LMFDB - Embedded newform 4027.2.a.c.1.17

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4027,2,Mod(1,4027)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4027, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("4027.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4027 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4027.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:	$32.1557568940$
Analytic rank:	$0$
Dimension:	$174$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.17
Character	$\chi$	$=$	4027.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.36541 q^{2} -2.31312 q^{3} +3.59515 q^{4} -1.08906 q^{5} +5.47148 q^{6} -0.547834 q^{7} -3.77318 q^{8} +2.35053 q^{9} +O(q^{10})$	\(q-2.36541 q^{2} -2.31312 q^{3} +3.59515 q^{4} -1.08906 q^{5} +5.47148 q^{6} -0.547834 q^{7} -3.77318 q^{8} +2.35053 q^{9} +2.57606 q^{10} +2.77385 q^{11} -8.31602 q^{12} +2.85967 q^{13} +1.29585 q^{14} +2.51912 q^{15} +1.73481 q^{16} -0.316358 q^{17} -5.55997 q^{18} -1.28066 q^{19} -3.91532 q^{20} +1.26721 q^{21} -6.56128 q^{22} +6.80143 q^{23} +8.72783 q^{24} -3.81396 q^{25} -6.76428 q^{26} +1.50229 q^{27} -1.96955 q^{28} -0.874497 q^{29} -5.95874 q^{30} +6.66183 q^{31} +3.44283 q^{32} -6.41625 q^{33} +0.748317 q^{34} +0.596622 q^{35} +8.45053 q^{36} +8.99003 q^{37} +3.02928 q^{38} -6.61477 q^{39} +4.10920 q^{40} +7.72982 q^{41} -2.99746 q^{42} +0.240790 q^{43} +9.97241 q^{44} -2.55986 q^{45} -16.0882 q^{46} +1.32538 q^{47} -4.01283 q^{48} -6.69988 q^{49} +9.02157 q^{50} +0.731776 q^{51} +10.2809 q^{52} +2.99865 q^{53} -3.55353 q^{54} -3.02088 q^{55} +2.06708 q^{56} +2.96232 q^{57} +2.06854 q^{58} +5.20806 q^{59} +9.05661 q^{60} -7.09930 q^{61} -15.7579 q^{62} -1.28770 q^{63} -11.6133 q^{64} -3.11434 q^{65} +15.1771 q^{66} +8.88197 q^{67} -1.13736 q^{68} -15.7325 q^{69} -1.41125 q^{70} +2.57084 q^{71} -8.86899 q^{72} +12.8924 q^{73} -21.2651 q^{74} +8.82215 q^{75} -4.60417 q^{76} -1.51961 q^{77} +15.6466 q^{78} +3.45817 q^{79} -1.88930 q^{80} -10.5266 q^{81} -18.2842 q^{82} +0.522288 q^{83} +4.55580 q^{84} +0.344532 q^{85} -0.569567 q^{86} +2.02282 q^{87} -10.4662 q^{88} -9.16932 q^{89} +6.05512 q^{90} -1.56663 q^{91} +24.4522 q^{92} -15.4096 q^{93} -3.13507 q^{94} +1.39471 q^{95} -7.96369 q^{96} +18.2279 q^{97} +15.8479 q^{98} +6.52003 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 174 q + 21 q^{2} + 17 q^{3} + 187 q^{4} + 72 q^{5} + 21 q^{6} + 24 q^{7} + 54 q^{8} + 197 q^{9}+O(q^{10}) $ 174 * q + 21 * q^2 + 17 * q^3 + 187 * q^4 + 72 * q^5 + 21 * q^6 + 24 * q^7 + 54 * q^8 + 197 * q^9	\( 174 q + 21 q^{2} + 17 q^{3} + 187 q^{4} + 72 q^{5} + 21 q^{6} + 24 q^{7} + 54 q^{8} + 197 q^{9} + 20 q^{10} + 35 q^{11} + 23 q^{12} + 91 q^{13} + 18 q^{14} + 16 q^{15} + 201 q^{16} + 148 q^{17} + 39 q^{18} + 36 q^{19} + 128 q^{20} + 57 q^{21} + 17 q^{22} + 96 q^{23} + 24 q^{24} + 226 q^{25} + 44 q^{26} + 62 q^{27} + 32 q^{28} + 122 q^{29} + 25 q^{30} + 23 q^{31} + 104 q^{32} + 91 q^{33} + 6 q^{34} + 80 q^{35} + 222 q^{36} + 71 q^{37} + 125 q^{38} + 16 q^{39} + 53 q^{40} + 97 q^{41} + 14 q^{42} + 38 q^{43} + 70 q^{44} + 185 q^{45} - 23 q^{46} + 110 q^{47} + 36 q^{48} + 210 q^{49} + 51 q^{50} + 33 q^{51} + 118 q^{52} + 214 q^{53} + 8 q^{54} + 37 q^{55} + 41 q^{56} + 76 q^{57} + 2 q^{58} + 66 q^{59} - 12 q^{60} + 114 q^{61} + 175 q^{62} + 62 q^{63} + 190 q^{64} + 128 q^{65} + 12 q^{66} - 6 q^{67} + 348 q^{68} + 115 q^{69} - 38 q^{70} + 54 q^{71} + 101 q^{72} + 107 q^{73} + 71 q^{74} - q^{75} + 31 q^{76} + 368 q^{77} - 14 q^{78} - 14 q^{79} + 205 q^{80} + 222 q^{81} + 26 q^{82} + 246 q^{83} + 41 q^{84} + 87 q^{85} + 33 q^{86} + 100 q^{87} - 6 q^{88} + 147 q^{89} + 50 q^{90} - 23 q^{91} + 189 q^{92} + 117 q^{93} + 23 q^{94} + 42 q^{95} + 38 q^{96} + 52 q^{97} + 148 q^{98} + 38 q^{99}+O(q^{100}) \) 174 * q + 21 * q^2 + 17 * q^3 + 187 * q^4 + 72 * q^5 + 21 * q^6 + 24 * q^7 + 54 * q^8 + 197 * q^9 + 20 * q^10 + 35 * q^11 + 23 * q^12 + 91 * q^13 + 18 * q^14 + 16 * q^15 + 201 * q^16 + 148 * q^17 + 39 * q^18 + 36 * q^19 + 128 * q^20 + 57 * q^21 + 17 * q^22 + 96 * q^23 + 24 * q^24 + 226 * q^25 + 44 * q^26 + 62 * q^27 + 32 * q^28 + 122 * q^29 + 25 * q^30 + 23 * q^31 + 104 * q^32 + 91 * q^33 + 6 * q^34 + 80 * q^35 + 222 * q^36 + 71 * q^37 + 125 * q^38 + 16 * q^39 + 53 * q^40 + 97 * q^41 + 14 * q^42 + 38 * q^43 + 70 * q^44 + 185 * q^45 - 23 * q^46 + 110 * q^47 + 36 * q^48 + 210 * q^49 + 51 * q^50 + 33 * q^51 + 118 * q^52 + 214 * q^53 + 8 * q^54 + 37 * q^55 + 41 * q^56 + 76 * q^57 + 2 * q^58 + 66 * q^59 - 12 * q^60 + 114 * q^61 + 175 * q^62 + 62 * q^63 + 190 * q^64 + 128 * q^65 + 12 * q^66 - 6 * q^67 + 348 * q^68 + 115 * q^69 - 38 * q^70 + 54 * q^71 + 101 * q^72 + 107 * q^73 + 71 * q^74 - q^75 + 31 * q^76 + 368 * q^77 - 14 * q^78 - 14 * q^79 + 205 * q^80 + 222 * q^81 + 26 * q^82 + 246 * q^83 + 41 * q^84 + 87 * q^85 + 33 * q^86 + 100 * q^87 - 6 * q^88 + 147 * q^89 + 50 * q^90 - 23 * q^91 + 189 * q^92 + 117 * q^93 + 23 * q^94 + 42 * q^95 + 38 * q^96 + 52 * q^97 + 148 * q^98 + 38 * 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 $

$2$	−2.36541	−1.67260	−0.836298	−	0.548276i	$-0.815284\pi$
$2$	−2.36541	−1.67260	−0.836298	+	0.548276i	$0.815284\pi$
$3$	−2.31312	−1.33548	−0.667741	−	0.744394i	$-0.732739\pi$
$3$	−2.31312	−1.33548	−0.667741	+	0.744394i	$0.732739\pi$
$4$	3.59515	1.79758
$5$	−1.08906	−0.487040	−0.243520	−	0.969896i	$-0.578302\pi$
$5$	−1.08906	−0.487040	−0.243520	+	0.969896i	$0.578302\pi$
$6$	5.47148	2.23372
$7$	−0.547834	−0.207062	−0.103531	−	0.994626i	$-0.533014\pi$
$7$	−0.547834	−0.207062	−0.103531	+	0.994626i	$0.533014\pi$
$8$	−3.77318	−1.33402
$9$	2.35053	0.783512
$10$	2.57606	0.814621
$11$	2.77385	0.836347	0.418174	−	0.908367i	$-0.362670\pi$
$11$	2.77385	0.836347	0.418174	+	0.908367i	$0.362670\pi$
$12$	−8.31602	−2.40063
$13$	2.85967	0.793130	0.396565	−	0.918007i	$-0.370202\pi$
$13$	2.85967	0.793130	0.396565	+	0.918007i	$0.370202\pi$
$14$	1.29585	0.346331
$15$	2.51912	0.650433
$16$	1.73481	0.433702
$17$	−0.316358	−0.0767282	−0.0383641	−	0.999264i	$-0.512215\pi$
$17$	−0.316358	−0.0767282	−0.0383641	+	0.999264i	$0.512215\pi$
$18$	−5.55997	−1.31050
$19$	−1.28066	−0.293804	−0.146902	−	0.989151i	$-0.546930\pi$
$19$	−1.28066	−0.293804	−0.146902	+	0.989151i	$0.546930\pi$
$20$	−3.91532	−0.875492
$21$	1.26721	0.276527
$22$	−6.56128	−1.39887
$23$	6.80143	1.41820	0.709098	−	0.705110i	$-0.249102\pi$
$23$	6.80143	1.41820	0.709098	+	0.705110i	$0.249102\pi$
$24$	8.72783	1.78156
$25$	−3.81396	−0.762792
$26$	−6.76428	−1.32659
$27$	1.50229	0.289116
$28$	−1.96955	−0.372210
$29$	−0.874497	−0.162390	−0.0811950	−	0.996698i	$-0.525874\pi$
$29$	−0.874497	−0.162390	−0.0811950	+	0.996698i	$0.525874\pi$
$30$	−5.95874	−1.08791
$31$	6.66183	1.19650	0.598250	−	0.801309i	$-0.295863\pi$
$31$	6.66183	1.19650	0.598250	+	0.801309i	$0.295863\pi$
$32$	3.44283	0.608613
$33$	−6.41625	−1.11693
$34$	0.748317	0.128335
$35$	0.596622	0.100848
$36$	8.45053	1.40842
$37$	8.99003	1.47795	0.738976	−	0.673732i	$-0.235310\pi$
$37$	8.99003	1.47795	0.738976	+	0.673732i	$0.235310\pi$
$38$	3.02928	0.491415
$39$	−6.61477	−1.05921
$40$	4.10920	0.649722
$41$	7.72982	1.20719	0.603597	−	0.797289i	$-0.293733\pi$
$41$	7.72982	1.20719	0.603597	+	0.797289i	$0.293733\pi$
$42$	−2.99746	−0.462519
$43$	0.240790	0.0367202	0.0183601	−	0.999831i	$-0.494155\pi$
$43$	0.240790	0.0367202	0.0183601	+	0.999831i	$0.494155\pi$
$44$	9.97241	1.50340
$45$	−2.55986	−0.381602
$46$	−16.0882	−2.37207
$47$	1.32538	0.193327	0.0966634	−	0.995317i	$-0.469183\pi$
$47$	1.32538	0.193327	0.0966634	+	0.995317i	$0.469183\pi$
$48$	−4.01283	−0.579202
$49$	−6.69988	−0.957125
$50$	9.02157	1.27584
$51$	0.731776	0.102469
$52$	10.2809	1.42571
$53$	2.99865	0.411896	0.205948	−	0.978563i	$-0.433972\pi$
$53$	2.99865	0.411896	0.205948	+	0.978563i	$0.433972\pi$
$54$	−3.55353	−0.483575
$55$	−3.02088	−0.407335
$56$	2.06708	0.276225
$57$	2.96232	0.392369
$58$	2.06854	0.271613
$59$	5.20806	0.678031	0.339016	−	0.940781i	$-0.389906\pi$
$59$	5.20806	0.678031	0.339016	+	0.940781i	$0.389906\pi$
$60$	9.05661	1.16920
$61$	−7.09930	−0.908972	−0.454486	−	0.890754i	$-0.650177\pi$
$61$	−7.09930	−0.908972	−0.454486	+	0.890754i	$0.650177\pi$
$62$	−15.7579	−2.00126
$63$	−1.28770	−0.162235
$64$	−11.6133	−1.45167
$65$	−3.11434	−0.386286
$66$	15.1771	1.86817
$67$	8.88197	1.08511	0.542553	−	0.840022i	$-0.317458\pi$
$67$	8.88197	1.08511	0.542553	+	0.840022i	$0.317458\pi$
$68$	−1.13736	−0.137925
$69$	−15.7325	−1.89398
$70$	−1.41125	−0.168677
$71$	2.57084	0.305103	0.152551	−	0.988296i	$-0.451251\pi$
$71$	2.57084	0.305103	0.152551	+	0.988296i	$0.451251\pi$
$72$	−8.86899	−1.04522
$73$	12.8924	1.50895	0.754473	−	0.656331i	$-0.227892\pi$
$73$	12.8924	1.50895	0.754473	+	0.656331i	$0.227892\pi$
$74$	−21.2651	−2.47202
$75$	8.82215	1.01869
$76$	−4.60417	−0.528134
$77$	−1.51961	−0.173176
$78$	15.6466	1.77163
$79$	3.45817	0.389075	0.194537	−	0.980895i	$-0.437679\pi$
$79$	3.45817	0.389075	0.194537	+	0.980895i	$0.437679\pi$
$80$	−1.88930	−0.211231
$81$	−10.5266	−1.16962
$82$	−18.2842	−2.01915
$83$	0.522288	0.0573286	0.0286643	−	0.999589i	$-0.490875\pi$
$83$	0.522288	0.0573286	0.0286643	+	0.999589i	$0.490875\pi$
$84$	4.55580	0.497079
$85$	0.344532	0.0373697
$86$	−0.569567	−0.0614180
$87$	2.02282	0.216869
$88$	−10.4662	−1.11570
$89$	−9.16932	−0.971946	−0.485973	−	0.873974i	$-0.661535\pi$
$89$	−9.16932	−0.971946	−0.485973	+	0.873974i	$0.661535\pi$
$90$	6.05512	0.638265
$91$	−1.56663	−0.164227
$92$	24.4522	2.54931
$93$	−15.4096	−1.59790
$94$	−3.13507	−0.323358
$95$	1.39471	0.143094
$96$	−7.96369	−0.812791
$97$	18.2279	1.85076	0.925379	−	0.379043i	$-0.123747\pi$
$97$	18.2279	1.85076	0.925379	+	0.379043i	$0.123747\pi$
$98$	15.8479	1.60088
$99$	6.52003	0.655288
$100$	−13.7118	−1.37118
$101$	5.55883	0.553124	0.276562	−	0.960996i	$-0.410805\pi$
$101$	5.55883	0.553124	0.276562	+	0.960996i	$0.410805\pi$
$102$	−1.73095	−0.171389
$103$	−17.4585	−1.72024	−0.860119	−	0.510094i	$-0.829611\pi$
$103$	−17.4585	−1.72024	−0.860119	+	0.510094i	$0.829611\pi$
$104$	−10.7901	−1.05805
$105$	−1.38006	−0.134680
$106$	−7.09303	−0.688936
$107$	−18.3147	−1.77055	−0.885275	−	0.465069i	$-0.846030\pi$
$107$	−18.3147	−1.77055	−0.885275	+	0.465069i	$0.846030\pi$
$108$	5.40097	0.519709
$109$	−9.80187	−0.938849	−0.469424	−	0.882973i	$-0.655539\pi$
$109$	−9.80187	−0.938849	−0.469424	+	0.882973i	$0.655539\pi$
$110$	7.14560	0.681306
$111$	−20.7950	−1.97378
$112$	−0.950388	−0.0898033
$113$	−5.24008	−0.492945	−0.246472	−	0.969150i	$-0.579271\pi$
$113$	−5.24008	−0.492945	−0.246472	+	0.969150i	$0.579271\pi$
$114$	−7.00710	−0.656275
$115$	−7.40713	−0.690719
$116$	−3.14395	−0.291908
$117$	6.72175	0.621426
$118$	−12.3192	−1.13407
$119$	0.173312	0.0158875
$120$	−9.50509	−0.867692
$121$	−3.30576	−0.300524
$122$	16.7927	1.52034
$123$	−17.8800	−1.61219
$124$	23.9503	2.15080
$125$	9.59889	0.858551
$126$	3.04594	0.271354
$127$	2.40309	0.213240	0.106620	−	0.994300i	$-0.465997\pi$
$127$	2.40309	0.213240	0.106620	+	0.994300i	$0.465997\pi$
$128$	20.5846	1.81944
$129$	−0.556977	−0.0490391
$130$	7.36668	0.646101
$131$	9.50901	0.830806	0.415403	−	0.909637i	$-0.363641\pi$
$131$	9.50901	0.830806	0.415403	+	0.909637i	$0.363641\pi$
$132$	−23.0674	−2.00776
$133$	0.701590	0.0608356
$134$	−21.0095	−1.81494
$135$	−1.63608	−0.140811
$136$	1.19368	0.102357
$137$	20.0984	1.71713	0.858563	−	0.512708i	$-0.171358\pi$
$137$	20.0984	1.71713	0.858563	+	0.512708i	$0.171358\pi$
$138$	37.2139	3.16785
$139$	−7.88681	−0.668951	−0.334475	−	0.942405i	$-0.608559\pi$
$139$	−7.88681	−0.668951	−0.334475	+	0.942405i	$0.608559\pi$
$140$	2.14495	0.181281
$141$	−3.06577	−0.258184
$142$	−6.08109	−0.510313
$143$	7.93230	0.663332
$144$	4.07773	0.339811
$145$	0.952376	0.0790905
$146$	−30.4959	−2.52386
$147$	15.4976	1.27822
$148$	32.3205	2.65673
$149$	−6.35011	−0.520221	−0.260111	−	0.965579i	$-0.583759\pi$
$149$	−6.35011	−0.520221	−0.260111	+	0.965579i	$0.583759\pi$
$150$	−20.8680	−1.70386
$151$	−2.33392	−0.189932	−0.0949658	−	0.995481i	$-0.530274\pi$
$151$	−2.33392	−0.189932	−0.0949658	+	0.995481i	$0.530274\pi$
$152$	4.83216	0.391940
$153$	−0.743611	−0.0601174
$154$	3.59450	0.289653
$155$	−7.25510	−0.582744
$156$	−23.7811	−1.90401
$157$	−17.0230	−1.35858	−0.679290	−	0.733870i	$-0.737712\pi$
$157$	−17.0230	−1.35858	−0.679290	+	0.733870i	$0.737712\pi$
$158$	−8.17999	−0.650765
$159$	−6.93625	−0.550080
$160$	−3.74944	−0.296419
$161$	−3.72606	−0.293654
$162$	24.8997	1.95630
$163$	−11.2962	−0.884790	−0.442395	−	0.896820i	$-0.645871\pi$
$163$	−11.2962	−0.884790	−0.442395	+	0.896820i	$0.645871\pi$
$164$	27.7899	2.17002
$165$	6.98765	0.543988
$166$	−1.23542	−0.0958876
$167$	4.40123	0.340577	0.170289	−	0.985394i	$-0.445530\pi$
$167$	4.40123	0.340577	0.170289	+	0.985394i	$0.445530\pi$
$168$	−4.78141	−0.368894
$169$	−4.82229	−0.370945
$170$	−0.814958	−0.0625044
$171$	−3.01024	−0.230199
$172$	0.865677	0.0660073
$173$	24.1657	1.83728	0.918641	−	0.395094i	$-0.129288\pi$
$173$	24.1657	1.83728	0.918641	+	0.395094i	$0.129288\pi$
$174$	−4.78479	−0.362734
$175$	2.08942	0.157945
$176$	4.81210	0.362726
$177$	−12.0469	−0.905498
$178$	21.6892	1.62567
$179$	5.95731	0.445271	0.222635	−	0.974902i	$-0.428534\pi$
$179$	5.95731	0.445271	0.222635	+	0.974902i	$0.428534\pi$
$180$	−9.20309	−0.685958
$181$	−24.0274	−1.78594	−0.892971	−	0.450115i	$-0.851383\pi$
$181$	−24.0274	−1.78594	−0.892971	+	0.450115i	$0.851383\pi$
$182$	3.70571	0.274685
$183$	16.4215	1.21392
$184$	−25.6630	−1.89190
$185$	−9.79064	−0.719822
$186$	36.4500	2.67265
$187$	−0.877531	−0.0641714
$188$	4.76495	0.347520
$189$	−0.823008	−0.0598650
$190$	−3.29906	−0.239339
$191$	20.1090	1.45504	0.727518	−	0.686088i	$-0.240674\pi$
$191$	20.1090	1.45504	0.727518	+	0.686088i	$0.240674\pi$
$192$	26.8630	1.93867
$193$	−21.2496	−1.52958	−0.764790	−	0.644280i	$-0.777157\pi$
$193$	−21.2496	−1.52958	−0.764790	+	0.644280i	$0.777157\pi$
$194$	−43.1163	−3.09557
$195$	7.20385	0.515878
$196$	−24.0871	−1.72051
$197$	11.4442	0.815364	0.407682	−	0.913124i	$-0.366337\pi$
$197$	11.4442	0.815364	0.407682	+	0.913124i	$0.366337\pi$
$198$	−15.4225	−1.09603
$199$	−24.9170	−1.76632	−0.883158	−	0.469075i	$-0.844587\pi$
$199$	−24.9170	−1.76632	−0.883158	+	0.469075i	$0.844587\pi$
$200$	14.3908	1.01758
$201$	−20.5451	−1.44914
$202$	−13.1489	−0.925153
$203$	0.479080	0.0336248
$204$	2.63084	0.184196
$205$	−8.41820	−0.587952
$206$	41.2965	2.87726
$207$	15.9870	1.11117
$208$	4.96098	0.343982
$209$	−3.55236	−0.245722
$210$	3.26440	0.225265
$211$	−11.8170	−0.813515	−0.406758	−	0.913536i	$-0.633341\pi$
$211$	−11.8170	−0.813515	−0.406758	+	0.913536i	$0.633341\pi$
$212$	10.7806	0.740415
$213$	−5.94667	−0.407459
$214$	43.3217	2.96141
$215$	−0.262234	−0.0178842
$216$	−5.66842	−0.385687
$217$	−3.64958	−0.247750
$218$	23.1854	1.57031
$219$	−29.8218	−2.01517
$220$	−10.8605	−0.732215
$221$	−0.904681	−0.0608554
$222$	49.1888	3.30133
$223$	−27.3889	−1.83410	−0.917048	−	0.398778i	$-0.869434\pi$
$223$	−27.3889	−1.83410	−0.917048	+	0.398778i	$0.869434\pi$
$224$	−1.88610	−0.126021
$225$	−8.96484	−0.597656
$226$	12.3949	0.824497
$227$	−23.3269	−1.54826	−0.774132	−	0.633024i	$-0.781813\pi$
$227$	−23.3269	−1.54826	−0.774132	+	0.633024i	$0.781813\pi$
$228$	10.6500	0.705314
$229$	28.3001	1.87012	0.935062	−	0.354485i	$-0.115344\pi$
$229$	28.3001	1.87012	0.935062	+	0.354485i	$0.115344\pi$
$230$	17.5209	1.15529
$231$	3.51504	0.231273
$232$	3.29964	0.216632
$233$	15.1106	0.989931	0.494965	−	0.868913i	$-0.335181\pi$
$233$	15.1106	0.989931	0.494965	+	0.868913i	$0.335181\pi$
$234$	−15.8997	−1.03939
$235$	−1.44341	−0.0941579
$236$	18.7238	1.21881
$237$	−7.99918	−0.519602
$238$	−0.409954	−0.0265733
$239$	−3.61373	−0.233753	−0.116876	−	0.993146i	$-0.537288\pi$
$239$	−3.61373	−0.233753	−0.116876	+	0.993146i	$0.537288\pi$
$240$	4.37019	0.282095
$241$	19.3535	1.24667	0.623334	−	0.781956i	$-0.285778\pi$
$241$	19.3535	1.24667	0.623334	+	0.781956i	$0.285778\pi$
$242$	7.81947	0.502654
$243$	19.8424	1.27289
$244$	−25.5231	−1.63395
$245$	7.29654	0.466159
$246$	42.2935	2.69654
$247$	−3.66227	−0.233024
$248$	−25.1363	−1.59616
$249$	−1.20812	−0.0765613
$250$	−22.7053	−1.43601
$251$	−20.2004	−1.27504	−0.637519	−	0.770435i	$-0.720039\pi$
$251$	−20.2004	−1.27504	−0.637519	+	0.770435i	$0.720039\pi$
$252$	−4.62949	−0.291630
$253$	18.8661	1.18610
$254$	−5.68428	−0.356664
$255$	−0.796944	−0.0499066
$256$	−25.4642	−1.59152
$257$	−19.2997	−1.20388	−0.601942	−	0.798540i	$-0.705606\pi$
$257$	−19.2997	−1.20388	−0.601942	+	0.798540i	$0.705606\pi$
$258$	1.31748	0.0820226
$259$	−4.92505	−0.306028
$260$	−11.1965	−0.694379
$261$	−2.05554	−0.127235
$262$	−22.4927	−1.38960
$263$	−12.1180	−0.747225	−0.373612	−	0.927585i	$-0.621881\pi$
$263$	−12.1180	−0.747225	−0.373612	+	0.927585i	$0.621881\pi$
$264$	24.2097	1.49000
$265$	−3.26570	−0.200610
$266$	−1.65955	−0.101753
$267$	21.2098	1.29802
$268$	31.9320	1.95056
$269$	18.9286	1.15410	0.577050	−	0.816709i	$-0.304204\pi$
$269$	18.9286	1.15410	0.577050	+	0.816709i	$0.304204\pi$
$270$	3.86999	0.235520
$271$	−3.01481	−0.183137	−0.0915684	−	0.995799i	$-0.529188\pi$
$271$	−3.01481	−0.183137	−0.0915684	+	0.995799i	$0.529188\pi$
$272$	−0.548822	−0.0332772
$273$	3.62380	0.219322
$274$	−47.5410	−2.87206
$275$	−10.5793	−0.637959
$276$	−56.5609	−3.40456
$277$	20.1710	1.21196	0.605979	−	0.795481i	$-0.292782\pi$
$277$	20.1710	1.21196	0.605979	+	0.795481i	$0.292782\pi$
$278$	18.6555	1.11888
$279$	15.6589	0.937472
$280$	−2.25116	−0.134533
$281$	20.3526	1.21414	0.607068	−	0.794650i	$-0.292345\pi$
$281$	20.3526	1.21414	0.607068	+	0.794650i	$0.292345\pi$
$282$	7.25180	0.431838
$283$	23.3486	1.38793	0.693966	−	0.720008i	$-0.255862\pi$
$283$	23.3486	1.38793	0.693966	+	0.720008i	$0.255862\pi$
$284$	9.24256	0.548445
$285$	−3.22613	−0.191100
$286$	−18.7631	−1.10949
$287$	−4.23466	−0.249964
$288$	8.09250	0.476855
$289$	−16.8999	−0.994113
$290$	−2.25276	−0.132286
$291$	−42.1633	−2.47165
$292$	46.3503	2.71245
$293$	4.82958	0.282147	0.141074	−	0.989999i	$-0.454945\pi$
$293$	4.82958	0.282147	0.141074	+	0.989999i	$0.454945\pi$
$294$	−36.6582	−2.13795
$295$	−5.67186	−0.330229
$296$	−33.9210	−1.97162
$297$	4.16713	0.241802
$298$	15.0206	0.870119
$299$	19.4498	1.12481
$300$	31.7170	1.83118
$301$	−0.131913	−0.00760335
$302$	5.52067	0.317679
$303$	−12.8583	−0.738687
$304$	−2.22170	−0.127423
$305$	7.73153	0.442706
$306$	1.75894	0.100552
$307$	22.3941	1.27810	0.639048	−	0.769167i	$-0.279328\pi$
$307$	22.3941	1.27810	0.639048	+	0.769167i	$0.279328\pi$
$308$	−5.46323	−0.311296
$309$	40.3837	2.29735
$310$	17.1613	0.974695
$311$	−5.85763	−0.332156	−0.166078	−	0.986113i	$-0.553110\pi$
$311$	−5.85763	−0.332156	−0.166078	+	0.986113i	$0.553110\pi$
$312$	24.9587	1.41301
$313$	−17.8198	−1.00723	−0.503616	−	0.863927i	$-0.667997\pi$
$313$	−17.8198	−1.00723	−0.503616	+	0.863927i	$0.667997\pi$
$314$	40.2662	2.27235
$315$	1.40238	0.0790152
$316$	12.4327	0.699392
$317$	32.2695	1.81244	0.906218	−	0.422812i	$-0.138957\pi$
$317$	32.2695	1.81244	0.906218	+	0.422812i	$0.138957\pi$
$318$	16.4070	0.920062
$319$	−2.42572	−0.135814
$320$	12.6475	0.707019
$321$	42.3641	2.36454
$322$	8.81364	0.491165
$323$	0.405148	0.0225430
$324$	−37.8447	−2.10248
$325$	−10.9067	−0.604993
$326$	26.7202	1.47990
$327$	22.6729	1.25382
$328$	−29.1660	−1.61042
$329$	−0.726090	−0.0400306
$330$	−16.5286	−0.909872
$331$	15.5168	0.852882	0.426441	−	0.904515i	$-0.359767\pi$
$331$	15.5168	0.852882	0.426441	+	0.904515i	$0.359767\pi$
$332$	1.87771	0.103053
$333$	21.1314	1.15799
$334$	−10.4107	−0.569648
$335$	−9.67296	−0.528490
$336$	2.19836	0.119931
$337$	23.2814	1.26822	0.634108	−	0.773244i	$-0.281367\pi$
$337$	23.2814	1.26822	0.634108	+	0.773244i	$0.281367\pi$
$338$	11.4067	0.620441
$339$	12.1209	0.658319
$340$	1.23864	0.0671749
$341$	18.4789	1.00069
$342$	7.12044	0.385029
$343$	7.50526	0.405246
$344$	−0.908545	−0.0489855
$345$	17.1336	0.922442
$346$	−57.1616	−3.07303
$347$	9.67063	0.519147	0.259573	−	0.965723i	$-0.416418\pi$
$347$	9.67063	0.519147	0.259573	+	0.965723i	$0.416418\pi$
$348$	7.27234	0.389838
$349$	−17.1839	−0.919833	−0.459917	−	0.887962i	$-0.652121\pi$
$349$	−17.1839	−0.919833	−0.459917	+	0.887962i	$0.652121\pi$
$350$	−4.94232	−0.264178
$351$	4.29606	0.229307
$352$	9.54990	0.509011
$353$	12.9193	0.687627	0.343813	−	0.939038i	$-0.388281\pi$
$353$	12.9193	0.687627	0.343813	+	0.939038i	$0.388281\pi$
$354$	28.4958	1.51453
$355$	−2.79979	−0.148597
$356$	−32.9651	−1.74715
$357$	−0.400892	−0.0212175
$358$	−14.0915	−0.744757
$359$	−8.09992	−0.427497	−0.213749	−	0.976889i	$-0.568567\pi$
$359$	−8.09992	−0.427497	−0.213749	+	0.976889i	$0.568567\pi$
$360$	9.65882	0.509065
$361$	−17.3599	−0.913679
$362$	56.8345	2.98716
$363$	7.64663	0.401344
$364$	−5.63226	−0.295210
$365$	−14.0406	−0.734918
$366$	−38.8436	−2.03039
$367$	31.2612	1.63182	0.815909	−	0.578180i	$-0.196237\pi$
$367$	31.2612	1.63182	0.815909	+	0.578180i	$0.196237\pi$
$368$	11.7992	0.615075
$369$	18.1692	0.945851
$370$	23.1589	1.20397
$371$	−1.64276	−0.0852881
$372$	−55.3999	−2.87235
$373$	−20.0718	−1.03928	−0.519639	−	0.854386i	$-0.673933\pi$
$373$	−20.0718	−1.03928	−0.519639	+	0.854386i	$0.673933\pi$
$374$	2.07572	0.107333
$375$	−22.2034	−1.14658
$376$	−5.00091	−0.257902
$377$	−2.50077	−0.128796
$378$	1.94675	0.100130
$379$	3.30672	0.169855	0.0849274	−	0.996387i	$-0.472934\pi$
$379$	3.30672	0.169855	0.0849274	+	0.996387i	$0.472934\pi$
$380$	5.01419	0.257223
$381$	−5.55864	−0.284778
$382$	−47.5660	−2.43369
$383$	−26.4297	−1.35050	−0.675248	−	0.737591i	$-0.735963\pi$
$383$	−26.4297	−1.35050	−0.675248	+	0.737591i	$0.735963\pi$
$384$	−47.6146	−2.42982
$385$	1.65494	0.0843435
$386$	50.2640	2.55837
$387$	0.565986	0.0287707
$388$	65.5319	3.32688
$389$	−18.0899	−0.917197	−0.458598	−	0.888644i	$-0.651648\pi$
$389$	−18.0899	−0.917197	−0.458598	+	0.888644i	$0.651648\pi$
$390$	−17.0400	−0.862855
$391$	−2.15169	−0.108816
$392$	25.2799	1.27683
$393$	−21.9955	−1.10953
$394$	−27.0701	−1.36377
$395$	−3.76614	−0.189495
$396$	23.4405	1.17793
$397$	31.0155	1.55662	0.778311	−	0.627879i	$-0.216077\pi$
$397$	31.0155	1.55662	0.778311	+	0.627879i	$0.216077\pi$
$398$	58.9388	2.95433
$399$	−1.62286	−0.0812448
$400$	−6.61649	−0.330825
$401$	31.0835	1.55224	0.776118	−	0.630587i	$-0.217186\pi$
$401$	31.0835	1.55224	0.776118	+	0.630587i	$0.217186\pi$
$402$	48.5975	2.42382
$403$	19.0506	0.948980
$404$	19.9848	0.994283
$405$	11.4640	0.569653
$406$	−1.13322	−0.0562407
$407$	24.9370	1.23608
$408$	−2.76112	−0.136696
$409$	31.7704	1.57094	0.785471	−	0.618898i	$-0.212421\pi$
$409$	31.7704	1.57094	0.785471	+	0.618898i	$0.212421\pi$
$410$	19.9125	0.983407
$411$	−46.4902	−2.29319
$412$	−62.7660	−3.09226
$413$	−2.85315	−0.140394
$414$	−37.8158	−1.85854
$415$	−0.568801	−0.0279213
$416$	9.84537	0.482709
$417$	18.2432	0.893371
$418$	8.40278	0.410993
$419$	26.0636	1.27329	0.636645	−	0.771157i	$-0.280322\pi$
$419$	26.0636	1.27329	0.636645	+	0.771157i	$0.280322\pi$
$420$	−4.96152	−0.242098
$421$	9.03188	0.440187	0.220094	−	0.975479i	$-0.429364\pi$
$421$	9.03188	0.440187	0.220094	+	0.975479i	$0.429364\pi$
$422$	27.9520	1.36068
$423$	3.11536	0.151474
$424$	−11.3145	−0.549479
$425$	1.20658	0.0585276
$426$	14.0663	0.681514
$427$	3.88924	0.188214
$428$	−65.8441	−3.18270
$429$	−18.3484	−0.885868
$430$	0.620290	0.0299130
$431$	0.822747	0.0396303	0.0198152	−	0.999804i	$-0.493692\pi$
$431$	0.822747	0.0396303	0.0198152	+	0.999804i	$0.493692\pi$
$432$	2.60619	0.125390
$433$	−3.98445	−0.191481	−0.0957403	−	0.995406i	$-0.530522\pi$
$433$	−3.98445	−0.191481	−0.0957403	+	0.995406i	$0.530522\pi$
$434$	8.63274	0.414385
$435$	−2.20296	−0.105624
$436$	−35.2392	−1.68765
$437$	−8.71032	−0.416671
$438$	70.5407	3.37057
$439$	27.9361	1.33332	0.666660	−	0.745362i	$-0.267723\pi$
$439$	27.9361	1.33332	0.666660	+	0.745362i	$0.267723\pi$
$440$	11.3983	0.543393
$441$	−15.7483	−0.749919
$442$	2.13994	0.101787
$443$	−20.5907	−0.978294	−0.489147	−	0.872201i	$-0.662692\pi$
$443$	−20.5907	−0.978294	−0.489147	+	0.872201i	$0.662692\pi$
$444$	−74.7613	−3.54802
$445$	9.98590	0.473377
$446$	64.7859	3.06770
$447$	14.6886	0.694746
$448$	6.36218	0.300585
$449$	32.3489	1.52664	0.763320	−	0.646020i	$-0.223568\pi$
$449$	32.3489	1.52664	0.763320	+	0.646020i	$0.223568\pi$
$450$	21.2055	0.999637
$451$	21.4413	1.00963
$452$	−18.8389	−0.886106
$453$	5.39864	0.253650
$454$	55.1777	2.58962
$455$	1.70614	0.0799852
$456$	−11.1774	−0.523429
$457$	4.80254	0.224653	0.112327	−	0.993671i	$-0.464170\pi$
$457$	4.80254	0.224653	0.112327	+	0.993671i	$0.464170\pi$
$458$	−66.9413	−3.12796
$459$	−0.475263	−0.0221834
$460$	−26.6298	−1.24162
$461$	−19.0427	−0.886908	−0.443454	−	0.896297i	$-0.646247\pi$
$461$	−19.0427	−0.886908	−0.443454	+	0.896297i	$0.646247\pi$
$462$	−8.31451	−0.386826
$463$	3.73829	0.173733	0.0868664	−	0.996220i	$-0.472315\pi$
$463$	3.73829	0.173733	0.0868664	+	0.996220i	$0.472315\pi$
$464$	−1.51709	−0.0704290
$465$	16.7819	0.778244
$466$	−35.7428	−1.65575
$467$	6.41151	0.296689	0.148345	−	0.988936i	$-0.452605\pi$
$467$	6.41151	0.296689	0.148345	+	0.988936i	$0.452605\pi$
$468$	24.1657	1.11706
$469$	−4.86585	−0.224684
$470$	3.41426	0.157488
$471$	39.3762	1.81436
$472$	−19.6509	−0.904508
$473$	0.667916	0.0307108
$474$	18.9213	0.869085
$475$	4.88439	0.224111
$476$	0.623083	0.0285590
$477$	7.04843	0.322726
$478$	8.54794	0.390974
$479$	12.2269	0.558661	0.279330	−	0.960195i	$-0.409888\pi$
$479$	12.2269	0.558661	0.279330	+	0.960195i	$0.409888\pi$
$480$	8.67290	0.395862
$481$	25.7085	1.17221
$482$	−45.7789	−2.08517
$483$	8.61883	0.392170
$484$	−11.8847	−0.540214
$485$	−19.8511	−0.901394
$486$	−46.9354	−2.12903
$487$	3.88616	0.176099	0.0880495	−	0.996116i	$-0.471937\pi$
$487$	3.88616	0.176099	0.0880495	+	0.996116i	$0.471937\pi$
$488$	26.7869	1.21259
$489$	26.1296	1.18162
$490$	−17.2593	−0.779695
$491$	−28.1469	−1.27025	−0.635125	−	0.772409i	$-0.719052\pi$
$491$	−28.1469	−1.27025	−0.635125	+	0.772409i	$0.719052\pi$
$492$	−64.2813	−2.89803
$493$	0.276655	0.0124599
$494$	8.66275	0.389756
$495$	−7.10067	−0.319151
$496$	11.5570	0.518925
$497$	−1.40840	−0.0631752
$498$	2.85769	0.128056
$499$	37.8077	1.69250	0.846252	−	0.532783i	$-0.178854\pi$
$499$	37.8077	1.69250	0.846252	+	0.532783i	$0.178854\pi$
$500$	34.5095	1.54331
$501$	−10.1806	−0.454835
$502$	47.7821	2.13262
$503$	1.92428	0.0857992	0.0428996	−	0.999079i	$-0.486340\pi$
$503$	1.92428	0.0857992	0.0428996	+	0.999079i	$0.486340\pi$
$504$	4.85874	0.216426
$505$	−6.05387	−0.269394
$506$	−44.6261	−1.98387
$507$	11.1545	0.495390
$508$	8.63947	0.383314
$509$	−20.3348	−0.901325	−0.450663	−	0.892694i	$-0.648812\pi$
$509$	−20.3348	−0.901325	−0.450663	+	0.892694i	$0.648812\pi$
$510$	1.88510	0.0834735
$511$	−7.06293	−0.312445
$512$	19.0642	0.842525
$513$	−1.92393	−0.0849434
$514$	45.6517	2.01361
$515$	19.0133	0.837825
$516$	−2.00242	−0.0881515
$517$	3.67641	0.161688
$518$	11.6497	0.511861
$519$	−55.8981	−2.45366
$520$	11.7510	0.515314
$521$	40.1443	1.75875	0.879376	−	0.476128i	$-0.157960\pi$
$521$	40.1443	1.75875	0.879376	+	0.476128i	$0.157960\pi$
$522$	4.86218	0.212812
$523$	−11.5864	−0.506638	−0.253319	−	0.967383i	$-0.581522\pi$
$523$	−11.5864	−0.506638	−0.253319	+	0.967383i	$0.581522\pi$
$524$	34.1863	1.49344
$525$	−4.83308	−0.210933
$526$	28.6639	1.24981
$527$	−2.10753	−0.0918053
$528$	−11.1310	−0.484414
$529$	23.2594	1.01128
$530$	7.72470	0.335540
$531$	12.2417	0.531245
$532$	2.52232	0.109357
$533$	22.1047	0.957462
$534$	−50.1697	−2.17106
$535$	19.9457	0.862329
$536$	−33.5133	−1.44755
$537$	−13.7800	−0.594651
$538$	−44.7739	−1.93034
$539$	−18.5845	−0.800489
$540$	−5.88195	−0.253119
$541$	−38.8930	−1.67214	−0.836069	−	0.548624i	$-0.815152\pi$
$541$	−38.8930	−1.67214	−0.836069	+	0.548624i	$0.815152\pi$
$542$	7.13126	0.306314
$543$	55.5783	2.38509
$544$	−1.08917	−0.0466978
$545$	10.6748	0.457257
$546$	−8.57176	−0.366837
$547$	14.5872	0.623702	0.311851	−	0.950131i	$-0.399051\pi$
$547$	14.5872	0.623702	0.311851	+	0.950131i	$0.399051\pi$
$548$	72.2569	3.08666
$549$	−16.6871	−0.712190
$550$	25.0245	1.06705
$551$	1.11993	0.0477108
$552$	59.3617	2.52660
$553$	−1.89451	−0.0805626
$554$	−47.7126	−2.02711
$555$	22.6470	0.961310
$556$	−28.3543	−1.20249
$557$	33.1462	1.40445	0.702225	−	0.711955i	$-0.252190\pi$
$557$	33.1462	1.40445	0.702225	+	0.711955i	$0.252190\pi$
$558$	−37.0396	−1.56801
$559$	0.688581	0.0291239
$560$	1.03503	0.0437378
$561$	2.02984	0.0856997
$562$	−48.1423	−2.03076
$563$	17.4849	0.736899	0.368450	−	0.929648i	$-0.379889\pi$
$563$	17.4849	0.736899	0.368450	+	0.929648i	$0.379889\pi$
$564$	−11.0219	−0.464106
$565$	5.70673	0.240084
$566$	−55.2290	−2.32145
$567$	5.76683	0.242184
$568$	−9.70025	−0.407014
$569$	−19.9990	−0.838400	−0.419200	−	0.907894i	$-0.637689\pi$
$569$	−19.9990	−0.838400	−0.419200	+	0.907894i	$0.637689\pi$
$570$	7.63112	0.319633
$571$	11.8504	0.495926	0.247963	−	0.968770i	$-0.420239\pi$
$571$	11.8504	0.495926	0.247963	+	0.968770i	$0.420239\pi$
$572$	28.5178	1.19239
$573$	−46.5146	−1.94318
$574$	10.0167	0.418089
$575$	−25.9404	−1.08179
$576$	−27.2975	−1.13740
$577$	−8.18442	−0.340722	−0.170361	−	0.985382i	$-0.554493\pi$
$577$	−8.18442	−0.340722	−0.170361	+	0.985382i	$0.554493\pi$
$578$	39.9752	1.66275
$579$	49.1529	2.04273
$580$	3.42394	0.142171
$581$	−0.286128	−0.0118706
$582$	99.7333	4.13408
$583$	8.31781	0.344488
$584$	−48.6456	−2.01297
$585$	−7.32036	−0.302660
$586$	−11.4239	−0.471918
$587$	21.6217	0.892422	0.446211	−	0.894928i	$-0.352773\pi$
$587$	21.6217	0.892422	0.446211	+	0.894928i	$0.352773\pi$
$588$	55.7163	2.29770
$589$	−8.53154	−0.351536
$590$	13.4163	0.552339
$591$	−26.4718	−1.08890
$592$	15.5960	0.640991
$593$	−33.0419	−1.35687	−0.678433	−	0.734662i	$-0.737341\pi$
$593$	−33.0419	−1.35687	−0.678433	+	0.734662i	$0.737341\pi$
$594$	−9.85697	−0.404436
$595$	−0.188746	−0.00773785
$596$	−22.8296	−0.935137
$597$	57.6360	2.35888
$598$	−46.0068	−1.88136
$599$	0.479515	0.0195925	0.00979623	−	0.999952i	$-0.496882\pi$
$599$	0.479515	0.0195925	0.00979623	+	0.999952i	$0.496882\pi$
$600$	−33.2876	−1.35896
$601$	44.1788	1.80209	0.901045	−	0.433725i	$-0.142801\pi$
$601$	44.1788	1.80209	0.901045	+	0.433725i	$0.142801\pi$
$602$	0.312028	0.0127173
$603$	20.8774	0.850193
$604$	−8.39079	−0.341417
$605$	3.60015	0.146367
$606$	30.4150	1.23553
$607$	−44.3551	−1.80032	−0.900159	−	0.435561i	$-0.856550\pi$
$607$	−44.3551	−1.80032	−0.900159	+	0.435561i	$0.856550\pi$
$608$	−4.40910	−0.178813
$609$	−1.10817	−0.0449053
$610$	−18.2882	−0.740468
$611$	3.79016	0.153333
$612$	−2.67340	−0.108066
$613$	7.59181	0.306630	0.153315	−	0.988177i	$-0.451005\pi$
$613$	7.59181	0.306630	0.153315	+	0.988177i	$0.451005\pi$
$614$	−52.9711	−2.13774
$615$	19.4723	0.785200
$616$	5.73377	0.231020
$617$	−8.77535	−0.353282	−0.176641	−	0.984275i	$-0.556523\pi$
$617$	−8.77535	−0.353282	−0.176641	+	0.984275i	$0.556523\pi$
$618$	−95.5238	−3.84253
$619$	−46.7829	−1.88036	−0.940181	−	0.340675i	$-0.889344\pi$
$619$	−46.7829	−1.88036	−0.940181	+	0.340675i	$0.889344\pi$
$620$	−26.0832	−1.04753
$621$	10.2177	0.410024
$622$	13.8557	0.555562
$623$	5.02327	0.201253
$624$	−11.4754	−0.459382
$625$	8.61608	0.344643
$626$	42.1510	1.68469
$627$	8.21704	0.328157
$628$	−61.2001	−2.44215
$629$	−2.84407	−0.113401
$630$	−3.31720	−0.132160
$631$	−27.9929	−1.11438	−0.557190	−	0.830385i	$-0.688120\pi$
$631$	−27.9929	−1.11438	−0.557190	+	0.830385i	$0.688120\pi$
$632$	−13.0483	−0.519034
$633$	27.3341	1.08643
$634$	−76.3305	−3.03147
$635$	−2.61710	−0.103856
$636$	−24.9369	−0.988811
$637$	−19.1594	−0.759125
$638$	5.73782	0.227163
$639$	6.04285	0.239052
$640$	−22.4177	−0.886139
$641$	−28.5804	−1.12886	−0.564428	−	0.825482i	$-0.690903\pi$
$641$	−28.5804	−1.12886	−0.564428	+	0.825482i	$0.690903\pi$
$642$	−100.208	−3.95491
$643$	36.4630	1.43796	0.718980	−	0.695031i	$-0.244609\pi$
$643$	36.4630	1.43796	0.718980	+	0.695031i	$0.244609\pi$
$644$	−13.3957	−0.527866
$645$	0.606579	0.0238840
$646$	−0.958339	−0.0377054
$647$	17.0334	0.669653	0.334826	−	0.942280i	$-0.391322\pi$
$647$	17.0334	0.669653	0.334826	+	0.942280i	$0.391322\pi$
$648$	39.7187	1.56030
$649$	14.4464	0.567069
$650$	25.7987	1.01191
$651$	8.44193	0.330865
$652$	−40.6117	−1.59048
$653$	5.00057	0.195688	0.0978438	−	0.995202i	$-0.468805\pi$
$653$	5.00057	0.195688	0.0978438	+	0.995202i	$0.468805\pi$
$654$	−53.6307	−2.09713
$655$	−10.3558	−0.404636
$656$	13.4098	0.523563
$657$	30.3041	1.18228
$658$	1.71750	0.0669551
$659$	−21.5461	−0.839315	−0.419658	−	0.907682i	$-0.637850\pi$
$659$	−21.5461	−0.839315	−0.419658	+	0.907682i	$0.637850\pi$
$660$	25.1217	0.977860
$661$	14.1922	0.552013	0.276006	−	0.961156i	$-0.410989\pi$
$661$	14.1922	0.552013	0.276006	+	0.961156i	$0.410989\pi$
$662$	−36.7036	−1.42653
$663$	2.09264	0.0812713
$664$	−1.97069	−0.0764776
$665$	−0.764070	−0.0296294
$666$	−49.9843	−1.93685
$667$	−5.94783	−0.230301
$668$	15.8231	0.612213
$669$	63.3538	2.44940
$670$	22.8805	0.883950
$671$	−19.6924	−0.760216
$672$	4.36279	0.168298
$673$	−7.86783	−0.303283	−0.151641	−	0.988436i	$-0.548456\pi$
$673$	−7.86783	−0.303283	−0.151641	+	0.988436i	$0.548456\pi$
$674$	−55.0699	−2.12121
$675$	−5.72968	−0.220536
$676$	−17.3368	−0.666802
$677$	−1.27452	−0.0489839	−0.0244920	−	0.999700i	$-0.507797\pi$
$677$	−1.27452	−0.0489839	−0.0244920	+	0.999700i	$0.507797\pi$
$678$	−28.6710	−1.10110
$679$	−9.98585	−0.383222
$680$	−1.29998	−0.0498520
$681$	53.9581	2.06768
$682$	−43.7102	−1.67375
$683$	−9.81181	−0.375439	−0.187719	−	0.982223i	$-0.560110\pi$
$683$	−9.81181	−0.375439	−0.187719	+	0.982223i	$0.560110\pi$
$684$	−10.8223	−0.413799
$685$	−21.8883	−0.836309
$686$	−17.7530	−0.677813
$687$	−65.4616	−2.49752
$688$	0.417725	0.0159256
$689$	8.57515	0.326687
$690$	−40.5279	−1.54287
$691$	−25.9826	−0.988426	−0.494213	−	0.869341i	$-0.664544\pi$
$691$	−25.9826	−0.988426	−0.494213	+	0.869341i	$0.664544\pi$
$692$	86.8792	3.30265
$693$	−3.57190	−0.135685
$694$	−22.8750	−0.868323
$695$	8.58917	0.325806
$696$	−7.63247	−0.289308
$697$	−2.44539	−0.0926259
$698$	40.6469	1.53851
$699$	−34.9527	−1.32203
$700$	7.51177	0.283918
$701$	31.6218	1.19434	0.597169	−	0.802115i	$-0.296292\pi$
$701$	31.6218	1.19434	0.597169	+	0.802115i	$0.296292\pi$
$702$	−10.1619	−0.383538
$703$	−11.5132	−0.434228
$704$	−32.2136	−1.21410
$705$	3.33879	0.125746
$706$	−30.5595	−1.15012
$707$	−3.04532	−0.114531
$708$	−43.3103	−1.62770
$709$	−43.9815	−1.65176	−0.825881	−	0.563845i	$-0.809322\pi$
$709$	−43.9815	−1.65176	−0.825881	+	0.563845i	$0.809322\pi$
$710$	6.62264	0.248543
$711$	8.12856	0.304845
$712$	34.5975	1.29660
$713$	45.3100	1.69687
$714$	0.948273	0.0354882
$715$	−8.63871	−0.323069
$716$	21.4174	0.800407
$717$	8.35900	0.312173
$718$	19.1596	0.715030
$719$	−7.66382	−0.285812	−0.142906	−	0.989736i	$-0.545645\pi$
$719$	−7.66382	−0.285812	−0.142906	+	0.989736i	$0.545645\pi$
$720$	−4.44087	−0.165502
$721$	9.56437	0.356196
$722$	41.0633	1.52822
$723$	−44.7670	−1.66490
$724$	−86.3821	−3.21036
$725$	3.33530	0.123870
$726$	−18.0874	−0.671286
$727$	−27.7274	−1.02835	−0.514177	−	0.857684i	$-0.671902\pi$
$727$	−27.7274	−1.02835	−0.514177	+	0.857684i	$0.671902\pi$
$728$	5.91116	0.219082
$729$	−14.3182	−0.530302
$730$	33.2117	1.22922
$731$	−0.0761760	−0.00281747
$732$	59.0379	2.18211
$733$	−23.0371	−0.850896	−0.425448	−	0.904983i	$-0.639883\pi$
$733$	−23.0371	−0.850896	−0.425448	+	0.904983i	$0.639883\pi$
$734$	−73.9454	−2.72937
$735$	−16.8778	−0.622546
$736$	23.4162	0.863132
$737$	24.6372	0.907525
$738$	−42.9776	−1.58203
$739$	−27.2446	−1.00221	−0.501105	−	0.865387i	$-0.667073\pi$
$739$	−27.2446	−1.00221	−0.501105	+	0.865387i	$0.667073\pi$
$740$	−35.1988	−1.29394
$741$	8.47127	0.311200
$742$	3.88581	0.142652
$743$	23.2996	0.854778	0.427389	−	0.904068i	$-0.359433\pi$
$743$	23.2996	0.854778	0.427389	+	0.904068i	$0.359433\pi$
$744$	58.1433	2.13164
$745$	6.91562	0.253369
$746$	47.4779	1.73829
$747$	1.22766	0.0449176
$748$	−3.15486	−0.115353
$749$	10.0334	0.366613
$750$	52.5201	1.91776
$751$	−21.9317	−0.800301	−0.400150	−	0.916449i	$-0.631042\pi$
$751$	−21.9317	−0.800301	−0.400150	+	0.916449i	$0.631042\pi$
$752$	2.29929	0.0838463
$753$	46.7260	1.70279
$754$	5.91535	0.215424
$755$	2.54177	0.0925044
$756$	−2.95884	−0.107612
$757$	−27.4099	−0.996229	−0.498114	−	0.867111i	$-0.665974\pi$
$757$	−27.4099	−0.996229	−0.498114	+	0.867111i	$0.665974\pi$
$758$	−7.82174	−0.284098
$759$	−43.6397	−1.58402
$760$	−5.26249	−0.190891
$761$	−11.0143	−0.399267	−0.199633	−	0.979871i	$-0.563975\pi$
$761$	−11.0143	−0.399267	−0.199633	+	0.979871i	$0.563975\pi$
$762$	13.1484	0.476318
$763$	5.36980	0.194400
$764$	72.2949	2.61554
$765$	0.809834	0.0292796
$766$	62.5171	2.25883
$767$	14.8933	0.537767
$768$	58.9019	2.12544
$769$	49.9880	1.80261	0.901306	−	0.433183i	$-0.142610\pi$
$769$	49.9880	1.80261	0.901306	+	0.433183i	$0.142610\pi$
$770$	−3.91461	−0.141073
$771$	44.6427	1.60777
$772$	−76.3955	−2.74954
$773$	32.4445	1.16695	0.583473	−	0.812132i	$-0.301693\pi$
$773$	32.4445	1.16695	0.583473	+	0.812132i	$0.301693\pi$
$774$	−1.33879	−0.0481217
$775$	−25.4079	−0.912680
$776$	−68.7770	−2.46895
$777$	11.3922	0.408694
$778$	42.7901	1.53410
$779$	−9.89927	−0.354678
$780$	25.8989	0.927330
$781$	7.13113	0.255172
$782$	5.08962	0.182005
$783$	−1.31375	−0.0469496
$784$	−11.6230	−0.415108
$785$	18.5389	0.661683
$786$	52.0283	1.85579
$787$	43.1828	1.53930	0.769650	−	0.638466i	$-0.220431\pi$
$787$	43.1828	1.53930	0.769650	+	0.638466i	$0.220431\pi$
$788$	41.1436	1.46568
$789$	28.0303	0.997905
$790$	8.90846	0.316949
$791$	2.87069	0.102070
$792$	−24.6013	−0.874168
$793$	−20.3017	−0.720933
$794$	−73.3642	−2.60360
$795$	7.55396	0.267911
$796$	−89.5803	−3.17509
$797$	−49.1787	−1.74200	−0.870999	−	0.491284i	$-0.836528\pi$
$797$	−49.1787	−1.74200	−0.870999	+	0.491284i	$0.836528\pi$
$798$	3.83873	0.135890
$799$	−0.419296	−0.0148336
$800$	−13.1308	−0.464245
$801$	−21.5528	−0.761531
$802$	−73.5252	−2.59626
$803$	35.7617	1.26200
$804$	−73.8627	−2.60494
$805$	4.05788	0.143022
$806$	−45.0625	−1.58726
$807$	−43.7842	−1.54128
$808$	−20.9745	−0.737880
$809$	31.6555	1.11295	0.556474	−	0.830865i	$-0.312154\pi$
$809$	31.6555	1.11295	0.556474	+	0.830865i	$0.312154\pi$
$810$	−27.1171	−0.952798
$811$	−27.1211	−0.952351	−0.476175	−	0.879350i	$-0.657977\pi$
$811$	−27.1211	−0.952351	−0.476175	+	0.879350i	$0.657977\pi$
$812$	1.72236	0.0604431
$813$	6.97363	0.244576
$814$	−58.9862	−2.06746
$815$	12.3022	0.430928
$816$	1.26949	0.0444411
$817$	−0.308370	−0.0107885
$818$	−75.1498	−2.62755
$819$	−3.68241	−0.128674
$820$	−30.2647	−1.05689
$821$	30.4777	1.06368	0.531839	−	0.846845i	$-0.321501\pi$
$821$	30.4777	1.06368	0.531839	+	0.846845i	$0.321501\pi$
$822$	109.968	3.83558
$823$	−8.32199	−0.290086	−0.145043	−	0.989425i	$-0.546332\pi$
$823$	−8.32199	−0.290086	−0.145043	+	0.989425i	$0.546332\pi$
$824$	65.8741	2.29483
$825$	24.4713	0.851982
$826$	6.74887	0.234823
$827$	−35.4349	−1.23219	−0.616096	−	0.787671i	$-0.711287\pi$
$827$	−35.4349	−1.23219	−0.616096	+	0.787671i	$0.711287\pi$
$828$	57.4757	1.99742
$829$	48.4583	1.68303	0.841513	−	0.540236i	$-0.181665\pi$
$829$	48.4583	1.68303	0.841513	+	0.540236i	$0.181665\pi$
$830$	1.34545	0.0467011
$831$	−46.6580	−1.61855
$832$	−33.2103	−1.15136
$833$	2.11956	0.0734385
$834$	−43.1525	−1.49425
$835$	−4.79318	−0.165875
$836$	−12.7713	−0.441704
$837$	10.0080	0.345928
$838$	−61.6510	−2.12970
$839$	−13.0366	−0.450073	−0.225037	−	0.974350i	$-0.572250\pi$
$839$	−13.0366	−0.450073	−0.225037	+	0.974350i	$0.572250\pi$
$840$	5.20722	0.179666
$841$	−28.2353	−0.973629
$842$	−21.3641	−0.736255
$843$	−47.0781	−1.62146
$844$	−42.4839	−1.46235
$845$	5.25173	0.180665
$846$	−7.36909	−0.253354
$847$	1.81101	0.0622270
$848$	5.20209	0.178640
$849$	−54.0082	−1.85356
$850$	−2.85405	−0.0978931
$851$	61.1451	2.09603
$852$	−21.3792	−0.732439
$853$	36.7517	1.25835	0.629177	−	0.777262i	$-0.283392\pi$
$853$	36.7517	1.25835	0.629177	+	0.777262i	$0.283392\pi$
$854$	−9.19964	−0.314805
$855$	3.27831	0.112116
$856$	69.1047	2.36195
$857$	−15.7392	−0.537640	−0.268820	−	0.963190i	$-0.586634\pi$
$857$	−15.7392	−0.537640	−0.268820	+	0.963190i	$0.586634\pi$
$858$	43.4014	1.48170
$859$	33.1034	1.12947	0.564737	−	0.825271i	$-0.308978\pi$
$859$	33.1034	1.12947	0.564737	+	0.825271i	$0.308978\pi$
$860$	−0.942770	−0.0321482
$861$	9.79529	0.333822
$862$	−1.94613	−0.0662855
$863$	−21.5795	−0.734575	−0.367287	−	0.930108i	$-0.619713\pi$
$863$	−21.5795	−0.734575	−0.367287	+	0.930108i	$0.619713\pi$
$864$	5.17214	0.175960
$865$	−26.3177	−0.894830
$866$	9.42486	0.320270
$867$	39.0916	1.32762
$868$	−13.1208	−0.445349
$869$	9.59245	0.325402
$870$	5.21090	0.176666
$871$	25.3995	0.860629
$872$	36.9842	1.25244
$873$	42.8452	1.45009
$874$	20.6035	0.696922
$875$	−5.25860	−0.177773
$876$	−107.214	−3.62242
$877$	25.8556	0.873081	0.436541	−	0.899685i	$-0.356204\pi$
$877$	25.8556	0.873081	0.436541	+	0.899685i	$0.356204\pi$
$878$	−66.0804	−2.23010
$879$	−11.1714	−0.376802
$880$	−5.24064	−0.176662
$881$	33.6747	1.13453	0.567265	−	0.823535i	$-0.308002\pi$
$881$	33.6747	1.13453	0.567265	+	0.823535i	$0.308002\pi$
$882$	37.2511	1.25431
$883$	13.9368	0.469012	0.234506	−	0.972115i	$-0.424653\pi$
$883$	13.9368	0.469012	0.234506	+	0.972115i	$0.424653\pi$
$884$	−3.25246	−0.109392
$885$	13.1197	0.441014
$886$	48.7054	1.63629
$887$	−39.1599	−1.31486	−0.657431	−	0.753515i	$-0.728357\pi$
$887$	−39.1599	−1.31486	−0.657431	+	0.753515i	$0.728357\pi$
$888$	78.4635	2.63306
$889$	−1.31649	−0.0441538
$890$	−23.6207	−0.791768
$891$	−29.1992	−0.978209
$892$	−98.4672	−3.29692
$893$	−1.69736	−0.0568001
$894$	−34.7445	−1.16203
$895$	−6.48784	−0.216865
$896$	−11.2769	−0.376736
$897$	−44.9899	−1.50217
$898$	−76.5184	−2.55345
$899$	−5.82575	−0.194300
$900$	−32.2300	−1.07433
$901$	−0.948649	−0.0316041
$902$	−50.7175	−1.68871
$903$	0.305131	0.0101541
$904$	19.7718	0.657599
$905$	26.1671	0.869825
$906$	−12.7700	−0.424254
$907$	26.2015	0.870006	0.435003	−	0.900429i	$-0.356747\pi$
$907$	26.2015	0.870006	0.435003	+	0.900429i	$0.356747\pi$
$908$	−83.8639	−2.78312
$909$	13.0662	0.433379
$910$	−4.03572	−0.133783
$911$	9.38362	0.310893	0.155447	−	0.987844i	$-0.450318\pi$
$911$	9.38362	0.310893	0.155447	+	0.987844i	$0.450318\pi$
$912$	5.13907	0.170172
$913$	1.44875	0.0479466
$914$	−11.3600	−0.375754
$915$	−17.8840	−0.591226
$916$	101.743	3.36169
$917$	−5.20936	−0.172028
$918$	1.12419	0.0371038
$919$	−25.1250	−0.828798	−0.414399	−	0.910095i	$-0.636008\pi$
$919$	−25.1250	−0.828798	−0.414399	+	0.910095i	$0.636008\pi$
$920$	27.9485	0.921433
$921$	−51.8002	−1.70687
$922$	45.0438	1.48344
$923$	7.35176	0.241986
$924$	12.6371	0.415731
$925$	−34.2876	−1.12737
$926$	−8.84257	−0.290585
$927$	−41.0368	−1.34783
$928$	−3.01075	−0.0988327
$929$	29.6383	0.972402	0.486201	−	0.873847i	$-0.338382\pi$
$929$	29.6383	0.972402	0.486201	+	0.873847i	$0.338382\pi$
$930$	−39.6961	−1.30169
$931$	8.58027	0.281207
$932$	54.3250	1.77948
$933$	13.5494	0.443588
$934$	−15.1658	−0.496241
$935$	0.955679	0.0312541
$936$	−25.3624	−0.828996
$937$	48.0486	1.56968	0.784839	−	0.619699i	$-0.212745\pi$
$937$	48.0486	1.56968	0.784839	+	0.619699i	$0.212745\pi$
$938$	11.5097	0.375805
$939$	41.2193	1.34514
$940$	−5.18929	−0.169256
$941$	−1.59630	−0.0520380	−0.0260190	−	0.999661i	$-0.508283\pi$
$941$	−1.59630	−0.0520380	−0.0260190	+	0.999661i	$0.508283\pi$
$942$	−93.1407	−3.03469
$943$	52.5738	1.71204
$944$	9.03499	0.294064
$945$	0.896301	0.0291567
$946$	−1.57989	−0.0513667
$947$	−31.4701	−1.02264	−0.511321	−	0.859390i	$-0.670844\pi$
$947$	−31.4701	−1.02264	−0.511321	+	0.859390i	$0.670844\pi$
$948$	−28.7583	−0.934025
$949$	36.8682	1.19679
$950$	−11.5536	−0.374847
$951$	−74.6433	−2.42047
$952$	−0.653938	−0.0211943
$953$	−45.2375	−1.46539	−0.732693	−	0.680559i	$-0.761737\pi$
$953$	−45.2375	−1.46539	−0.732693	+	0.680559i	$0.761737\pi$
$954$	−16.6724	−0.539789
$955$	−21.8998	−0.708661
$956$	−12.9919	−0.420188
$957$	5.61100	0.181378
$958$	−28.9216	−0.934413
$959$	−11.0106	−0.355551
$960$	−29.2553	−0.944212
$961$	13.3800	0.431613
$962$	−60.8111	−1.96063
$963$	−43.0493	−1.38725
$964$	69.5787	2.24098
$965$	23.1420	0.744967
$966$	−20.3870	−0.655942
$967$	−41.4454	−1.33280	−0.666398	−	0.745596i	$-0.732165\pi$
$967$	−41.4454	−1.33280	−0.666398	+	0.745596i	$0.732165\pi$
$968$	12.4732	0.400905
$969$	−0.937156	−0.0301058
$970$	46.9560	1.50767
$971$	−10.6681	−0.342356	−0.171178	−	0.985240i	$-0.554757\pi$
$971$	−10.6681	−0.342356	−0.171178	+	0.985240i	$0.554757\pi$
$972$	71.3365	2.28812
$973$	4.32067	0.138514
$974$	−9.19236	−0.294542
$975$	25.2285	0.807957
$976$	−12.3159	−0.394223
$977$	−40.0480	−1.28125	−0.640624	−	0.767855i	$-0.721324\pi$
$977$	−40.0480	−1.28125	−0.640624	+	0.767855i	$0.721324\pi$
$978$	−61.8071	−1.97637
$979$	−25.4343	−0.812884
$980$	26.2322	0.837955
$981$	−23.0396	−0.735599
$982$	66.5788	2.12462
$983$	34.7661	1.10887	0.554433	−	0.832229i	$-0.312935\pi$
$983$	34.7661	1.10887	0.554433	+	0.832229i	$0.312935\pi$
$984$	67.4645	2.15069
$985$	−12.4633	−0.397115
$986$	−0.654401	−0.0208404
$987$	1.67953	0.0534602
$988$	−13.1664	−0.418879
$989$	1.63772	0.0520764
$990$	16.7960	0.533811
$991$	11.1677	0.354754	0.177377	−	0.984143i	$-0.443239\pi$
$991$	11.1677	0.354754	0.177377	+	0.984143i	$0.443239\pi$
$992$	22.9356	0.728205
$993$	−35.8923	−1.13901
$994$	3.33143	0.105667
$995$	27.1360	0.860268
$996$	−4.34336	−0.137625
$997$	−33.0502	−1.04671	−0.523356	−	0.852114i	$-0.675320\pi$
$997$	−33.0502	−1.04671	−0.523356	+	0.852114i	$0.675320\pi$
$998$	−89.4305	−2.83087
$999$	13.5057	0.427300

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4027.2.a.c.1.17	✓	174

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4027.2.a.c.1.17	✓	174	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:	\( N \)	\(=\)	\( 4027 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4027.a (trivial)

\(f(q)\)	\(=\)	\(q-2.36541 q^{2} -2.31312 q^{3} +3.59515 q^{4} -1.08906 q^{5} +5.47148 q^{6} -0.547834 q^{7} -3.77318 q^{8} +2.35053 q^{9} +O(q^{10})\)	\(q-2.36541 q^{2} -2.31312 q^{3} +3.59515 q^{4} -1.08906 q^{5} +5.47148 q^{6} -0.547834 q^{7} -3.77318 q^{8} +2.35053 q^{9} +2.57606 q^{10} +2.77385 q^{11} -8.31602 q^{12} +2.85967 q^{13} +1.29585 q^{14} +2.51912 q^{15} +1.73481 q^{16} -0.316358 q^{17} -5.55997 q^{18} -1.28066 q^{19} -3.91532 q^{20} +1.26721 q^{21} -6.56128 q^{22} +6.80143 q^{23} +8.72783 q^{24} -3.81396 q^{25} -6.76428 q^{26} +1.50229 q^{27} -1.96955 q^{28} -0.874497 q^{29} -5.95874 q^{30} +6.66183 q^{31} +3.44283 q^{32} -6.41625 q^{33} +0.748317 q^{34} +0.596622 q^{35} +8.45053 q^{36} +8.99003 q^{37} +3.02928 q^{38} -6.61477 q^{39} +4.10920 q^{40} +7.72982 q^{41} -2.99746 q^{42} +0.240790 q^{43} +9.97241 q^{44} -2.55986 q^{45} -16.0882 q^{46} +1.32538 q^{47} -4.01283 q^{48} -6.69988 q^{49} +9.02157 q^{50} +0.731776 q^{51} +10.2809 q^{52} +2.99865 q^{53} -3.55353 q^{54} -3.02088 q^{55} +2.06708 q^{56} +2.96232 q^{57} +2.06854 q^{58} +5.20806 q^{59} +9.05661 q^{60} -7.09930 q^{61} -15.7579 q^{62} -1.28770 q^{63} -11.6133 q^{64} -3.11434 q^{65} +15.1771 q^{66} +8.88197 q^{67} -1.13736 q^{68} -15.7325 q^{69} -1.41125 q^{70} +2.57084 q^{71} -8.86899 q^{72} +12.8924 q^{73} -21.2651 q^{74} +8.82215 q^{75} -4.60417 q^{76} -1.51961 q^{77} +15.6466 q^{78} +3.45817 q^{79} -1.88930 q^{80} -10.5266 q^{81} -18.2842 q^{82} +0.522288 q^{83} +4.55580 q^{84} +0.344532 q^{85} -0.569567 q^{86} +2.02282 q^{87} -10.4662 q^{88} -9.16932 q^{89} +6.05512 q^{90} -1.56663 q^{91} +24.4522 q^{92} -15.4096 q^{93} -3.13507 q^{94} +1.39471 q^{95} -7.96369 q^{96} +18.2279 q^{97} +15.8479 q^{98} +6.52003 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 174 q + 21 q^{2} + 17 q^{3} + 187 q^{4} + 72 q^{5} + 21 q^{6} + 24 q^{7} + 54 q^{8} + 197 q^{9}+O(q^{10}) \) 174 * q + 21 * q^2 + 17 * q^3 + 187 * q^4 + 72 * q^5 + 21 * q^6 + 24 * q^7 + 54 * q^8 + 197 * q^9	\( 174 q + 21 q^{2} + 17 q^{3} + 187 q^{4} + 72 q^{5} + 21 q^{6} + 24 q^{7} + 54 q^{8} + 197 q^{9} + 20 q^{10} + 35 q^{11} + 23 q^{12} + 91 q^{13} + 18 q^{14} + 16 q^{15} + 201 q^{16} + 148 q^{17} + 39 q^{18} + 36 q^{19} + 128 q^{20} + 57 q^{21} + 17 q^{22} + 96 q^{23} + 24 q^{24} + 226 q^{25} + 44 q^{26} + 62 q^{27} + 32 q^{28} + 122 q^{29} + 25 q^{30} + 23 q^{31} + 104 q^{32} + 91 q^{33} + 6 q^{34} + 80 q^{35} + 222 q^{36} + 71 q^{37} + 125 q^{38} + 16 q^{39} + 53 q^{40} + 97 q^{41} + 14 q^{42} + 38 q^{43} + 70 q^{44} + 185 q^{45} - 23 q^{46} + 110 q^{47} + 36 q^{48} + 210 q^{49} + 51 q^{50} + 33 q^{51} + 118 q^{52} + 214 q^{53} + 8 q^{54} + 37 q^{55} + 41 q^{56} + 76 q^{57} + 2 q^{58} + 66 q^{59} - 12 q^{60} + 114 q^{61} + 175 q^{62} + 62 q^{63} + 190 q^{64} + 128 q^{65} + 12 q^{66} - 6 q^{67} + 348 q^{68} + 115 q^{69} - 38 q^{70} + 54 q^{71} + 101 q^{72} + 107 q^{73} + 71 q^{74} - q^{75} + 31 q^{76} + 368 q^{77} - 14 q^{78} - 14 q^{79} + 205 q^{80} + 222 q^{81} + 26 q^{82} + 246 q^{83} + 41 q^{84} + 87 q^{85} + 33 q^{86} + 100 q^{87} - 6 q^{88} + 147 q^{89} + 50 q^{90} - 23 q^{91} + 189 q^{92} + 117 q^{93} + 23 q^{94} + 42 q^{95} + 38 q^{96} + 52 q^{97} + 148 q^{98} + 38 q^{99}+O(q^{100}) \) 174 * q + 21 * q^2 + 17 * q^3 + 187 * q^4 + 72 * q^5 + 21 * q^6 + 24 * q^7 + 54 * q^8 + 197 * q^9 + 20 * q^10 + 35 * q^11 + 23 * q^12 + 91 * q^13 + 18 * q^14 + 16 * q^15 + 201 * q^16 + 148 * q^17 + 39 * q^18 + 36 * q^19 + 128 * q^20 + 57 * q^21 + 17 * q^22 + 96 * q^23 + 24 * q^24 + 226 * q^25 + 44 * q^26 + 62 * q^27 + 32 * q^28 + 122 * q^29 + 25 * q^30 + 23 * q^31 + 104 * q^32 + 91 * q^33 + 6 * q^34 + 80 * q^35 + 222 * q^36 + 71 * q^37 + 125 * q^38 + 16 * q^39 + 53 * q^40 + 97 * q^41 + 14 * q^42 + 38 * q^43 + 70 * q^44 + 185 * q^45 - 23 * q^46 + 110 * q^47 + 36 * q^48 + 210 * q^49 + 51 * q^50 + 33 * q^51 + 118 * q^52 + 214 * q^53 + 8 * q^54 + 37 * q^55 + 41 * q^56 + 76 * q^57 + 2 * q^58 + 66 * q^59 - 12 * q^60 + 114 * q^61 + 175 * q^62 + 62 * q^63 + 190 * q^64 + 128 * q^65 + 12 * q^66 - 6 * q^67 + 348 * q^68 + 115 * q^69 - 38 * q^70 + 54 * q^71 + 101 * q^72 + 107 * q^73 + 71 * q^74 - q^75 + 31 * q^76 + 368 * q^77 - 14 * q^78 - 14 * q^79 + 205 * q^80 + 222 * q^81 + 26 * q^82 + 246 * q^83 + 41 * q^84 + 87 * q^85 + 33 * q^86 + 100 * q^87 - 6 * q^88 + 147 * q^89 + 50 * q^90 - 23 * q^91 + 189 * q^92 + 117 * q^93 + 23 * q^94 + 42 * q^95 + 38 * q^96 + 52 * q^97 + 148 * q^98 + 38 * q^99

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