−	0.999996i	$-0.500879\pi$
$401$	−44.3454	−0.00552245	−0.00276123	+	0.999996i	$0.500879\pi$
$402$	1678.76	0.208281
$403$	174.714	0.0215959
$404$	−12327.1	−1.51806
$405$	5650.21	0.693238
$406$	3369.25	0.411854
$407$	8904.96	1.08453
$408$	1441.10	0.174865
$409$	−5691.85	−0.688127	−0.344064	−	0.938946i	$-0.611804\pi$
$409$	−5691.85	−0.688127	−0.344064	+	0.938946i	$0.611804\pi$
$410$	351.258	0.0423107
$411$	−670.314	−0.0804481
$412$	16049.4	1.91916
$413$	−14105.8	−1.68063
$414$	11782.1	1.39870
$415$	−9105.37	−1.07702
$416$	181.821	0.0214291
$417$	997.132	0.117098
$418$	3215.20	0.376222
$419$	−8659.32	−1.00963	−0.504816	−	0.863227i	$-0.668440\pi$
$419$	−8659.32	−1.00963	−0.504816	+	0.863227i	$0.668440\pi$
$420$	1001.07	0.116303
$421$	11238.9	1.30107	0.650534	−	0.759477i	$-0.274545\pi$
$421$	11238.9	1.30107	0.650534	+	0.759477i	$0.274545\pi$
$422$	−8008.65	−0.923827
$423$	−6983.65	−0.802734
$424$	−23895.5	−2.73696
$425$	4400.30	0.502226
$426$	−1630.98	−0.185496
$427$	−1305.35	−0.147940
$428$	8142.01	0.919530
$429$	−5.65286	−0.000636183 0
$430$	−21603.0	−2.42276
$431$	−9909.46	−1.10748	−0.553738	−	0.832691i	$-0.686799\pi$
$431$	−9909.46	−1.10748	−0.553738	+	0.832691i	$0.686799\pi$
$432$	−2817.94	−0.313839
$433$	−6001.95	−0.666132	−0.333066	−	0.942903i	$-0.608083\pi$
$433$	−6001.95	−0.666132	−0.333066	+	0.942903i	$0.608083\pi$
$434$	27024.4	2.98897
$435$	−100.138	−0.0110374
$436$	−9806.68	−1.07719
$437$	−2082.81	−0.227996
$438$	1064.94	0.116175
$439$	−15347.5	−1.66855	−0.834276	−	0.551348i	$-0.814114\pi$
$439$	−15347.5	−1.66855	−0.834276	+	0.551348i	$0.814114\pi$
$440$	−11249.0	−1.21881
$441$	31.3819	0.00338860
$442$	−224.709	−0.0241817
$443$	−5971.50	−0.640439	−0.320220	−	0.947343i	$-0.603757\pi$
$443$	−5971.50	−0.640439	−0.320220	+	0.947343i	$0.603757\pi$
$444$	2442.54	0.261076
$445$	−4774.27	−0.508589
$446$	−20603.3	−2.18743
$447$	428.288	0.0453184
$448$	6798.75	0.716989
$449$	−1077.91	−0.113295	−0.0566476	−	0.998394i	$-0.518041\pi$
$449$	−1077.91	−0.113295	−0.0566476	+	0.998394i	$0.518041\pi$
$450$	−8809.76	−0.922880
$451$	−215.805	−0.0225318
$452$	21438.3	2.23092
$453$	136.443	0.0141516
$454$	−12294.1	−1.27090
$455$	−90.2760	−0.00930154
$456$	510.034	0.0523784
$457$	−10761.4	−1.10153	−0.550764	−	0.834661i	$-0.685663\pi$
$457$	−10761.4	−1.10153	−0.550764	+	0.834661i	$0.685663\pi$
$458$	22936.8	2.34010
$459$	1362.26	0.138529
$460$	12600.1	1.27714
$461$	10446.0	1.05535	0.527676	−	0.849446i	$-0.323063\pi$
$461$	10446.0	1.05535	0.527676	+	0.849446i	$0.323063\pi$
$462$	−874.371	−0.0880507
$463$	3139.17	0.315097	0.157548	−	0.987511i	$-0.449641\pi$
$463$	3139.17	0.315097	0.157548	+	0.987511i	$0.449641\pi$
$464$	−5058.91	−0.506151
$465$	−803.198	−0.0801020
$466$	−14133.6	−1.40499
$467$	11975.0	1.18658	0.593292	−	0.804987i	$-0.297828\pi$
$467$	11975.0	1.18658	0.593292	+	0.804987i	$0.297828\pi$
$468$	316.450	0.0312562
$469$	−16471.3	−1.62169
$470$	−10617.6	−1.04203
$471$	−1140.72	−0.111596
$472$	43476.8	4.23979
$473$	13272.4	1.29020
$474$	1446.32	0.140151
$475$	1557.36	0.150435
$476$	−24448.4	−2.35418
$477$	−11266.6	−1.08147
$478$	17945.0	1.71713
$479$	15602.4	1.48829	0.744145	−	0.668018i	$-0.232857\pi$
$479$	15602.4	1.48829	0.744145	+	0.668018i	$0.232857\pi$
$480$	−835.871	−0.0794835
$481$	−220.267	−0.0208801
$482$	1586.92	0.149964
$483$	566.417	0.0533600
$484$	−13302.4	−1.24929
$485$	−4092.19	−0.383127
$486$	−4094.37	−0.382149
$487$	−4750.50	−0.442024	−0.221012	−	0.975271i	$-0.570936\pi$
$487$	−4750.50	−0.442024	−0.221012	+	0.975271i	$0.570936\pi$
$488$	4023.33	0.373213
$489$	−643.830	−0.0595399
$490$	47.7117	0.00439876
$491$	1595.84	0.146679	0.0733395	−	0.997307i	$-0.476634\pi$
$491$	1595.84	0.146679	0.0733395	+	0.997307i	$0.476634\pi$
$492$	−59.1930	−0.00542404
$493$	2445.60	0.223417
$494$	−79.5292	−0.00724329
$495$	−5303.82	−0.481594
$496$	−40577.0	−3.67331
$497$	16002.5	1.44429
$498$	2181.42	0.196288
$499$	20470.2	1.83642	0.918210	−	0.396094i	$-0.129635\pi$
$499$	20470.2	1.83642	0.918210	+	0.396094i	$0.129635\pi$
$500$	−28074.9	−2.51109
$501$	312.990	0.0279109
$502$	22254.9	1.97866
$503$	−11110.0	−0.984833	−0.492417	−	0.870360i	$-0.663886\pi$
$503$	−11110.0	−0.984833	−0.492417	+	0.870360i	$0.663886\pi$
$504$	28308.3	2.50189
$505$	−5110.50	−0.450325
$506$	−11005.4	−0.966894
$507$	−797.005	−0.0698151
$508$	−36606.7	−3.19717
$509$	−7325.33	−0.637897	−0.318949	−	0.947772i	$-0.603330\pi$
$509$	−7325.33	−0.637897	−0.318949	+	0.947772i	$0.603330\pi$
$510$	1033.03	0.0896930
$511$	−10448.7	−0.904547
$512$	23499.9	2.02843
$513$	482.134	0.0414946
$514$	29697.3	2.54843
$515$	6653.65	0.569310
$516$	3640.48	0.310587
$517$	6523.23	0.554916
$518$	−34070.4	−2.88990
$519$	−1252.99	−0.105973
$520$	278.248	0.0234653
$521$	−14723.5	−1.23809	−0.619047	−	0.785354i	$-0.712481\pi$
$521$	−14723.5	−1.23809	−0.619047	+	0.785354i	$0.712481\pi$
$522$	−4896.29	−0.410546
$523$	−19270.5	−1.61116	−0.805582	−	0.592485i	$-0.798147\pi$
$523$	−19270.5	−1.61116	−0.805582	+	0.592485i	$0.798147\pi$
$524$	31105.4	2.59322
$525$	−423.522	−0.0352077
$526$	−29660.1	−2.45864
$527$	19615.9	1.62141
$528$	1312.87	0.108210
$529$	−5037.72	−0.414048
$530$	−17129.2	−1.40386
$531$	20499.0	1.67529
$532$	−8652.81	−0.705163
$533$	5.33800	0.000433798 0
$534$	1143.79	0.0926907
$535$	3375.46	0.272774
$536$	50767.7	4.09110
$537$	629.964	0.0506238
$538$	−35391.2	−2.83611
$539$	−29.3129	−0.00234248
$540$	−2916.70	−0.232435
$541$	22576.4	1.79415	0.897076	−	0.441876i	$-0.145687\pi$
$541$	22576.4	1.79415	0.897076	+	0.441876i	$0.145687\pi$
$542$	34213.5	2.71143
$543$	617.250	0.0487822
$544$	20413.9	1.60889
$545$	−4065.59	−0.319543
$546$	21.6279	0.00169521
$547$	21770.0	1.70168	0.850840	−	0.525424i	$-0.176093\pi$
$547$	21770.0	1.70168	0.850840	+	0.525424i	$0.176093\pi$
$548$	−35050.6	−2.73228
$549$	1896.97	0.147470
$550$	8228.95	0.637970
$551$	865.550	0.0669214
$552$	−1745.80	−0.134613
$553$	−14190.7	−1.09123
$554$	36718.1	2.81589
$555$	1012.61	0.0774470
$556$	52139.8	3.97702
$557$	−3206.15	−0.243894	−0.121947	−	0.992537i	$-0.538914\pi$
$557$	−3206.15	−0.243894	−0.121947	+	0.992537i	$0.538914\pi$
$558$	−39272.7	−2.97947
$559$	−328.296	−0.0248398
$560$	20966.4	1.58213
$561$	−634.671	−0.0477644
$562$	−3130.12	−0.234940
$563$	11921.2	0.892393	0.446197	−	0.894935i	$-0.352778\pi$
$563$	11921.2	0.892393	0.446197	+	0.894935i	$0.352778\pi$
$564$	1789.26	0.133584
$565$	8887.77	0.661790
$566$	−11075.3	−0.822487
$567$	13281.4	0.983717
$568$	−49322.7	−3.64355
$569$	−964.747	−0.0710796	−0.0355398	−	0.999368i	$-0.511315\pi$
$569$	−964.747	−0.0710796	−0.0355398	+	0.999368i	$0.511315\pi$
$570$	365.612	0.0268663
$571$	10359.7	0.759267	0.379634	−	0.925137i	$-0.376050\pi$
$571$	10359.7	0.759267	0.379634	+	0.925137i	$0.376050\pi$
$572$	−295.587	−0.0216068
$573$	229.546	0.0167355
$574$	825.669	0.0600397
$575$	−5330.71	−0.386619
$576$	−9880.16	−0.714711
$577$	−12232.0	−0.882535	−0.441268	−	0.897376i	$-0.645471\pi$
$577$	−12232.0	−0.882535	−0.441268	+	0.897376i	$0.645471\pi$
$578$	286.678	0.0206301
$579$	−868.569	−0.0623428
$580$	−5236.21	−0.374865
$581$	−21403.1	−1.52832
$582$	980.385	0.0698252
$583$	10523.8	0.747600
$584$	32204.9	2.28193
$585$	131.192	0.00927199
$586$	29253.8	2.06222
$587$	6388.94	0.449233	0.224616	−	0.974447i	$-0.427887\pi$
$587$	6388.94	0.449233	0.224616	+	0.974447i	$0.427887\pi$
$588$	−8.04024	−0.000563901 0
$589$	6942.49	0.485671
$590$	31165.8	2.17471
$591$	59.1304	0.00411557
$592$	51156.6	3.55156
$593$	7789.90	0.539449	0.269724	−	0.962938i	$-0.413067\pi$
$593$	7789.90	0.539449	0.269724	+	0.962938i	$0.413067\pi$
$594$	2547.55	0.175972
$595$	−10135.7	−0.698357
$596$	22395.1	1.53916
$597$	1486.71	0.101921
$598$	272.222	0.0186153
$599$	4263.36	0.290812	0.145406	−	0.989372i	$-0.453551\pi$
$599$	4263.36	0.290812	0.145406	+	0.989372i	$0.453551\pi$
$600$	1305.38	0.0888196
$601$	−22486.4	−1.52619	−0.763095	−	0.646286i	$-0.776321\pi$
$601$	−22486.4	−1.52619	−0.763095	+	0.646286i	$0.776321\pi$
$602$	−50780.1	−3.43795
$603$	23936.6	1.61654
$604$	7134.58	0.480632
$605$	−5514.83	−0.370595
$606$	1224.35	0.0820722
$607$	−10555.3	−0.705808	−0.352904	−	0.935659i	$-0.614806\pi$
$607$	−10555.3	−0.705808	−0.352904	+	0.935659i	$0.614806\pi$
$608$	7224.90	0.481922
$609$	−235.385	−0.0156622
$610$	2884.08	0.191431
$611$	−161.354	−0.0106836
$612$	35529.2	2.34670
$613$	−10091.5	−0.664915	−0.332458	−	0.943118i	$-0.607878\pi$
$613$	−10091.5	−0.664915	−0.332458	+	0.943118i	$0.607878\pi$
$614$	−32503.2	−2.13635
$615$	−24.5399	−0.00160901
$616$	−26442.0	−1.72951
$617$	−13970.7	−0.911570	−0.455785	−	0.890090i	$-0.650641\pi$
$617$	−13970.7	−0.911570	−0.455785	+	0.890090i	$0.650641\pi$
$618$	−1594.05	−0.103757
$619$	−27946.7	−1.81466	−0.907328	−	0.420424i	$-0.861881\pi$
$619$	−27946.7	−1.81466	−0.907328	+	0.420424i	$0.861881\pi$
$620$	−41999.1	−2.72052
$621$	−1650.30	−0.106641
$622$	12453.1	0.802771
$623$	−11222.4	−0.721697
$624$	−32.4742	−0.00208334
$625$	−3747.38	−0.239832
$626$	38370.5	2.44983
$627$	−224.623	−0.0143072
$628$	−59648.2	−3.79016
$629$	−24730.4	−1.56767
$630$	20292.4	1.28329
$631$	−20854.3	−1.31568	−0.657842	−	0.753156i	$-0.728531\pi$
$631$	−20854.3	−1.31568	−0.657842	+	0.753156i	$0.728531\pi$
$632$	43738.3	2.75288
$633$	559.507	0.0351318
$634$	−33354.9	−2.08942
$635$	−15176.2	−0.948423
$636$	2886.57	0.179969
$637$	0.725065	4.50991e−5 0
$638$	4573.49	0.283803
$639$	−23255.3	−1.43970
$640$	3408.47	0.210518
$641$	21374.4	1.31707	0.658533	−	0.752552i	$-0.271177\pi$
$641$	21374.4	1.31707	0.658533	+	0.752552i	$0.271177\pi$
$642$	−808.676	−0.0497133
$643$	−22471.3	−1.37820	−0.689100	−	0.724666i	$-0.741994\pi$
$643$	−22471.3	−1.37820	−0.689100	+	0.724666i	$0.741994\pi$
$644$	29617.8	1.81228
$645$	1509.25	0.0921342
$646$	−8929.08	−0.543824
$647$	−25239.0	−1.53361	−0.766805	−	0.641880i	$-0.778155\pi$
$647$	−25239.0	−1.53361	−0.766805	+	0.641880i	$0.778155\pi$
$648$	−40935.9	−2.48166
$649$	−19147.6	−1.15810
$650$	−203.546	−0.0122827
$651$	−1888.00	−0.113666
$652$	−33665.8	−2.02217
$653$	−16709.4	−1.00136	−0.500680	−	0.865633i	$-0.666917\pi$
$653$	−16709.4	−1.00136	−0.500680	+	0.865633i	$0.666917\pi$
$654$	974.014	0.0582369
$655$	12895.5	0.769266
$656$	−1239.74	−0.0737861
$657$	15184.4	0.901673
$658$	−24957.9	−1.47866
$659$	7835.10	0.463144	0.231572	−	0.972818i	$-0.425613\pi$
$659$	7835.10	0.463144	0.231572	+	0.972818i	$0.425613\pi$
$660$	1358.87	0.0801426
$661$	−17245.6	−1.01479	−0.507396	−	0.861713i	$-0.669392\pi$
$661$	−17245.6	−1.01479	−0.507396	+	0.861713i	$0.669392\pi$
$662$	−35832.8	−2.10375
$663$	15.6988	0.000919594 0
$664$	65968.5	3.85553
$665$	−3587.23	−0.209183
$666$	49512.2	2.88072
$667$	−2962.71	−0.171989
$668$	16366.2	0.947943
$669$	1439.41	0.0831849
$670$	36392.2	2.09844
$671$	−1771.91	−0.101943
$672$	−1964.80	−0.112789
$673$	−6288.78	−0.360200	−0.180100	−	0.983648i	$-0.557642\pi$
$673$	−6288.78	−0.360200	−0.180100	+	0.983648i	$0.557642\pi$
$674$	27519.2	1.57270
$675$	1233.97	0.0703635
$676$	−41675.3	−2.37115
$677$	6308.07	0.358108	0.179054	−	0.983839i	$-0.442696\pi$
$677$	6308.07	0.358108	0.179054	+	0.983839i	$0.442696\pi$
$678$	−2129.29	−0.120612
$679$	−9619.12	−0.543664
$680$	31240.1	1.76177
$681$	858.902	0.0483307
$682$	36683.5	2.05965
$683$	−18594.9	−1.04175	−0.520874	−	0.853634i	$-0.674394\pi$
$683$	−18594.9	−1.04175	−0.520874	+	0.853634i	$0.674394\pi$
$684$	12574.5	0.702923
$685$	−14531.1	−0.810516
$686$	33047.4	1.83929
$687$	−1602.43	−0.0889907
$688$	76246.2	4.22509
$689$	−260.310	−0.0143933
$690$	−1251.46	−0.0690468
$691$	8122.15	0.447151	0.223575	−	0.974687i	$-0.428227\pi$
$691$	8122.15	0.447151	0.223575	+	0.974687i	$0.428227\pi$
$692$	−65518.7	−3.59920
$693$	−12467.2	−0.683391
$694$	27659.4	1.51288
$695$	21615.8	1.17976
$696$	725.502	0.0395116
$697$	599.320	0.0325694
$698$	65083.2	3.52928
$699$	987.416	0.0534299
$700$	−22145.9	−1.19577
$701$	−21868.2	−1.17825	−0.589123	−	0.808043i	$-0.700527\pi$
$701$	−21868.2	−1.17825	−0.589123	+	0.808043i	$0.700527\pi$
$702$	−63.0145	−0.00338794
$703$	−8752.60	−0.469574
$704$	9228.78	0.494067
$705$	741.780	0.0396270
$706$	−20180.7	−1.07580
$707$	−12012.8	−0.639020
$708$	−5251.98	−0.278788
$709$	33154.9	1.75622	0.878108	−	0.478462i	$-0.158806\pi$
$709$	33154.9	1.75622	0.878108	+	0.478462i	$0.158806\pi$
$710$	−35356.4	−1.86888
$711$	20622.3	1.08776
$712$	34589.7	1.82065
$713$	−23763.6	−1.24818
$714$	2428.25	0.127276
$715$	−122.543	−0.00640956
$716$	32940.7	1.71935
$717$	−1253.69	−0.0652998
$718$	−61344.9	−3.18854
$719$	4552.65	0.236141	0.118070	−	0.993005i	$-0.462329\pi$
$719$	4552.65	0.236141	0.118070	+	0.993005i	$0.462329\pi$
$720$	−30469.0	−1.57710
$721$	15640.1	0.807861
$722$	32462.1	1.67329
$723$	−110.867	−0.00570290
$724$	32275.9	1.65680
$725$	2215.28	0.113481
$726$	1321.21	0.0675411
$727$	−1426.27	−0.0727613	−0.0363806	−	0.999338i	$-0.511583\pi$
$727$	−1426.27	−0.0727613	−0.0363806	+	0.999338i	$0.511583\pi$
$728$	654.051	0.0332977
$729$	−19109.5	−0.970864
$730$	23085.7	1.17047
$731$	−36859.3	−1.86496
$732$	−486.017	−0.0245406
$733$	−5871.35	−0.295857	−0.147928	−	0.988998i	$-0.547261\pi$
$733$	−5871.35	−0.295857	−0.147928	+	0.988998i	$0.547261\pi$
$734$	67316.1	3.38512
$735$	−3.33328	−0.000167279 0
$736$	−24730.2	−1.23854
$737$	−22358.5	−1.11748
$738$	−1199.89	−0.0598489
$739$	−10596.8	−0.527482	−0.263741	−	0.964594i	$-0.584956\pi$
$739$	−10596.8	−0.527482	−0.263741	+	0.964594i	$0.584956\pi$
$740$	52949.4	2.63035
$741$	5.55614	0.000275452 0
$742$	−40264.1	−1.99210
$743$	−32246.9	−1.59223	−0.796114	−	0.605147i	$-0.793114\pi$
$743$	−32246.9	−1.59223	−0.796114	+	0.605147i	$0.793114\pi$
$744$	5819.18	0.286749
$745$	9284.44	0.456584
$746$	−55598.9	−2.72871
$747$	31103.7	1.52346
$748$	−33186.8	−1.62223
$749$	7934.39	0.387071
$750$	2788.44	0.135759
$751$	40004.5	1.94379	0.971895	−	0.235415i	$-0.0756448\pi$
$751$	40004.5	1.94379	0.971895	+	0.235415i	$0.0756448\pi$
$752$	37474.2	1.81721
$753$	−1554.79	−0.0752455
$754$	−113.127	−0.00546398
$755$	2957.81	0.142577
$756$	−6856.01	−0.329829
$757$	13567.0	0.651389	0.325695	−	0.945475i	$-0.394402\pi$
$757$	13567.0	0.651389	0.325695	+	0.945475i	$0.394402\pi$
$758$	−33027.0	−1.58258
$759$	768.867	0.0367696
$760$	11056.5	0.527714
$761$	−20245.6	−0.964394	−0.482197	−	0.876063i	$-0.660161\pi$
$761$	−20245.6	−0.964394	−0.482197	+	0.876063i	$0.660161\pi$
$762$	3635.83	0.172851
$763$	−9556.61	−0.453437
$764$	12002.9	0.568391
$765$	14729.5	0.696138
$766$	−2206.18	−0.104064
$767$	473.621	0.0222966
$768$	−1883.97	−0.0885179
$769$	21343.8	1.00088	0.500441	−	0.865771i	$-0.333171\pi$
$769$	21343.8	1.00088	0.500441	+	0.865771i	$0.333171\pi$
$770$	−18954.6	−0.887113
$771$	−2074.74	−0.0969130
$772$	−45417.3	−2.11736
$773$	31605.3	1.47059	0.735294	−	0.677748i	$-0.237044\pi$
$773$	31605.3	1.47059	0.735294	+	0.677748i	$0.237044\pi$
$774$	73795.2	3.42702
$775$	17768.5	0.823567
$776$	29647.9	1.37152
$777$	2380.26	0.109899
$778$	61599.9	2.83864
$779$	212.112	0.00975572
$780$	−33.6122	−0.00154296
$781$	21722.1	0.995236
$782$	30563.5	1.39763
$783$	685.815	0.0313014
$784$	−168.395	−0.00767105
$785$	−24728.6	−1.12433
$786$	−3089.44	−0.140199
$787$	21696.8	0.982729	0.491364	−	0.870954i	$-0.336498\pi$
$787$	21696.8	0.982729	0.491364	+	0.870954i	$0.336498\pi$
$788$	3091.92	0.139778
$789$	2072.14	0.0934984
$790$	31353.3	1.41203
$791$	20891.7	0.939092
$792$	38426.3	1.72401
$793$	43.8288	0.00196268
$794$	1890.14	0.0844819
$795$	1196.70	0.0533868
$796$	77740.0	3.46158
$797$	9559.84	0.424877	0.212438	−	0.977174i	$-0.431860\pi$
$797$	9559.84	0.424877	0.212438	+	0.977174i	$0.431860\pi$
$798$	859.410	0.0381238
$799$	−18116.0	−0.802123
$800$	18491.3	0.817209
$801$	16308.8	0.719404
$802$	230.308	0.0101402
$803$	−14183.3	−0.623310
$804$	−6132.72	−0.269010
$805$	12278.8	0.537603
$806$	−907.379	−0.0396539
$807$	2472.53	0.107853
$808$	37025.7	1.61208
$809$	−6818.40	−0.296319	−0.148160	−	0.988963i	$-0.547335\pi$
$809$	−6818.40	−0.296319	−0.148160	+	0.988963i	$0.547335\pi$
$810$	−29344.4	−1.27291
$811$	−25893.1	−1.12112	−0.560560	−	0.828114i	$-0.689414\pi$
$811$	−25893.1	−1.12112	−0.560560	+	0.828114i	$0.689414\pi$
$812$	−12308.3	−0.531940
$813$	−2390.25	−0.103112
$814$	−46248.0	−1.99139
$815$	−13957.0	−0.599866
$816$	−3646.01	−0.156417
$817$	−13045.3	−0.558625
$818$	29560.7	1.26353
$819$	308.380	0.0131571
$820$	−1283.19	−0.0546474
$821$	27170.7	1.15501	0.577505	−	0.816387i	$-0.304026\pi$
$821$	27170.7	1.15501	0.577505	+	0.816387i	$0.304026\pi$
$822$	3481.28	0.147717
$823$	7351.39	0.311365	0.155683	−	0.987807i	$-0.450242\pi$
$823$	7351.39	0.311365	0.155683	+	0.987807i	$0.450242\pi$
$824$	−48205.8	−2.03802
$825$	−574.898	−0.0242611
$826$	73258.6	3.08595
$827$	29800.5	1.25304	0.626520	−	0.779405i	$-0.284479\pi$
$827$	29800.5	1.25304	0.626520	+	0.779405i	$0.284479\pi$
$828$	−43041.6	−1.80652
$829$	35471.1	1.48608	0.743041	−	0.669246i	$-0.233383\pi$
$829$	35471.1	1.48608	0.743041	+	0.669246i	$0.233383\pi$
$830$	47288.8	1.97761
$831$	−2565.23	−0.107084
$832$	−228.277	−0.00951212
$833$	81.4062	0.00338602
$834$	−5178.61	−0.215013
$835$	6784.99	0.281203
$836$	−11745.5	−0.485918
$837$	5500.85	0.227165
$838$	44972.2	1.85387
$839$	−45372.1	−1.86701	−0.933504	−	0.358567i	$-0.883265\pi$
$839$	−45372.1	−1.86701	−0.933504	+	0.358567i	$0.883265\pi$
$840$	−3006.81	−0.123506
$841$	−23157.8	−0.949518
$842$	−58369.2	−2.38900
$843$	218.679	0.00893442
$844$	29256.5	1.19319
$845$	−17277.5	−0.703388
$846$	36269.6	1.47397
$847$	−12963.2	−0.525880
$848$	60456.4	2.44821
$849$	773.749	0.0312780
$850$	−22853.0	−0.922177
$851$	29959.4	1.20681
$852$	5958.17	0.239582
$853$	22098.5	0.887030	0.443515	−	0.896267i	$-0.353731\pi$
$853$	22098.5	0.887030	0.443515	+	0.896267i	$0.353731\pi$
$854$	6779.34	0.271644
$855$	5213.07	0.208519
$856$	−24455.3	−0.976477
$857$	−46076.6	−1.83658	−0.918289	−	0.395911i	$-0.870429\pi$
$857$	−46076.6	−1.83658	−0.918289	+	0.395911i	$0.870429\pi$
$858$	29.3581	0.00116815
$859$	−7484.40	−0.297281	−0.148640	−	0.988891i	$-0.547490\pi$
$859$	−7484.40	−0.297281	−0.148640	+	0.988891i	$0.547490\pi$
$860$	78918.3	3.12918
$861$	−57.6836	−0.00228322
$862$	51464.8	2.03353
$863$	9779.90	0.385761	0.192880	−	0.981222i	$-0.438217\pi$
$863$	9779.90	0.385761	0.192880	+	0.981222i	$0.438217\pi$
$864$	5724.61	0.225411
$865$	−27162.3	−1.06768
$866$	31171.1	1.22314
$867$	−20.0281	−0.000784534 0
$868$	−98723.3	−3.86047
$869$	−19262.7	−0.751949
$870$	520.068	0.0202666
$871$	553.045	0.0215146
$872$	29455.3	1.14390
$873$	13978.8	0.541937
$874$	10817.1	0.418642
$875$	−27359.0	−1.05703
$876$	−3890.34	−0.150048
$877$	46840.6	1.80353	0.901765	−	0.432227i	$-0.142272\pi$
$877$	46840.6	1.80353	0.901765	+	0.432227i	$0.142272\pi$
$878$	79707.1	3.06376
$879$	−2043.75	−0.0784234
$880$	28460.3	1.09022
$881$	3603.07	0.137787	0.0688937	−	0.997624i	$-0.478053\pi$
$881$	3603.07	0.137787	0.0688937	+	0.997624i	$0.478053\pi$
$882$	−162.982	−0.00622209
$883$	15710.5	0.598755	0.299377	−	0.954135i	$-0.403221\pi$
$883$	15710.5	0.598755	0.299377	+	0.954135i	$0.403221\pi$
$884$	820.887	0.0312324
$885$	−2177.34	−0.0827010
$886$	31013.0	1.17596
$887$	−24507.4	−0.927710	−0.463855	−	0.885911i	$-0.653534\pi$
$887$	−24507.4	−0.927710	−0.463855	+	0.885911i	$0.653534\pi$
$888$	−7336.41	−0.277245
$889$	−35673.3	−1.34583
$890$	24795.2	0.933861
$891$	18028.5	0.677865
$892$	75266.3	2.82523
$893$	−6411.62	−0.240265
$894$	−2224.32	−0.0832129
$895$	13656.4	0.510036
$896$	8011.97	0.298729
$897$	−19.0182	−0.000707914 0
$898$	5598.11	0.208031
$899$	9875.40	0.366366
$900$	32183.1	1.19197
$901$	−29226.1	−1.08065
$902$	1120.78	0.0413725
$903$	3547.65	0.130740
$904$	−64392.1	−2.36908
$905$	13380.7	0.491482
$906$	−708.617	−0.0259848
$907$	−27754.2	−1.01606	−0.508028	−	0.861341i	$-0.669625\pi$
$907$	−27754.2	−1.01606	−0.508028	+	0.861341i	$0.669625\pi$
$908$	44911.8	1.64147
$909$	17457.3	0.636990
$910$	468.849	0.0170793
$911$	15888.7	0.577844	0.288922	−	0.957353i	$-0.406703\pi$
$911$	15888.7	0.577844	0.288922	+	0.957353i	$0.406703\pi$
$912$	−1290.40	−0.0468525
$913$	−29053.1	−1.05314
$914$	55889.5	2.02260
$915$	−201.490	−0.00727985
$916$	−83791.0	−3.02241
$917$	30312.3	1.09160
$918$	−7074.91	−0.254365
$919$	3301.54	0.118507	0.0592535	−	0.998243i	$-0.481128\pi$
$919$	3301.54	0.118507	0.0592535	+	0.998243i	$0.481128\pi$
$920$	−37845.6	−1.35623
$921$	2270.77	0.0812425
$922$	−54251.2	−1.93782
$923$	−537.305	−0.0191610
$924$	3194.18	0.113724
$925$	−22401.3	−0.796270
$926$	−16303.3	−0.578574
$927$	−22728.7	−0.805295
$928$	10277.1	0.363538
$929$	24340.0	0.859601	0.429801	−	0.902924i	$-0.358584\pi$
$929$	24340.0	0.859601	0.429801	+	0.902924i	$0.358584\pi$
$930$	4171.41	0.147082
$931$	28.8114	0.00101424
$932$	51631.8	1.81465
$933$	−870.009	−0.0305282
$934$	−62192.0	−2.17878
$935$	−13758.4	−0.481227
$936$	−950.487	−0.0331919
$937$	10596.6	0.369451	0.184726	−	0.982790i	$-0.440860\pi$
$937$	10596.6	0.369451	0.184726	+	0.982790i	$0.440860\pi$
$938$	85543.7	2.97772
$939$	−2680.68	−0.0931635
$940$	38787.5	1.34586
$941$	203.663	0.00705550	0.00352775	−	0.999994i	$-0.498877\pi$
$941$	203.663	0.00705550	0.00352775	+	0.999994i	$0.498877\pi$
$942$	5924.35	0.204911
$943$	−726.042	−0.0250723
$944$	−109998.	−3.79250
$945$	−2842.32	−0.0978421
$946$	−68930.1	−2.36904
$947$	−40703.6	−1.39671	−0.698357	−	0.715750i	$-0.746085\pi$
$947$	−40703.6	−1.39671	−0.698357	+	0.715750i	$0.746085\pi$
$948$	−5283.58	−0.181015
$949$	350.829	0.0120004
$950$	−8088.16	−0.276226
$951$	2330.27	0.0794576
$952$	73433.1	2.49998
$953$	10927.8	0.371443	0.185722	−	0.982602i	$-0.440538\pi$
$953$	10927.8	0.371443	0.185722	+	0.982602i	$0.440538\pi$
$954$	58513.0	1.98577
$955$	4976.10	0.168610
$956$	−65555.3	−2.21779
$957$	−319.517	−0.0107926
$958$	−81031.0	−2.73277
$959$	−34156.8	−1.15014
$960$	1049.44	0.0352817
$961$	49418.6	1.65884
$962$	1143.96	0.0383396
$963$	−11530.5	−0.385841
$964$	−5797.22	−0.193689
$965$	−18828.8	−0.628105
$966$	−2941.69	−0.0979786
$967$	−54397.7	−1.80901	−0.904504	−	0.426464i	$-0.859759\pi$
$967$	−54397.7	−1.80901	−0.904504	+	0.426464i	$0.859759\pi$
$968$	39955.0	1.32666
$969$	623.812	0.0206808
$970$	21252.8	0.703491
$971$	42582.2	1.40734	0.703671	−	0.710526i	$-0.251543\pi$
$971$	42582.2	1.40734	0.703671	+	0.710526i	$0.251543\pi$
$972$	14957.2	0.493573
$973$	50810.3	1.67410
$974$	24671.7	0.811636
$975$	14.2203	0.000467092 0
$976$	−10179.2	−0.333839
$977$	56510.2	1.85048	0.925242	−	0.379378i	$-0.123862\pi$
$977$	56510.2	1.85048	0.925242	+	0.379378i	$0.123862\pi$
$978$	3343.74	0.109326
$979$	−15233.6	−0.497311
$980$	−174.296	−0.00568132
$981$	13888.0	0.451996
$982$	−8288.01	−0.269329
$983$	16711.3	0.542224	0.271112	−	0.962548i	$-0.412609\pi$
$983$	16711.3	0.542224	0.271112	+	0.962548i	$0.412609\pi$
$984$	177.792	0.00575996
$985$	1281.83	0.0414645
$986$	−12701.2	−0.410233
$987$	1743.63	0.0562314
$988$	290.529	0.00935524
$989$	44652.9	1.43567
$990$	27545.4	0.884294
$991$	−17369.3	−0.556764	−0.278382	−	0.960470i	$-0.589798\pi$
$991$	−17369.3	−0.556764	−0.278382	+	0.960470i	$0.589798\pi$
$992$	82431.7	2.63832
$993$	2503.39	0.0800026
$994$	−83109.0	−2.65197
$995$	32229.0	1.02686
$996$	−7968.97	−0.253521
$997$	22929.2	0.728359	0.364180	−	0.931329i	$-0.381349\pi$
$997$	22929.2	0.728359	0.364180	+	0.931329i	$0.381349\pi$
$998$	−106312.	−3.37200
$999$	−6935.08	−0.219636

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1997.4.a.a.1.12	✓	239

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1997.4.a.a.1.12	✓	239	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:	\( N \)	\(=\)	\( 1997 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1997.a (trivial)

\(f(q)\)	\(=\)	\(q-5.19350 q^{2} +0.362833 q^{3} +18.9725 q^{4} +7.86550 q^{5} -1.88438 q^{6} +18.4887 q^{7} -56.9857 q^{8} -26.8684 q^{9} +O(q^{10})\)	\(q-5.19350 q^{2} +0.362833 q^{3} +18.9725 q^{4} +7.86550 q^{5} -1.88438 q^{6} +18.4887 q^{7} -56.9857 q^{8} -26.8684 q^{9} -40.8495 q^{10} +25.0970 q^{11} +6.88385 q^{12} -0.620782 q^{13} -96.0211 q^{14} +2.85387 q^{15} +144.175 q^{16} -69.6979 q^{17} +139.541 q^{18} -24.6676 q^{19} +149.228 q^{20} +6.70832 q^{21} -130.341 q^{22} +84.4351 q^{23} -20.6763 q^{24} -63.1339 q^{25} +3.22404 q^{26} -19.5452 q^{27} +350.777 q^{28} -35.0886 q^{29} -14.8216 q^{30} -281.442 q^{31} -292.891 q^{32} +9.10602 q^{33} +361.977 q^{34} +145.423 q^{35} -509.760 q^{36} +354.822 q^{37} +128.111 q^{38} -0.225241 q^{39} -448.221 q^{40} -8.59883 q^{41} -34.8397 q^{42} +528.843 q^{43} +476.152 q^{44} -211.333 q^{45} -438.514 q^{46} +259.921 q^{47} +52.3117 q^{48} -1.16799 q^{49} +327.886 q^{50} -25.2887 q^{51} -11.7778 q^{52} +419.325 q^{53} +101.508 q^{54} +197.400 q^{55} -1053.59 q^{56} -8.95022 q^{57} +182.233 q^{58} -762.943 q^{59} +54.1449 q^{60} -70.6025 q^{61} +1461.67 q^{62} -496.761 q^{63} +367.725 q^{64} -4.88276 q^{65} -47.2922 q^{66} -890.884 q^{67} -1322.34 q^{68} +30.6359 q^{69} -755.254 q^{70} +865.528 q^{71} +1531.11 q^{72} -565.140 q^{73} -1842.77 q^{74} -22.9071 q^{75} -468.005 q^{76} +464.011 q^{77} +1.16979 q^{78} -767.532 q^{79} +1134.01 q^{80} +718.354 q^{81} +44.6580 q^{82} -1157.63 q^{83} +127.273 q^{84} -548.209 q^{85} -2746.55 q^{86} -12.7313 q^{87} -1430.17 q^{88} -606.989 q^{89} +1097.56 q^{90} -11.4775 q^{91} +1601.94 q^{92} -102.117 q^{93} -1349.90 q^{94} -194.023 q^{95} -106.270 q^{96} -520.270 q^{97} +6.06594 q^{98} -674.315 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 239 q - 16 q^{2} - 106 q^{3} + 872 q^{4} - 85 q^{5} - 111 q^{6} - 352 q^{7} - 210 q^{8} + 1961 q^{9}+O(q^{10}) \) 239 * q - 16 * q^2 - 106 * q^3 + 872 * q^4 - 85 * q^5 - 111 * q^6 - 352 * q^7 - 210 * q^8 + 1961 * q^9	\( 239 q - 16 q^{2} - 106 q^{3} + 872 q^{4} - 85 q^{5} - 111 q^{6} - 352 q^{7} - 210 q^{8} + 1961 q^{9} - 273 q^{10} - 294 q^{11} - 864 q^{12} - 797 q^{13} - 220 q^{14} - 580 q^{15} + 2816 q^{16} - 439 q^{17} - 536 q^{18} - 1704 q^{19} - 933 q^{20} - 596 q^{21} - 1046 q^{22} - 829 q^{23} - 1237 q^{24} + 4364 q^{25} - 818 q^{26} - 3670 q^{27} - 3690 q^{28} - 316 q^{29} - 888 q^{30} - 2595 q^{31} - 1881 q^{32} - 2066 q^{33} - 2605 q^{34} - 2450 q^{35} + 5863 q^{36} - 1912 q^{37} - 1709 q^{38} - 914 q^{39} - 3582 q^{40} - 1064 q^{41} - 3228 q^{42} - 5184 q^{43} - 2656 q^{44} - 3967 q^{45} - 2521 q^{46} - 4909 q^{47} - 7461 q^{48} + 7193 q^{49} - 1906 q^{50} - 3240 q^{51} - 9614 q^{52} - 2722 q^{53} - 3754 q^{54} - 6018 q^{55} - 2347 q^{56} - 2032 q^{57} - 6709 q^{58} - 6318 q^{59} - 5821 q^{60} - 2990 q^{61} - 2117 q^{62} - 8738 q^{63} + 6866 q^{64} - 1738 q^{65} - 3080 q^{66} - 14729 q^{67} - 3897 q^{68} - 2080 q^{69} - 7445 q^{70} - 3240 q^{71} - 8263 q^{72} - 8828 q^{73} - 3103 q^{74} - 12716 q^{75} - 14843 q^{76} - 3818 q^{77} - 8029 q^{78} - 4794 q^{79} - 10336 q^{80} + 11899 q^{81} - 13447 q^{82} - 11434 q^{83} - 7957 q^{84} - 8188 q^{85} - 5196 q^{86} - 11266 q^{87} - 11861 q^{88} - 4845 q^{89} - 7759 q^{90} - 12734 q^{91} - 8644 q^{92} - 10130 q^{93} - 6909 q^{94} - 3686 q^{95} - 11958 q^{96} - 16108 q^{97} - 6845 q^{98} - 12372 q^{99}+O(q^{100}) \) 239 * q - 16 * q^2 - 106 * q^3 + 872 * q^4 - 85 * q^5 - 111 * q^6 - 352 * q^7 - 210 * q^8 + 1961 * q^9 - 273 * q^10 - 294 * q^11 - 864 * q^12 - 797 * q^13 - 220 * q^14 - 580 * q^15 + 2816 * q^16 - 439 * q^17 - 536 * q^18 - 1704 * q^19 - 933 * q^20 - 596 * q^21 - 1046 * q^22 - 829 * q^23 - 1237 * q^24 + 4364 * q^25 - 818 * q^26 - 3670 * q^27 - 3690 * q^28 - 316 * q^29 - 888 * q^30 - 2595 * q^31 - 1881 * q^32 - 2066 * q^33 - 2605 * q^34 - 2450 * q^35 + 5863 * q^36 - 1912 * q^37 - 1709 * q^38 - 914 * q^39 - 3582 * q^40 - 1064 * q^41 - 3228 * q^42 - 5184 * q^43 - 2656 * q^44 - 3967 * q^45 - 2521 * q^46 - 4909 * q^47 - 7461 * q^48 + 7193 * q^49 - 1906 * q^50 - 3240 * q^51 - 9614 * q^52 - 2722 * q^53 - 3754 * q^54 - 6018 * q^55 - 2347 * q^56 - 2032 * q^57 - 6709 * q^58 - 6318 * q^59 - 5821 * q^60 - 2990 * q^61 - 2117 * q^62 - 8738 * q^63 + 6866 * q^64 - 1738 * q^65 - 3080 * q^66 - 14729 * q^67 - 3897 * q^68 - 2080 * q^69 - 7445 * q^70 - 3240 * q^71 - 8263 * q^72 - 8828 * q^73 - 3103 * q^74 - 12716 * q^75 - 14843 * q^76 - 3818 * q^77 - 8029 * q^78 - 4794 * q^79 - 10336 * q^80 + 11899 * q^81 - 13447 * q^82 - 11434 * q^83 - 7957 * q^84 - 8188 * q^85 - 5196 * q^86 - 11266 * q^87 - 11861 * q^88 - 4845 * q^89 - 7759 * q^90 - 12734 * q^91 - 8644 * q^92 - 10130 * q^93 - 6909 * q^94 - 3686 * q^95 - 11958 * q^96 - 16108 * q^97 - 6845 * q^98 - 12372 * q^99

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