LMFDB - Embedded newform 4001.2.a.a.1.13

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4001,2,Mod(1,4001)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4001, 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("4001.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4001 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4001.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:	$31.9481458487$
Analytic rank:	$1$
Dimension:	$149$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.13
Character	$\chi$	$=$	4001.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.41976 q^{2} -1.68750 q^{3} +3.85525 q^{4} -0.342140 q^{5} +4.08335 q^{6} +2.33028 q^{7} -4.48927 q^{8} -0.152339 q^{9} +O(q^{10})$	\(q-2.41976 q^{2} -1.68750 q^{3} +3.85525 q^{4} -0.342140 q^{5} +4.08335 q^{6} +2.33028 q^{7} -4.48927 q^{8} -0.152339 q^{9} +0.827898 q^{10} +1.44102 q^{11} -6.50574 q^{12} -1.29517 q^{13} -5.63872 q^{14} +0.577362 q^{15} +3.15247 q^{16} +0.986005 q^{17} +0.368624 q^{18} -0.0757367 q^{19} -1.31904 q^{20} -3.93235 q^{21} -3.48692 q^{22} +2.87604 q^{23} +7.57565 q^{24} -4.88294 q^{25} +3.13401 q^{26} +5.31958 q^{27} +8.98381 q^{28} +0.748536 q^{29} -1.39708 q^{30} +5.91336 q^{31} +1.35032 q^{32} -2.43172 q^{33} -2.38590 q^{34} -0.797282 q^{35} -0.587306 q^{36} -3.84784 q^{37} +0.183265 q^{38} +2.18560 q^{39} +1.53596 q^{40} -4.65625 q^{41} +9.51535 q^{42} -8.59010 q^{43} +5.55548 q^{44} +0.0521213 q^{45} -6.95934 q^{46} +3.47831 q^{47} -5.31979 q^{48} -1.56981 q^{49} +11.8156 q^{50} -1.66388 q^{51} -4.99321 q^{52} -0.153372 q^{53} -12.8721 q^{54} -0.493030 q^{55} -10.4612 q^{56} +0.127806 q^{57} -1.81128 q^{58} -2.50685 q^{59} +2.22588 q^{60} -3.96388 q^{61} -14.3089 q^{62} -0.354992 q^{63} -9.57239 q^{64} +0.443130 q^{65} +5.88418 q^{66} +1.64267 q^{67} +3.80130 q^{68} -4.85333 q^{69} +1.92923 q^{70} +8.90909 q^{71} +0.683891 q^{72} -5.75235 q^{73} +9.31086 q^{74} +8.23997 q^{75} -0.291984 q^{76} +3.35797 q^{77} -5.28864 q^{78} -9.14863 q^{79} -1.07859 q^{80} -8.51978 q^{81} +11.2670 q^{82} +3.02346 q^{83} -15.1602 q^{84} -0.337352 q^{85} +20.7860 q^{86} -1.26316 q^{87} -6.46911 q^{88} -7.91527 q^{89} -0.126121 q^{90} -3.01811 q^{91} +11.0879 q^{92} -9.97881 q^{93} -8.41669 q^{94} +0.0259126 q^{95} -2.27866 q^{96} -12.9207 q^{97} +3.79856 q^{98} -0.219523 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 149 q - 6 q^{2} - 28 q^{3} + 116 q^{4} - 19 q^{5} - 31 q^{6} - 47 q^{7} - 15 q^{8} + 115 q^{9}+O(q^{10}) $ 149 * q - 6 * q^2 - 28 * q^3 + 116 * q^4 - 19 * q^5 - 31 * q^6 - 47 * q^7 - 15 * q^8 + 115 * q^9	\( 149 q - 6 q^{2} - 28 q^{3} + 116 q^{4} - 19 q^{5} - 31 q^{6} - 47 q^{7} - 15 q^{8} + 115 q^{9} - 48 q^{10} - 31 q^{11} - 61 q^{12} - 54 q^{13} - 44 q^{14} - 65 q^{15} + 58 q^{16} - 26 q^{17} - 23 q^{18} - 86 q^{19} - 52 q^{20} - 30 q^{21} - 56 q^{22} - 63 q^{23} - 90 q^{24} + 92 q^{25} - 38 q^{26} - 103 q^{27} - 77 q^{28} - 51 q^{29} - 22 q^{30} - 256 q^{31} - 21 q^{32} - 36 q^{33} - 124 q^{34} - 50 q^{35} + 45 q^{36} - 42 q^{37} - 14 q^{38} - 119 q^{39} - 131 q^{40} - 55 q^{41} - 5 q^{42} - 55 q^{43} - 54 q^{44} - 68 q^{45} - 59 q^{46} - 82 q^{47} - 89 q^{48} + 30 q^{49} + 13 q^{50} - 83 q^{51} - 126 q^{52} - 23 q^{53} - 83 q^{54} - 244 q^{55} - 94 q^{56} - 14 q^{57} - 60 q^{58} - 93 q^{59} - 70 q^{60} - 139 q^{61} - 10 q^{62} - 120 q^{63} - 45 q^{64} - 37 q^{65} - 28 q^{66} - 110 q^{67} - 27 q^{68} - 79 q^{69} - 78 q^{70} - 123 q^{71} - 74 q^{73} - 25 q^{74} - 146 q^{75} - 192 q^{76} - q^{77} + 26 q^{78} - 273 q^{79} - 51 q^{80} + 41 q^{81} - 120 q^{82} - 27 q^{83} - 14 q^{84} - 71 q^{85} + 3 q^{86} - 98 q^{87} - 143 q^{88} - 70 q^{89} - 24 q^{90} - 256 q^{91} - 47 q^{92} + 16 q^{93} - 151 q^{94} - 83 q^{95} - 137 q^{96} - 108 q^{97} + 17 q^{98} - 131 q^{99}+O(q^{100}) \) 149 * q - 6 * q^2 - 28 * q^3 + 116 * q^4 - 19 * q^5 - 31 * q^6 - 47 * q^7 - 15 * q^8 + 115 * q^9 - 48 * q^10 - 31 * q^11 - 61 * q^12 - 54 * q^13 - 44 * q^14 - 65 * q^15 + 58 * q^16 - 26 * q^17 - 23 * q^18 - 86 * q^19 - 52 * q^20 - 30 * q^21 - 56 * q^22 - 63 * q^23 - 90 * q^24 + 92 * q^25 - 38 * q^26 - 103 * q^27 - 77 * q^28 - 51 * q^29 - 22 * q^30 - 256 * q^31 - 21 * q^32 - 36 * q^33 - 124 * q^34 - 50 * q^35 + 45 * q^36 - 42 * q^37 - 14 * q^38 - 119 * q^39 - 131 * q^40 - 55 * q^41 - 5 * q^42 - 55 * q^43 - 54 * q^44 - 68 * q^45 - 59 * q^46 - 82 * q^47 - 89 * q^48 + 30 * q^49 + 13 * q^50 - 83 * q^51 - 126 * q^52 - 23 * q^53 - 83 * q^54 - 244 * q^55 - 94 * q^56 - 14 * q^57 - 60 * q^58 - 93 * q^59 - 70 * q^60 - 139 * q^61 - 10 * q^62 - 120 * q^63 - 45 * q^64 - 37 * q^65 - 28 * q^66 - 110 * q^67 - 27 * q^68 - 79 * q^69 - 78 * q^70 - 123 * q^71 - 74 * q^73 - 25 * q^74 - 146 * q^75 - 192 * q^76 - q^77 + 26 * q^78 - 273 * q^79 - 51 * q^80 + 41 * q^81 - 120 * q^82 - 27 * q^83 - 14 * q^84 - 71 * q^85 + 3 * q^86 - 98 * q^87 - 143 * q^88 - 70 * q^89 - 24 * q^90 - 256 * q^91 - 47 * q^92 + 16 * q^93 - 151 * q^94 - 83 * q^95 - 137 * q^96 - 108 * q^97 + 17 * q^98 - 131 * 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 $

$2$	−2.41976	−1.71103	−0.855515	−	0.517777i	$-0.826760\pi$
$2$	−2.41976	−1.71103	−0.855515	+	0.517777i	$0.826760\pi$
$3$	−1.68750	−0.974279	−0.487140	−	0.873324i	$-0.661960\pi$
$3$	−1.68750	−0.974279	−0.487140	+	0.873324i	$0.661960\pi$
$4$	3.85525	1.92763
$5$	−0.342140	−0.153010	−0.0765049	−	0.997069i	$-0.524376\pi$
$5$	−0.342140	−0.153010	−0.0765049	+	0.997069i	$0.524376\pi$
$6$	4.08335	1.66702
$7$	2.33028	0.880762	0.440381	−	0.897811i	$-0.354843\pi$
$7$	2.33028	0.880762	0.440381	+	0.897811i	$0.354843\pi$
$8$	−4.48927	−1.58720
$9$	−0.152339	−0.0507797
$10$	0.827898	0.261804
$11$	1.44102	0.434483	0.217241	−	0.976118i	$-0.430294\pi$
$11$	1.44102	0.434483	0.217241	+	0.976118i	$0.430294\pi$
$12$	−6.50574	−1.87805
$13$	−1.29517	−0.359216	−0.179608	−	0.983738i	$-0.557483\pi$
$13$	−1.29517	−0.359216	−0.179608	+	0.983738i	$0.557483\pi$
$14$	−5.63872	−1.50701
$15$	0.577362	0.149074
$16$	3.15247	0.788117
$17$	0.986005	0.239141	0.119571	−	0.992826i	$-0.461848\pi$
$17$	0.986005	0.239141	0.119571	+	0.992826i	$0.461848\pi$
$18$	0.368624	0.0868856
$19$	−0.0757367	−0.0173752	−0.00868760	−	0.999962i	$-0.502765\pi$
$19$	−0.0757367	−0.0173752	−0.00868760	+	0.999962i	$0.502765\pi$
$20$	−1.31904	−0.294946
$21$	−3.93235	−0.858108
$22$	−3.48692	−0.743413
$23$	2.87604	0.599697	0.299848	−	0.953987i	$-0.403064\pi$
$23$	2.87604	0.599697	0.299848	+	0.953987i	$0.403064\pi$
$24$	7.57565	1.54637
$25$	−4.88294	−0.976588
$26$	3.13401	0.614629
$27$	5.31958	1.02375
$28$	8.98381	1.69778
$29$	0.748536	0.139000	0.0694998	−	0.997582i	$-0.477860\pi$
$29$	0.748536	0.139000	0.0694998	+	0.997582i	$0.477860\pi$
$30$	−1.39708	−0.255071
$31$	5.91336	1.06207	0.531036	−	0.847349i	$-0.321803\pi$
$31$	5.91336	1.06207	0.531036	+	0.847349i	$0.321803\pi$
$32$	1.35032	0.238705
$33$	−2.43172	−0.423308
$34$	−2.38590	−0.409178
$35$	−0.797282	−0.134765
$36$	−0.587306	−0.0978843
$37$	−3.84784	−0.632581	−0.316290	−	0.948662i	$-0.602437\pi$
$37$	−3.84784	−0.632581	−0.316290	+	0.948662i	$0.602437\pi$
$38$	0.183265	0.0297295
$39$	2.18560	0.349977
$40$	1.53596	0.242857
$41$	−4.65625	−0.727185	−0.363592	−	0.931558i	$-0.618450\pi$
$41$	−4.65625	−0.727185	−0.363592	+	0.931558i	$0.618450\pi$
$42$	9.51535	1.46825
$43$	−8.59010	−1.30998	−0.654989	−	0.755638i	$-0.727327\pi$
$43$	−8.59010	−1.30998	−0.654989	+	0.755638i	$0.727327\pi$
$44$	5.55548	0.837520
$45$	0.0521213	0.00776979
$46$	−6.95934	−1.02610
$47$	3.47831	0.507364	0.253682	−	0.967288i	$-0.418358\pi$
$47$	3.47831	0.507364	0.253682	+	0.967288i	$0.418358\pi$
$48$	−5.31979	−0.767846
$49$	−1.56981	−0.224258
$50$	11.8156	1.67097
$51$	−1.66388	−0.232990
$52$	−4.99321	−0.692434
$53$	−0.153372	−0.0210673	−0.0105337	−	0.999945i	$-0.503353\pi$
$53$	−0.153372	−0.0210673	−0.0105337	+	0.999945i	$0.503353\pi$
$54$	−12.8721	−1.75167
$55$	−0.493030	−0.0664801
$56$	−10.4612	−1.39794
$57$	0.127806	0.0169283
$58$	−1.81128	−0.237833
$59$	−2.50685	−0.326364	−0.163182	−	0.986596i	$-0.552176\pi$
$59$	−2.50685	−0.326364	−0.163182	+	0.986596i	$0.552176\pi$
$60$	2.22588	0.287359
$61$	−3.96388	−0.507522	−0.253761	−	0.967267i	$-0.581668\pi$
$61$	−3.96388	−0.507522	−0.253761	+	0.967267i	$0.581668\pi$
$62$	−14.3089	−1.81724
$63$	−0.354992	−0.0447248
$64$	−9.57239	−1.19655
$65$	0.443130	0.0549635
$66$	5.88418	0.724292
$67$	1.64267	0.200684	0.100342	−	0.994953i	$-0.468006\pi$
$67$	1.64267	0.200684	0.100342	+	0.994953i	$0.468006\pi$
$68$	3.80130	0.460975
$69$	−4.85333	−0.584272
$70$	1.92923	0.230587
$71$	8.90909	1.05731	0.528657	−	0.848835i	$-0.322696\pi$
$71$	8.90909	1.05731	0.528657	+	0.848835i	$0.322696\pi$
$72$	0.683891	0.0805974
$73$	−5.75235	−0.673262	−0.336631	−	0.941637i	$-0.609288\pi$
$73$	−5.75235	−0.673262	−0.336631	+	0.941637i	$0.609288\pi$
$74$	9.31086	1.08237
$75$	8.23997	0.951470
$76$	−0.291984	−0.0334929
$77$	3.35797	0.382676
$78$	−5.28864	−0.598821
$79$	−9.14863	−1.02930	−0.514651	−	0.857400i	$-0.672078\pi$
$79$	−9.14863	−1.02930	−0.514651	+	0.857400i	$0.672078\pi$
$80$	−1.07859	−0.120590
$81$	−8.51978	−0.946642
$82$	11.2670	1.24424
$83$	3.02346	0.331868	0.165934	−	0.986137i	$-0.446936\pi$
$83$	3.02346	0.331868	0.165934	+	0.986137i	$0.446936\pi$
$84$	−15.1602	−1.65411
$85$	−0.337352	−0.0365909
$86$	20.7860	2.24141
$87$	−1.26316	−0.135425
$88$	−6.46911	−0.689610
$89$	−7.91527	−0.839017	−0.419508	−	0.907751i	$-0.637798\pi$
$89$	−7.91527	−0.839017	−0.419508	+	0.907751i	$0.637798\pi$
$90$	−0.126121	−0.0132943
$91$	−3.01811	−0.316384
$92$	11.0879	1.15599
$93$	−9.97881	−1.03475
$94$	−8.41669	−0.868115
$95$	0.0259126	0.00265858
$96$	−2.27866	−0.232565
$97$	−12.9207	−1.31190	−0.655951	−	0.754803i	$-0.727732\pi$
$97$	−12.9207	−1.31190	−0.655951	+	0.754803i	$0.727732\pi$
$98$	3.79856	0.383713
$99$	−0.219523	−0.0220629
$100$	−18.8250	−1.88250
$101$	6.00771	0.597789	0.298895	−	0.954286i	$-0.403382\pi$
$101$	6.00771	0.597789	0.298895	+	0.954286i	$0.403382\pi$
$102$	4.02620	0.398654
$103$	−2.54395	−0.250663	−0.125332	−	0.992115i	$-0.539999\pi$
$103$	−2.54395	−0.250663	−0.125332	+	0.992115i	$0.539999\pi$
$104$	5.81437	0.570146
$105$	1.34541	0.131299
$106$	0.371125	0.0360468
$107$	12.8176	1.23912	0.619562	−	0.784948i	$-0.287310\pi$
$107$	12.8176	1.23912	0.619562	+	0.784948i	$0.287310\pi$
$108$	20.5083	1.97341
$109$	0.915428	0.0876821	0.0438411	−	0.999039i	$-0.486040\pi$
$109$	0.915428	0.0876821	0.0438411	+	0.999039i	$0.486040\pi$
$110$	1.19301	0.113749
$111$	6.49323	0.616310
$112$	7.34613	0.694144
$113$	−0.423552	−0.0398444	−0.0199222	−	0.999802i	$-0.506342\pi$
$113$	−0.423552	−0.0398444	−0.0199222	+	0.999802i	$0.506342\pi$
$114$	−0.309260	−0.0289648
$115$	−0.984010	−0.0917594
$116$	2.88580	0.267939
$117$	0.197305	0.0182409
$118$	6.06599	0.558419
$119$	2.29766	0.210627
$120$	−2.59194	−0.236610
$121$	−8.92347	−0.811225
$122$	9.59164	0.868386
$123$	7.85743	0.708481
$124$	22.7975	2.04728
$125$	3.38135	0.302437
$126$	0.858997	0.0765255
$127$	21.6280	1.91918	0.959588	−	0.281409i	$-0.0908016\pi$
$127$	21.6280	1.91918	0.959588	+	0.281409i	$0.0908016\pi$
$128$	20.4623	1.80863
$129$	14.4958	1.27628
$130$	−1.07227	−0.0940443
$131$	−2.12233	−0.185429	−0.0927144	−	0.995693i	$-0.529554\pi$
$131$	−2.12233	−0.185429	−0.0927144	+	0.995693i	$0.529554\pi$
$132$	−9.37488	−0.815979
$133$	−0.176488	−0.0153034
$134$	−3.97487	−0.343377
$135$	−1.82004	−0.156644
$136$	−4.42644	−0.379564
$137$	18.9165	1.61615	0.808073	−	0.589083i	$-0.200511\pi$
$137$	18.9165	1.61615	0.808073	+	0.589083i	$0.200511\pi$
$138$	11.7439	0.999708
$139$	22.0143	1.86723	0.933616	−	0.358276i	$-0.116635\pi$
$139$	22.0143	1.86723	0.933616	+	0.358276i	$0.116635\pi$
$140$	−3.07372	−0.259777
$141$	−5.86966	−0.494314
$142$	−21.5579	−1.80910
$143$	−1.86636	−0.156073
$144$	−0.480244	−0.0400203
$145$	−0.256104	−0.0212683
$146$	13.9193	1.15197
$147$	2.64905	0.218490
$148$	−14.8344	−1.21938
$149$	2.15645	0.176663	0.0883317	−	0.996091i	$-0.471846\pi$
$149$	2.15645	0.176663	0.0883317	+	0.996091i	$0.471846\pi$
$150$	−19.9388	−1.62799
$151$	−17.4924	−1.42351	−0.711757	−	0.702425i	$-0.752100\pi$
$151$	−17.4924	−1.42351	−0.711757	+	0.702425i	$0.752100\pi$
$152$	0.340003	0.0275779
$153$	−0.150207	−0.0121435
$154$	−8.12548	−0.654770
$155$	−2.02320	−0.162507
$156$	8.42605	0.674624
$157$	11.0152	0.879108	0.439554	−	0.898216i	$-0.355137\pi$
$157$	11.0152	0.879108	0.439554	+	0.898216i	$0.355137\pi$
$158$	22.1375	1.76117
$159$	0.258816	0.0205254
$160$	−0.461998	−0.0365241
$161$	6.70198	0.528190
$162$	20.6158	1.61973
$163$	−11.7765	−0.922409	−0.461205	−	0.887294i	$-0.652583\pi$
$163$	−11.7765	−0.922409	−0.461205	+	0.887294i	$0.652583\pi$
$164$	−17.9510	−1.40174
$165$	0.831988	0.0647702
$166$	−7.31606	−0.567836
$167$	−12.5667	−0.972437	−0.486218	−	0.873837i	$-0.661624\pi$
$167$	−12.5667	−0.972437	−0.486218	+	0.873837i	$0.661624\pi$
$168$	17.6534	1.36199
$169$	−11.3225	−0.870964
$170$	0.816311	0.0626082
$171$	0.0115377	0.000882307 0
$172$	−33.1170	−2.52515
$173$	−11.2296	−0.853772	−0.426886	−	0.904305i	$-0.640389\pi$
$173$	−11.2296	−0.853772	−0.426886	+	0.904305i	$0.640389\pi$
$174$	3.05654	0.231716
$175$	−11.3786	−0.860142
$176$	4.54276	0.342423
$177$	4.23032	0.317970
$178$	19.1531	1.43558
$179$	−16.8701	−1.26093	−0.630465	−	0.776218i	$-0.717136\pi$
$179$	−16.8701	−1.26093	−0.630465	+	0.776218i	$0.717136\pi$
$180$	0.200941	0.0149772
$181$	25.8017	1.91783	0.958913	−	0.283702i	$-0.0915626\pi$
$181$	25.8017	1.91783	0.958913	+	0.283702i	$0.0915626\pi$
$182$	7.30311	0.541342
$183$	6.68905	0.494469
$184$	−12.9113	−0.951837
$185$	1.31650	0.0967910
$186$	24.1463	1.77050
$187$	1.42085	0.103903
$188$	13.4098	0.978008
$189$	12.3961	0.901683
$190$	−0.0627023	−0.00454890
$191$	−21.3117	−1.54206	−0.771029	−	0.636800i	$-0.780258\pi$
$191$	−21.3117	−1.54206	−0.771029	+	0.636800i	$0.780258\pi$
$192$	16.1534	1.16577
$193$	20.2191	1.45540	0.727702	−	0.685894i	$-0.240588\pi$
$193$	20.2191	1.45540	0.727702	+	0.685894i	$0.240588\pi$
$194$	31.2651	2.24471
$195$	−0.747783	−0.0535498
$196$	−6.05200	−0.432286
$197$	−2.05454	−0.146380	−0.0731901	−	0.997318i	$-0.523318\pi$
$197$	−2.05454	−0.146380	−0.0731901	+	0.997318i	$0.523318\pi$
$198$	0.531194	0.0377503
$199$	0.707387	0.0501454	0.0250727	−	0.999686i	$-0.492018\pi$
$199$	0.707387	0.0501454	0.0250727	+	0.999686i	$0.492018\pi$
$200$	21.9208	1.55004
$201$	−2.77201	−0.195522
$202$	−14.5372	−1.02284
$203$	1.74430	0.122426
$204$	−6.41469	−0.449118
$205$	1.59309	0.111266
$206$	6.15576	0.428892
$207$	−0.438134	−0.0304524
$208$	−4.08299	−0.283104
$209$	−0.109138	−0.00754922
$210$	−3.25558	−0.224657
$211$	−4.34078	−0.298832	−0.149416	−	0.988774i	$-0.547739\pi$
$211$	−4.34078	−0.298832	−0.149416	+	0.988774i	$0.547739\pi$
$212$	−0.591289	−0.0406099
$213$	−15.0341	−1.03012
$214$	−31.0156	−2.12018
$215$	2.93902	0.200439
$216$	−23.8810	−1.62490
$217$	13.7798	0.935432
$218$	−2.21512	−0.150027
$219$	9.70711	0.655945
$220$	−1.90075	−0.128149
$221$	−1.27704	−0.0859033
$222$	−15.7121	−1.05453
$223$	15.8308	1.06011	0.530054	−	0.847964i	$-0.322171\pi$
$223$	15.8308	1.06011	0.530054	+	0.847964i	$0.322171\pi$
$224$	3.14661	0.210242
$225$	0.743862	0.0495908
$226$	1.02490	0.0681751
$227$	5.33287	0.353955	0.176978	−	0.984215i	$-0.443368\pi$
$227$	5.33287	0.353955	0.176978	+	0.984215i	$0.443368\pi$
$228$	0.492724	0.0326314
$229$	−3.25933	−0.215383	−0.107691	−	0.994184i	$-0.534346\pi$
$229$	−3.25933	−0.215383	−0.107691	+	0.994184i	$0.534346\pi$
$230$	2.38107	0.157003
$231$	−5.66657	−0.372833
$232$	−3.36038	−0.220620
$233$	5.15303	0.337586	0.168793	−	0.985651i	$-0.446013\pi$
$233$	5.15303	0.337586	0.168793	+	0.985651i	$0.446013\pi$
$234$	−0.477432	−0.0312107
$235$	−1.19007	−0.0776316
$236$	−9.66455	−0.629109
$237$	15.4383	1.00283
$238$	−5.55980	−0.360388
$239$	23.3109	1.50786	0.753929	−	0.656956i	$-0.228156\pi$
$239$	23.3109	1.50786	0.753929	+	0.656956i	$0.228156\pi$
$240$	1.82012	0.117488
$241$	−22.6795	−1.46092	−0.730458	−	0.682958i	$-0.760693\pi$
$241$	−22.6795	−1.46092	−0.730458	+	0.682958i	$0.760693\pi$
$242$	21.5927	1.38803
$243$	−1.58160	−0.101459
$244$	−15.2817	−0.978314
$245$	0.537094	0.0343137
$246$	−19.0131	−1.21223
$247$	0.0980920	0.00624145
$248$	−26.5467	−1.68572
$249$	−5.10209	−0.323332
$250$	−8.18207	−0.517479
$251$	−5.29252	−0.334061	−0.167031	−	0.985952i	$-0.553418\pi$
$251$	−5.29252	−0.334061	−0.167031	+	0.985952i	$0.553418\pi$
$252$	−1.36858	−0.0862127
$253$	4.14443	0.260558
$254$	−52.3347	−3.28377
$255$	0.569282	0.0356498
$256$	−30.3691	−1.89807
$257$	−0.769143	−0.0479778	−0.0239889	−	0.999712i	$-0.507637\pi$
$257$	−0.769143	−0.0479778	−0.0239889	+	0.999712i	$0.507637\pi$
$258$	−35.0764	−2.18376
$259$	−8.96653	−0.557153
$260$	1.70838	0.105949
$261$	−0.114031	−0.00705836
$262$	5.13554	0.317274
$263$	28.8009	1.77594	0.887969	−	0.459904i	$-0.152116\pi$
$263$	28.8009	1.77594	0.887969	+	0.459904i	$0.152116\pi$
$264$	10.9166	0.671873
$265$	0.0524748	0.00322350
$266$	0.427058	0.0261846
$267$	13.3570	0.817437
$268$	6.33291	0.386844
$269$	9.75327	0.594667	0.297334	−	0.954774i	$-0.403903\pi$
$269$	9.75327	0.594667	0.297334	+	0.954774i	$0.403903\pi$
$270$	4.40407	0.268023
$271$	26.3599	1.60125	0.800624	−	0.599168i	$-0.204502\pi$
$271$	26.3599	1.60125	0.800624	+	0.599168i	$0.204502\pi$
$272$	3.10835	0.188471
$273$	5.09306	0.308246
$274$	−45.7734	−2.76527
$275$	−7.03639	−0.424311
$276$	−18.7108	−1.12626
$277$	9.51355	0.571614	0.285807	−	0.958287i	$-0.407738\pi$
$277$	9.51355	0.571614	0.285807	+	0.958287i	$0.407738\pi$
$278$	−53.2695	−3.19489
$279$	−0.900836	−0.0539316
$280$	3.57921	0.213899
$281$	−15.7402	−0.938984	−0.469492	−	0.882937i	$-0.655563\pi$
$281$	−15.7402	−0.938984	−0.469492	+	0.882937i	$0.655563\pi$
$282$	14.2032	0.845787
$283$	−13.4654	−0.800434	−0.400217	−	0.916420i	$-0.631065\pi$
$283$	−13.4654	−0.800434	−0.400217	+	0.916420i	$0.631065\pi$
$284$	34.3468	2.03811
$285$	−0.0437275	−0.00259019
$286$	4.51615	0.267046
$287$	−10.8504	−0.640477
$288$	−0.205706	−0.0121213
$289$	−16.0278	−0.942811
$290$	0.619712	0.0363907
$291$	21.8038	1.27816
$292$	−22.1768	−1.29780
$293$	−23.2132	−1.35613	−0.678065	−	0.735002i	$-0.737181\pi$
$293$	−23.2132	−1.35613	−0.678065	+	0.735002i	$0.737181\pi$
$294$	−6.41008	−0.373843
$295$	0.857695	0.0499369
$296$	17.2740	1.00403
$297$	7.66560	0.444803
$298$	−5.21810	−0.302277
$299$	−3.72497	−0.215421
$300$	31.7672	1.83408
$301$	−20.0173	−1.15378
$302$	42.3276	2.43568
$303$	−10.1380	−0.582414
$304$	−0.238758	−0.0136937
$305$	1.35620	0.0776559
$306$	0.363465	0.0207779
$307$	−32.8276	−1.87357	−0.936786	−	0.349902i	$-0.886215\pi$
$307$	−32.8276	−1.87357	−0.936786	+	0.349902i	$0.886215\pi$
$308$	12.9458	0.737656
$309$	4.29292	0.244216
$310$	4.89566	0.278055
$311$	−15.8592	−0.899292	−0.449646	−	0.893207i	$-0.648450\pi$
$311$	−15.8592	−0.899292	−0.449646	+	0.893207i	$0.648450\pi$
$312$	−9.81177	−0.555482
$313$	−8.04991	−0.455008	−0.227504	−	0.973777i	$-0.573056\pi$
$313$	−8.04991	−0.455008	−0.227504	+	0.973777i	$0.573056\pi$
$314$	−26.6541	−1.50418
$315$	0.121457	0.00684333
$316$	−35.2703	−1.98411
$317$	14.1990	0.797497	0.398748	−	0.917060i	$-0.369445\pi$
$317$	14.1990	0.797497	0.398748	+	0.917060i	$0.369445\pi$
$318$	−0.626273	−0.0351197
$319$	1.07865	0.0603929
$320$	3.27510	0.183084
$321$	−21.6297	−1.20725
$322$	−16.2172	−0.903749
$323$	−0.0746768	−0.00415513
$324$	−32.8459	−1.82477
$325$	6.32424	0.350806
$326$	28.4964	1.57827
$327$	−1.54479	−0.0854269
$328$	20.9032	1.15419
$329$	8.10543	0.446867
$330$	−2.01321	−0.110824
$331$	−3.63482	−0.199788	−0.0998938	−	0.994998i	$-0.531850\pi$
$331$	−3.63482	−0.199788	−0.0998938	+	0.994998i	$0.531850\pi$
$332$	11.6562	0.639717
$333$	0.586176	0.0321223
$334$	30.4083	1.66387
$335$	−0.562024	−0.0307066
$336$	−12.3966	−0.676290
$337$	−16.4949	−0.898536	−0.449268	−	0.893397i	$-0.648315\pi$
$337$	−16.4949	−0.898536	−0.449268	+	0.893397i	$0.648315\pi$
$338$	27.3978	1.49025
$339$	0.714745	0.0388196
$340$	−1.30058	−0.0705337
$341$	8.52125	0.461452
$342$	−0.0279184	−0.00150965
$343$	−19.9700	−1.07828
$344$	38.5633	2.07919
$345$	1.66052	0.0893993
$346$	27.1730	1.46083
$347$	−6.79819	−0.364946	−0.182473	−	0.983211i	$-0.558410\pi$
$347$	−6.79819	−0.364946	−0.182473	+	0.983211i	$0.558410\pi$
$348$	−4.86978	−0.261048
$349$	−20.7716	−1.11188	−0.555939	−	0.831223i	$-0.687641\pi$
$349$	−20.7716	−1.11188	−0.555939	+	0.831223i	$0.687641\pi$
$350$	27.5335	1.47173
$351$	−6.88976	−0.367748
$352$	1.94583	0.103713
$353$	−2.84109	−0.151216	−0.0756079	−	0.997138i	$-0.524090\pi$
$353$	−2.84109	−0.151216	−0.0756079	+	0.997138i	$0.524090\pi$
$354$	−10.2364	−0.544057
$355$	−3.04816	−0.161779
$356$	−30.5154	−1.61731
$357$	−3.87731	−0.205209
$358$	40.8216	2.15749
$359$	−12.6061	−0.665327	−0.332663	−	0.943046i	$-0.607947\pi$
$359$	−12.6061	−0.665327	−0.332663	+	0.943046i	$0.607947\pi$
$360$	−0.233987	−0.0123322
$361$	−18.9943	−0.999698
$362$	−62.4340	−3.28146
$363$	15.0584	0.790360
$364$	−11.6356	−0.609870
$365$	1.96811	0.103016
$366$	−16.1859	−0.846051
$367$	6.87447	0.358845	0.179422	−	0.983772i	$-0.442577\pi$
$367$	6.87447	0.358845	0.179422	+	0.983772i	$0.442577\pi$
$368$	9.06664	0.472631
$369$	0.709329	0.0369262
$370$	−3.18562	−0.165612
$371$	−0.357400	−0.0185553
$372$	−38.4708	−1.99462
$373$	−2.77923	−0.143903	−0.0719516	−	0.997408i	$-0.522923\pi$
$373$	−2.77923	−0.143903	−0.0719516	+	0.997408i	$0.522923\pi$
$374$	−3.43812	−0.177781
$375$	−5.70604	−0.294658
$376$	−15.6151	−0.805287
$377$	−0.969482	−0.0499309
$378$	−29.9956	−1.54281
$379$	−28.5793	−1.46802	−0.734011	−	0.679138i	$-0.762354\pi$
$379$	−28.5793	−1.46802	−0.734011	+	0.679138i	$0.762354\pi$
$380$	0.0998996	0.00512474
$381$	−36.4973	−1.86981
$382$	51.5692	2.63851
$383$	28.0272	1.43212	0.716062	−	0.698037i	$-0.245943\pi$
$383$	28.0272	1.43212	0.716062	+	0.698037i	$0.245943\pi$
$384$	−34.5301	−1.76211
$385$	−1.14890	−0.0585531
$386$	−48.9255	−2.49024
$387$	1.30861	0.0665203
$388$	−49.8127	−2.52886
$389$	18.5590	0.940980	0.470490	−	0.882405i	$-0.344077\pi$
$389$	18.5590	0.940980	0.470490	+	0.882405i	$0.344077\pi$
$390$	1.80946	0.0916254
$391$	2.83579	0.143412
$392$	7.04729	0.355942
$393$	3.58143	0.180659
$394$	4.97151	0.250461
$395$	3.13011	0.157493
$396$	−0.846317	−0.0425290
$397$	2.15579	0.108196	0.0540979	−	0.998536i	$-0.482772\pi$
$397$	2.15579	0.108196	0.0540979	+	0.998536i	$0.482772\pi$
$398$	−1.71171	−0.0858002
$399$	0.297823	0.0149098
$400$	−15.3933	−0.769666
$401$	−22.1324	−1.10524	−0.552619	−	0.833434i	$-0.686371\pi$
$401$	−22.1324	−1.10524	−0.552619	+	0.833434i	$0.686371\pi$
$402$	6.70761	0.334545
$403$	−7.65882	−0.381513
$404$	23.1612	1.15231
$405$	2.91496	0.144845
$406$	−4.22078	−0.209474
$407$	−5.54480	−0.274845
$408$	7.46963	0.369802
$409$	−10.7923	−0.533646	−0.266823	−	0.963746i	$-0.585974\pi$
$409$	−10.7923	−0.533646	−0.266823	+	0.963746i	$0.585974\pi$
$410$	−3.85490	−0.190380
$411$	−31.9216	−1.57458
$412$	−9.80758	−0.483185
$413$	−5.84166	−0.287449
$414$	1.06018	0.0521050
$415$	−1.03445	−0.0507790
$416$	−1.74889	−0.0857465
$417$	−37.1492	−1.81921
$418$	0.264088	0.0129170
$419$	15.9116	0.777333	0.388666	−	0.921379i	$-0.372936\pi$
$419$	15.9116	0.777333	0.388666	+	0.921379i	$0.372936\pi$
$420$	5.18691	0.253095
$421$	−18.8529	−0.918834	−0.459417	−	0.888221i	$-0.651942\pi$
$421$	−18.8529	−0.918834	−0.459417	+	0.888221i	$0.651942\pi$
$422$	10.5037	0.511310
$423$	−0.529883	−0.0257638
$424$	0.688530	0.0334380
$425$	−4.81460	−0.233542
$426$	36.3790	1.76257
$427$	−9.23693	−0.447006
$428$	49.4151	2.38857
$429$	3.14949	0.152059
$430$	−7.11173	−0.342958
$431$	−23.1410	−1.11466	−0.557331	−	0.830290i	$-0.688175\pi$
$431$	−23.1410	−1.11466	−0.557331	+	0.830290i	$0.688175\pi$
$432$	16.7698	0.806837
$433$	−20.9750	−1.00799	−0.503997	−	0.863706i	$-0.668138\pi$
$433$	−20.9750	−1.00799	−0.503997	+	0.863706i	$0.668138\pi$
$434$	−33.3438	−1.60055
$435$	0.432176	0.0207213
$436$	3.52921	0.169018
$437$	−0.217822	−0.0104198
$438$	−23.4889	−1.12234
$439$	−24.1481	−1.15252	−0.576262	−	0.817265i	$-0.695489\pi$
$439$	−24.1481	−1.15252	−0.576262	+	0.817265i	$0.695489\pi$
$440$	2.21334	0.105517
$441$	0.239143	0.0113878
$442$	3.09014	0.146983
$443$	20.5771	0.977645	0.488823	−	0.872383i	$-0.337427\pi$
$443$	20.5771	0.977645	0.488823	+	0.872383i	$0.337427\pi$
$444$	25.0331	1.18802
$445$	2.70813	0.128378
$446$	−38.3068	−1.81388
$447$	−3.63902	−0.172120
$448$	−22.3063	−1.05387
$449$	−24.0514	−1.13505	−0.567527	−	0.823355i	$-0.692100\pi$
$449$	−24.0514	−1.13505	−0.567527	+	0.823355i	$0.692100\pi$
$450$	−1.79997	−0.0848514
$451$	−6.70974	−0.315949
$452$	−1.63290	−0.0768052
$453$	29.5185	1.38690
$454$	−12.9043	−0.605628
$455$	1.03262	0.0484098
$456$	−0.573755	−0.0268686
$457$	4.79497	0.224299	0.112150	−	0.993691i	$-0.464226\pi$
$457$	4.79497	0.224299	0.112150	+	0.993691i	$0.464226\pi$
$458$	7.88680	0.368526
$459$	5.24513	0.244822
$460$	−3.79361	−0.176878
$461$	21.3398	0.993893	0.496946	−	0.867781i	$-0.334455\pi$
$461$	21.3398	0.993893	0.496946	+	0.867781i	$0.334455\pi$
$462$	13.7118	0.637929
$463$	−38.6824	−1.79772	−0.898861	−	0.438233i	$-0.855605\pi$
$463$	−38.6824	−1.79772	−0.898861	+	0.438233i	$0.855605\pi$
$464$	2.35974	0.109548
$465$	3.41415	0.158327
$466$	−12.4691	−0.577620
$467$	−23.7718	−1.10003	−0.550015	−	0.835155i	$-0.685378\pi$
$467$	−23.7718	−1.10003	−0.550015	+	0.835155i	$0.685378\pi$
$468$	0.760661	0.0351616
$469$	3.82788	0.176755
$470$	2.87969	0.132830
$471$	−18.5881	−0.856497
$472$	11.2539	0.518005
$473$	−12.3785	−0.569163
$474$	−37.3571	−1.71587
$475$	0.369818	0.0169684
$476$	8.85808	0.406009
$477$	0.0233646	0.00106979
$478$	−56.4069	−2.57999
$479$	−36.5452	−1.66979	−0.834897	−	0.550407i	$-0.814473\pi$
$479$	−36.5452	−1.66979	−0.834897	+	0.550407i	$0.814473\pi$
$480$	0.779622	0.0355847
$481$	4.98361	0.227233
$482$	54.8790	2.49967
$483$	−11.3096	−0.514605
$484$	−34.4022	−1.56374
$485$	4.42071	0.200734
$486$	3.82709	0.173600
$487$	13.3147	0.603346	0.301673	−	0.953412i	$-0.402455\pi$
$487$	13.3147	0.603346	0.301673	+	0.953412i	$0.402455\pi$
$488$	17.7949	0.805538
$489$	19.8729	0.898684
$490$	−1.29964	−0.0587118
$491$	4.30808	0.194421	0.0972105	−	0.995264i	$-0.469008\pi$
$491$	4.30808	0.194421	0.0972105	+	0.995264i	$0.469008\pi$
$492$	30.2924	1.36569
$493$	0.738060	0.0332406
$494$	−0.237359	−0.0106793
$495$	0.0751077	0.00337584
$496$	18.6417	0.837037
$497$	20.7607	0.931243
$498$	12.3459	0.553231
$499$	−0.108783	−0.00486981	−0.00243490	−	0.999997i	$-0.500775\pi$
$499$	−0.108783	−0.00486981	−0.00243490	+	0.999997i	$0.500775\pi$
$500$	13.0360	0.582986
$501$	21.2062	0.947425
$502$	12.8067	0.571589
$503$	−21.1780	−0.944279	−0.472140	−	0.881524i	$-0.656518\pi$
$503$	−21.1780	−0.944279	−0.472140	+	0.881524i	$0.656518\pi$
$504$	1.59366	0.0709871
$505$	−2.05548	−0.0914676
$506$	−10.0285	−0.445822
$507$	19.1068	0.848562
$508$	83.3815	3.69945
$509$	−18.0516	−0.800125	−0.400063	−	0.916488i	$-0.631012\pi$
$509$	−18.0516	−0.800125	−0.400063	+	0.916488i	$0.631012\pi$
$510$	−1.37753	−0.0609979
$511$	−13.4046	−0.592984
$512$	32.5614	1.43902
$513$	−0.402887	−0.0177879
$514$	1.86114	0.0820916
$515$	0.870388	0.0383539
$516$	55.8850	2.46020
$517$	5.01230	0.220441
$518$	21.6969	0.953306
$519$	18.9500	0.831812
$520$	−1.98933	−0.0872380
$521$	9.79969	0.429332	0.214666	−	0.976687i	$-0.431134\pi$
$521$	9.79969	0.429332	0.214666	+	0.976687i	$0.431134\pi$
$522$	0.275929	0.0120771
$523$	10.6777	0.466905	0.233453	−	0.972368i	$-0.424998\pi$
$523$	10.6777	0.466905	0.233453	+	0.972368i	$0.424998\pi$
$524$	−8.18212	−0.357437
$525$	19.2014	0.838018
$526$	−69.6913	−3.03868
$527$	5.83060	0.253985
$528$	−7.66591	−0.333616
$529$	−14.7284	−0.640364
$530$	−0.126977	−0.00551551
$531$	0.381892	0.0165727
$532$	−0.680404	−0.0294993
$533$	6.03065	0.261216
$534$	−32.3208	−1.39866
$535$	−4.38542	−0.189598
$536$	−7.37440	−0.318525
$537$	28.4683	1.22850
$538$	−23.6006	−1.01749
$539$	−2.26212	−0.0974363
$540$	−7.01672	−0.301952
$541$	17.8818	0.768800	0.384400	−	0.923167i	$-0.374408\pi$
$541$	17.8818	0.768800	0.384400	+	0.923167i	$0.374408\pi$
$542$	−63.7846	−2.73978
$543$	−43.5404	−1.86850
$544$	1.33142	0.0570841
$545$	−0.313205	−0.0134162
$546$	−12.3240	−0.527419
$547$	2.79753	0.119614	0.0598069	−	0.998210i	$-0.480951\pi$
$547$	2.79753	0.119614	0.0598069	+	0.998210i	$0.480951\pi$
$548$	72.9279	3.11532
$549$	0.603853	0.0257718
$550$	17.0264	0.726008
$551$	−0.0566917	−0.00241515
$552$	21.7879	0.927355
$553$	−21.3188	−0.906570
$554$	−23.0205	−0.978049
$555$	−2.22160	−0.0943015
$556$	84.8708	3.59932
$557$	14.3539	0.608194	0.304097	−	0.952641i	$-0.401645\pi$
$557$	14.3539	0.608194	0.304097	+	0.952641i	$0.401645\pi$
$558$	2.17981	0.0922787
$559$	11.1256	0.470565
$560$	−2.51341	−0.106211
$561$	−2.39768	−0.101230
$562$	38.0876	1.60663
$563$	−23.2963	−0.981822	−0.490911	−	0.871210i	$-0.663336\pi$
$563$	−23.2963	−0.981822	−0.490911	+	0.871210i	$0.663336\pi$
$564$	−22.6290	−0.952853
$565$	0.144914	0.00609659
$566$	32.5830	1.36957
$567$	−19.8534	−0.833766
$568$	−39.9953	−1.67817
$569$	−34.0255	−1.42642	−0.713211	−	0.700949i	$-0.752760\pi$
$569$	−34.0255	−1.42642	−0.713211	+	0.700949i	$0.752760\pi$
$570$	0.105810	0.00443190
$571$	−7.42503	−0.310728	−0.155364	−	0.987857i	$-0.549655\pi$
$571$	−7.42503	−0.310728	−0.155364	+	0.987857i	$0.549655\pi$
$572$	−7.19530	−0.300851
$573$	35.9635	1.50240
$574$	26.2553	1.09588
$575$	−14.0436	−0.585657
$576$	1.45825	0.0607603
$577$	34.6600	1.44291	0.721457	−	0.692459i	$-0.243473\pi$
$577$	34.6600	1.44291	0.721457	+	0.692459i	$0.243473\pi$
$578$	38.7835	1.61318
$579$	−34.1198	−1.41797
$580$	−0.987347	−0.0409973
$581$	7.04550	0.292297
$582$	−52.7600	−2.18697
$583$	−0.221012	−0.00915338
$584$	25.8239	1.06860
$585$	−0.0675060	−0.00279103
$586$	56.1704	2.32038
$587$	−16.1171	−0.665225	−0.332612	−	0.943064i	$-0.607930\pi$
$587$	−16.1171	−0.665225	−0.332612	+	0.943064i	$0.607930\pi$
$588$	10.2128	0.421167
$589$	−0.447859	−0.0184537
$590$	−2.07542	−0.0854436
$591$	3.46705	0.142615
$592$	−12.1302	−0.498548
$593$	−24.7820	−1.01767	−0.508837	−	0.860863i	$-0.669924\pi$
$593$	−24.7820	−1.01767	−0.508837	+	0.860863i	$0.669924\pi$
$594$	−18.5489	−0.761072
$595$	−0.786123	−0.0322279
$596$	8.31367	0.340541
$597$	−1.19372	−0.0488556
$598$	9.01354	0.368591
$599$	−33.4117	−1.36516	−0.682582	−	0.730809i	$-0.739143\pi$
$599$	−33.4117	−1.36516	−0.682582	+	0.730809i	$0.739143\pi$
$600$	−36.9915	−1.51017
$601$	14.3110	0.583759	0.291880	−	0.956455i	$-0.405719\pi$
$601$	14.3110	0.583759	0.291880	+	0.956455i	$0.405719\pi$
$602$	48.4372	1.97415
$603$	−0.250243	−0.0101907
$604$	−67.4378	−2.74400
$605$	3.05308	0.124125
$606$	24.5316	0.996528
$607$	1.45525	0.0590668	0.0295334	−	0.999564i	$-0.490598\pi$
$607$	1.45525	0.0590668	0.0295334	+	0.999564i	$0.490598\pi$
$608$	−0.102269	−0.00414754
$609$	−2.94350	−0.119277
$610$	−3.28169	−0.132872
$611$	−4.50501	−0.182253
$612$	−0.579086	−0.0234082
$613$	45.3803	1.83289	0.916447	−	0.400155i	$-0.131044\pi$
$613$	45.3803	1.83289	0.916447	+	0.400155i	$0.131044\pi$
$614$	79.4351	3.20574
$615$	−2.68834	−0.108405
$616$	−15.0748	−0.607382
$617$	−28.9157	−1.16410	−0.582050	−	0.813153i	$-0.697749\pi$
$617$	−28.9157	−1.16410	−0.582050	+	0.813153i	$0.697749\pi$
$618$	−10.3879	−0.417861
$619$	−9.51170	−0.382307	−0.191154	−	0.981560i	$-0.561223\pi$
$619$	−9.51170	−0.382307	−0.191154	+	0.981560i	$0.561223\pi$
$620$	−7.79994	−0.313253
$621$	15.2993	0.613941
$622$	38.3755	1.53872
$623$	−18.4448	−0.738974
$624$	6.89004	0.275823
$625$	23.2578	0.930312
$626$	19.4789	0.778533
$627$	0.184170	0.00735505
$628$	42.4663	1.69459
$629$	−3.79399	−0.151276
$630$	−0.293897	−0.0117092
$631$	11.7218	0.466636	0.233318	−	0.972400i	$-0.425042\pi$
$631$	11.7218	0.466636	0.233318	+	0.972400i	$0.425042\pi$
$632$	41.0707	1.63370
$633$	7.32508	0.291146
$634$	−34.3583	−1.36454
$635$	−7.39981	−0.293653
$636$	0.997801	0.0395654
$637$	2.03317	0.0805571
$638$	−2.61008	−0.103334
$639$	−1.35720	−0.0536901
$640$	−7.00097	−0.276737
$641$	40.4402	1.59729	0.798645	−	0.601802i	$-0.205550\pi$
$641$	40.4402	1.59729	0.798645	+	0.601802i	$0.205550\pi$
$642$	52.3388	2.06565
$643$	26.9338	1.06217	0.531083	−	0.847320i	$-0.321785\pi$
$643$	26.9338	1.06217	0.531083	+	0.847320i	$0.321785\pi$
$644$	25.8378	1.01815
$645$	−4.95960	−0.195284
$646$	0.180700	0.00710955
$647$	4.72114	0.185607	0.0928035	−	0.995684i	$-0.470417\pi$
$647$	4.72114	0.185607	0.0928035	+	0.995684i	$0.470417\pi$
$648$	38.2476	1.50251
$649$	−3.61241	−0.141800
$650$	−15.3032	−0.600240
$651$	−23.2534	−0.911372
$652$	−45.4015	−1.77806
$653$	−6.76517	−0.264742	−0.132371	−	0.991200i	$-0.542259\pi$
$653$	−6.76517	−0.264742	−0.132371	+	0.991200i	$0.542259\pi$
$654$	3.73802	0.146168
$655$	0.726134	0.0283724
$656$	−14.6787	−0.573107
$657$	0.876308	0.0341880
$658$	−19.6132	−0.764603
$659$	−16.4811	−0.642011	−0.321006	−	0.947077i	$-0.604021\pi$
$659$	−16.4811	−0.642011	−0.321006	+	0.947077i	$0.604021\pi$
$660$	3.20752	0.124853
$661$	−11.8073	−0.459252	−0.229626	−	0.973279i	$-0.573750\pi$
$661$	−11.8073	−0.459252	−0.229626	+	0.973279i	$0.573750\pi$
$662$	8.79540	0.341843
$663$	2.15501	0.0836938
$664$	−13.5731	−0.526740
$665$	0.0603835	0.00234157
$666$	−1.41841	−0.0549622
$667$	2.15282	0.0833576
$668$	−48.4476	−1.87449
$669$	−26.7145	−1.03284
$670$	1.35996	0.0525400
$671$	−5.71201	−0.220510
$672$	−5.30992	−0.204834
$673$	−43.8812	−1.69149	−0.845747	−	0.533584i	$-0.820845\pi$
$673$	−43.8812	−1.69149	−0.845747	+	0.533584i	$0.820845\pi$
$674$	39.9138	1.53742
$675$	−25.9752	−0.999785
$676$	−43.6512	−1.67889
$677$	9.02433	0.346833	0.173417	−	0.984849i	$-0.444519\pi$
$677$	9.02433	0.346833	0.173417	+	0.984849i	$0.444519\pi$
$678$	−1.72951	−0.0664215
$679$	−30.1089	−1.15547
$680$	1.51446	0.0580770
$681$	−8.99923	−0.344851
$682$	−20.6194	−0.789558
$683$	10.3892	0.397530	0.198765	−	0.980047i	$-0.436307\pi$
$683$	10.3892	0.397530	0.198765	+	0.980047i	$0.436307\pi$
$684$	0.0444806	0.00170076
$685$	−6.47209	−0.247286
$686$	48.3227	1.84497
$687$	5.50012	0.209843
$688$	−27.0800	−1.03242
$689$	0.198643	0.00756771
$690$	−4.01806	−0.152965
$691$	−4.22334	−0.160663	−0.0803317	−	0.996768i	$-0.525598\pi$
$691$	−4.22334	−0.160663	−0.0803317	+	0.996768i	$0.525598\pi$
$692$	−43.2930	−1.64575
$693$	−0.511550	−0.0194322
$694$	16.4500	0.624434
$695$	−7.53199	−0.285705
$696$	5.67065	0.214945
$697$	−4.59109	−0.173900
$698$	50.2624	1.90246
$699$	−8.69575	−0.328903
$700$	−43.8674	−1.65803
$701$	12.3182	0.465252	0.232626	−	0.972566i	$-0.425268\pi$
$701$	12.3182	0.465252	0.232626	+	0.972566i	$0.425268\pi$
$702$	16.6716	0.629229
$703$	0.291423	0.0109912
$704$	−13.7940	−0.519879
$705$	2.00825	0.0756349
$706$	6.87476	0.258735
$707$	13.9996	0.526510
$708$	16.3089	0.612928
$709$	11.6659	0.438122	0.219061	−	0.975711i	$-0.429701\pi$
$709$	11.6659	0.438122	0.219061	+	0.975711i	$0.429701\pi$
$710$	7.37582	0.276810
$711$	1.39369	0.0522676
$712$	35.5338	1.33169
$713$	17.0071	0.636921
$714$	9.38217	0.351119
$715$	0.638558	0.0238807
$716$	−65.0385	−2.43060
$717$	−39.3372	−1.46908
$718$	30.5039	1.13839
$719$	−3.44097	−0.128326	−0.0641632	−	0.997939i	$-0.520438\pi$
$719$	−3.44097	−0.128326	−0.0641632	+	0.997939i	$0.520438\pi$
$720$	0.164311	0.00612350
$721$	−5.92811	−0.220774
$722$	45.9616	1.71051
$723$	38.2717	1.42334
$724$	99.4721	3.69685
$725$	−3.65506	−0.135745
$726$	−36.4377	−1.35233
$727$	−3.13100	−0.116122	−0.0580612	−	0.998313i	$-0.518492\pi$
$727$	−3.13100	−0.116122	−0.0580612	+	0.998313i	$0.518492\pi$
$728$	13.5491	0.502163
$729$	28.2283	1.04549
$730$	−4.76236	−0.176263
$731$	−8.46988	−0.313270
$732$	25.7880	0.953151
$733$	−5.88260	−0.217279	−0.108639	−	0.994081i	$-0.534649\pi$
$733$	−5.88260	−0.217279	−0.108639	+	0.994081i	$0.534649\pi$
$734$	−16.6346	−0.613994
$735$	−0.906347	−0.0334311
$736$	3.88357	0.143150
$737$	2.36711	0.0871938
$738$	−1.71641	−0.0631819
$739$	−6.71975	−0.247190	−0.123595	−	0.992333i	$-0.539442\pi$
$739$	−6.71975	−0.247190	−0.123595	+	0.992333i	$0.539442\pi$
$740$	5.07544	0.186577
$741$	−0.165530	−0.00608091
$742$	0.864823	0.0317487
$743$	−38.7116	−1.42019	−0.710096	−	0.704105i	$-0.751348\pi$
$743$	−38.7116	−1.42019	−0.710096	+	0.704105i	$0.751348\pi$
$744$	44.7976	1.64236
$745$	−0.737809	−0.0270312
$746$	6.72508	0.246223
$747$	−0.460591	−0.0168521
$748$	5.47773	0.200286
$749$	29.8686	1.09137
$750$	13.8073	0.504170
$751$	29.0953	1.06170	0.530852	−	0.847464i	$-0.321872\pi$
$751$	29.0953	1.06170	0.530852	+	0.847464i	$0.321872\pi$
$752$	10.9653	0.399862
$753$	8.93114	0.325469
$754$	2.34592	0.0854333
$755$	5.98487	0.217812
$756$	47.7901	1.73811
$757$	−24.9763	−0.907780	−0.453890	−	0.891058i	$-0.649964\pi$
$757$	−24.9763	−0.907780	−0.453890	+	0.891058i	$0.649964\pi$
$758$	69.1552	2.51183
$759$	−6.99372	−0.253856
$760$	−0.116329	−0.00421968
$761$	3.59941	0.130479	0.0652393	−	0.997870i	$-0.479219\pi$
$761$	3.59941	0.130479	0.0652393	+	0.997870i	$0.479219\pi$
$762$	88.3148	3.19931
$763$	2.13320	0.0772271
$764$	−82.1619	−2.97251
$765$	0.0513919	0.00185808
$766$	−67.8192	−2.45041
$767$	3.24680	0.117235
$768$	51.2478	1.84925
$769$	−36.1733	−1.30444	−0.652221	−	0.758029i	$-0.726162\pi$
$769$	−36.1733	−1.30444	−0.652221	+	0.758029i	$0.726162\pi$
$770$	2.78006	0.100186
$771$	1.29793	0.0467438
$772$	77.9498	2.80547
$773$	−15.4107	−0.554283	−0.277141	−	0.960829i	$-0.589387\pi$
$773$	−15.4107	−0.554283	−0.277141	+	0.960829i	$0.589387\pi$
$774$	−3.16652	−0.113818
$775$	−28.8746	−1.03721
$776$	58.0047	2.08225
$777$	15.1310	0.542823
$778$	−44.9085	−1.61005
$779$	0.352649	0.0126350
$780$	−2.88289	−0.103224
$781$	12.8381	0.459385
$782$	−6.86195	−0.245383
$783$	3.98190	0.142301
$784$	−4.94877	−0.176742
$785$	−3.76874	−0.134512
$786$	−8.66622	−0.309114
$787$	−13.8746	−0.494576	−0.247288	−	0.968942i	$-0.579539\pi$
$787$	−13.8746	−0.494576	−0.247288	+	0.968942i	$0.579539\pi$
$788$	−7.92079	−0.282166
$789$	−48.6015	−1.73026
$790$	−7.57414	−0.269476
$791$	−0.986994	−0.0350935
$792$	0.985499	0.0350182
$793$	5.13390	0.182310
$794$	−5.21649	−0.185126
$795$	−0.0885513	−0.00314059
$796$	2.72716	0.0966615
$797$	−21.9090	−0.776058	−0.388029	−	0.921647i	$-0.626844\pi$
$797$	−21.9090	−0.776058	−0.388029	+	0.921647i	$0.626844\pi$
$798$	−0.720661	−0.0255111
$799$	3.42963	0.121332
$800$	−6.59352	−0.233116
$801$	1.20580	0.0426050
$802$	53.5551	1.89109
$803$	−8.28923	−0.292521
$804$	−10.6868	−0.376894
$805$	−2.29302	−0.0808182
$806$	18.5325	0.652780
$807$	−16.4587	−0.579372
$808$	−26.9702	−0.948810
$809$	−20.9806	−0.737639	−0.368819	−	0.929501i	$-0.620238\pi$
$809$	−20.9806	−0.737639	−0.368819	+	0.929501i	$0.620238\pi$
$810$	−7.05351	−0.247835
$811$	−31.4983	−1.10605	−0.553027	−	0.833163i	$-0.686527\pi$
$811$	−31.4983	−1.10605	−0.553027	+	0.833163i	$0.686527\pi$
$812$	6.72471	0.235991
$813$	−44.4823	−1.56006
$814$	13.4171	0.470269
$815$	4.02923	0.141138
$816$	−5.24534	−0.183624
$817$	0.650586	0.0227611
$818$	26.1148	0.913084
$819$	0.459776	0.0160659
$820$	6.14177	0.214480
$821$	−42.3421	−1.47775	−0.738874	−	0.673843i	$-0.764642\pi$
$821$	−42.3421	−1.47775	−0.738874	+	0.673843i	$0.764642\pi$
$822$	77.2427	2.69415
$823$	24.6925	0.860725	0.430362	−	0.902656i	$-0.358386\pi$
$823$	24.6925	0.860725	0.430362	+	0.902656i	$0.358386\pi$
$824$	11.4205	0.397852
$825$	11.8739	0.413397
$826$	14.1354	0.491835
$827$	20.4673	0.711719	0.355860	−	0.934539i	$-0.384188\pi$
$827$	20.4673	0.711719	0.355860	+	0.934539i	$0.384188\pi$
$828$	−1.68912	−0.0587009
$829$	−10.2301	−0.355306	−0.177653	−	0.984093i	$-0.556850\pi$
$829$	−10.2301	−0.355306	−0.177653	+	0.984093i	$0.556850\pi$
$830$	2.50312	0.0868845
$831$	−16.0541	−0.556912
$832$	12.3979	0.429819
$833$	−1.54784	−0.0536294
$834$	89.8923	3.11272
$835$	4.29956	0.148792
$836$	−0.420754	−0.0145521
$837$	31.4566	1.08730
$838$	−38.5023	−1.33004
$839$	34.1005	1.17728	0.588640	−	0.808395i	$-0.299664\pi$
$839$	34.1005	1.17728	0.588640	+	0.808395i	$0.299664\pi$
$840$	−6.03993	−0.208397
$841$	−28.4397	−0.980679
$842$	45.6195	1.57215
$843$	26.5617	0.914833
$844$	−16.7348	−0.576036
$845$	3.87389	0.133266
$846$	1.28219	0.0440826
$847$	−20.7942	−0.714496
$848$	−0.483501	−0.0166035
$849$	22.7228	0.779846
$850$	11.6502	0.399598
$851$	−11.0666	−0.379357
$852$	−57.9603	−1.98569
$853$	13.1270	0.449461	0.224730	−	0.974421i	$-0.427850\pi$
$853$	13.1270	0.449461	0.224730	+	0.974421i	$0.427850\pi$
$854$	22.3512	0.764842
$855$	−0.00394750	−0.000135002 0
$856$	−57.5417	−1.96673
$857$	15.3193	0.523296	0.261648	−	0.965163i	$-0.415734\pi$
$857$	15.3193	0.523296	0.261648	+	0.965163i	$0.415734\pi$
$858$	−7.62102	−0.260177
$859$	19.0102	0.648619	0.324309	−	0.945951i	$-0.394868\pi$
$859$	19.0102	0.648619	0.324309	+	0.945951i	$0.394868\pi$
$860$	11.3307	0.386372
$861$	18.3100	0.624003
$862$	55.9957	1.90722
$863$	−20.2656	−0.689850	−0.344925	−	0.938630i	$-0.612096\pi$
$863$	−20.2656	−0.689850	−0.344925	+	0.938630i	$0.612096\pi$
$864$	7.18312	0.244375
$865$	3.84210	0.130635
$866$	50.7545	1.72471
$867$	27.0469	0.918562
$868$	53.1245	1.80316
$869$	−13.1833	−0.447214
$870$	−1.04576	−0.0354547
$871$	−2.12754	−0.0720889
$872$	−4.10961	−0.139169
$873$	1.96833	0.0666180
$874$	0.527078	0.0178287
$875$	7.87949	0.266375
$876$	37.4233	1.26442
$877$	5.02492	0.169679	0.0848397	−	0.996395i	$-0.472962\pi$
$877$	5.02492	0.169679	0.0848397	+	0.996395i	$0.472962\pi$
$878$	58.4326	1.97200
$879$	39.1723	1.32125
$880$	−1.55426	−0.0523941
$881$	−17.6467	−0.594531	−0.297266	−	0.954795i	$-0.596075\pi$
$881$	−17.6467	−0.594531	−0.297266	+	0.954795i	$0.596075\pi$
$882$	−0.578669	−0.0194848
$883$	0.990211	0.0333232	0.0166616	−	0.999861i	$-0.494696\pi$
$883$	0.990211	0.0333232	0.0166616	+	0.999861i	$0.494696\pi$
$884$	−4.92333	−0.165589
$885$	−1.44736	−0.0486525
$886$	−49.7916	−1.67278
$887$	20.3402	0.682959	0.341479	−	0.939889i	$-0.389072\pi$
$887$	20.3402	0.682959	0.341479	+	0.939889i	$0.389072\pi$
$888$	−29.1499	−0.978206
$889$	50.3993	1.69034
$890$	−6.55304	−0.219658
$891$	−12.2771	−0.411299
$892$	61.0317	2.04349
$893$	−0.263436	−0.00881555
$894$	8.80556	0.294502
$895$	5.77194	0.192935
$896$	47.6828	1.59297
$897$	6.28589	0.209880
$898$	58.1986	1.94211
$899$	4.42637	0.147628
$900$	2.86778	0.0955926
$901$	−0.151226	−0.00503806
$902$	16.2360	0.540599
$903$	33.7792	1.12410
$904$	1.90144	0.0632410
$905$	−8.82780	−0.293446
$906$	−71.4278	−2.37303
$907$	9.51318	0.315880	0.157940	−	0.987449i	$-0.449515\pi$
$907$	9.51318	0.315880	0.157940	+	0.987449i	$0.449515\pi$
$908$	20.5596	0.682293
$909$	−0.915209	−0.0303556
$910$	−2.49869	−0.0828306
$911$	−17.1220	−0.567279	−0.283639	−	0.958931i	$-0.591542\pi$
$911$	−17.1220	−0.567279	−0.283639	+	0.958931i	$0.591542\pi$
$912$	0.402904	0.0133415
$913$	4.35685	0.144191
$914$	−11.6027	−0.383783
$915$	−2.28859	−0.0756585
$916$	−12.5655	−0.415177
$917$	−4.94562	−0.163319
$918$	−12.6920	−0.418897
$919$	−55.4428	−1.82889	−0.914445	−	0.404710i	$-0.867372\pi$
$919$	−55.4428	−1.82889	−0.914445	+	0.404710i	$0.867372\pi$
$920$	4.41749	0.145640
$921$	55.3967	1.82538
$922$	−51.6372	−1.70058
$923$	−11.5388	−0.379804
$924$	−21.8461	−0.718683
$925$	18.7888	0.617771
$926$	93.6022	3.07596
$927$	0.387543	0.0127286
$928$	1.01076	0.0331799
$929$	25.3058	0.830257	0.415128	−	0.909763i	$-0.363737\pi$
$929$	25.3058	0.830257	0.415128	+	0.909763i	$0.363737\pi$
$930$	−8.26144	−0.270903
$931$	0.118892	0.00389653
$932$	19.8662	0.650740
$933$	26.7624	0.876162
$934$	57.5222	1.88218
$935$	−0.486129	−0.0158981
$936$	−0.885756	−0.0289519
$937$	36.5197	1.19305	0.596524	−	0.802595i	$-0.296548\pi$
$937$	36.5197	1.19305	0.596524	+	0.802595i	$0.296548\pi$
$938$	−9.26256	−0.302433
$939$	13.5842	0.443305
$940$	−4.58802	−0.149645
$941$	13.3528	0.435289	0.217644	−	0.976028i	$-0.430163\pi$
$941$	13.3528	0.435289	0.217644	+	0.976028i	$0.430163\pi$
$942$	44.9789	1.46549
$943$	−13.3916	−0.436090
$944$	−7.90277	−0.257213
$945$	−4.24120	−0.137966
$946$	29.9530	0.973855
$947$	43.8561	1.42513	0.712566	−	0.701605i	$-0.247533\pi$
$947$	43.8561	1.42513	0.712566	+	0.701605i	$0.247533\pi$
$948$	59.5187	1.93308
$949$	7.45028	0.241846
$950$	−0.894872	−0.0290335
$951$	−23.9609	−0.776985
$952$	−10.3148	−0.334306
$953$	5.63359	0.182490	0.0912449	−	0.995828i	$-0.470915\pi$
$953$	5.63359	0.182490	0.0912449	+	0.995828i	$0.470915\pi$
$954$	−0.0565368	−0.00183045
$955$	7.29158	0.235950
$956$	89.8695	2.90659
$957$	−1.82023	−0.0588396
$958$	88.4308	2.85707
$959$	44.0807	1.42344
$960$	−5.52673	−0.178375
$961$	3.96786	0.127995
$962$	−12.0592	−0.388803
$963$	−1.95262	−0.0629223
$964$	−87.4352	−2.81610
$965$	−6.91777	−0.222691
$966$	27.3666	0.880504
$967$	−30.4208	−0.978268	−0.489134	−	0.872209i	$-0.662687\pi$
$967$	−30.4208	−0.978268	−0.489134	+	0.872209i	$0.662687\pi$
$968$	40.0599	1.28757
$969$	0.126017	0.00404825
$970$	−10.6971	−0.343462
$971$	11.7030	0.375568	0.187784	−	0.982210i	$-0.439869\pi$
$971$	11.7030	0.375568	0.187784	+	0.982210i	$0.439869\pi$
$972$	−6.09745	−0.195576
$973$	51.2995	1.64459
$974$	−32.2184	−1.03234
$975$	−10.6722	−0.341783
$976$	−12.4960	−0.399987
$977$	−48.6552	−1.55662	−0.778309	−	0.627881i	$-0.783922\pi$
$977$	−48.6552	−1.55662	−0.778309	+	0.627881i	$0.783922\pi$
$978$	−48.0877	−1.53768
$979$	−11.4060	−0.364538
$980$	2.07063	0.0661440
$981$	−0.139456	−0.00445247
$982$	−10.4245	−0.332660
$983$	−38.8094	−1.23783	−0.618914	−	0.785459i	$-0.712427\pi$
$983$	−38.8094	−1.23783	−0.618914	+	0.785459i	$0.712427\pi$
$984$	−35.2742	−1.12450
$985$	0.702942	0.0223976
$986$	−1.78593	−0.0568756
$987$	−13.6779	−0.435373
$988$	0.378170	0.0120312
$989$	−24.7055	−0.785589
$990$	−0.181743	−0.00577616
$991$	4.97668	0.158090	0.0790448	−	0.996871i	$-0.474813\pi$
$991$	4.97668	0.158090	0.0790448	+	0.996871i	$0.474813\pi$
$992$	7.98492	0.253521
$993$	6.13376	0.194649
$994$	−50.2359	−1.59338
$995$	−0.242026	−0.00767273
$996$	−19.6699	−0.623263
$997$	40.4846	1.28216	0.641080	−	0.767474i	$-0.278487\pi$
$997$	40.4846	1.28216	0.641080	+	0.767474i	$0.278487\pi$
$998$	0.263230	0.00833239
$999$	−20.4689	−0.647606

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4001.2.a.a.1.13	✓	149

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4001.2.a.a.1.13	✓	149	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:	\( N \)	\(=\)	\( 4001 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4001.a (trivial)

\(f(q)\)	\(=\)	\(q-2.41976 q^{2} -1.68750 q^{3} +3.85525 q^{4} -0.342140 q^{5} +4.08335 q^{6} +2.33028 q^{7} -4.48927 q^{8} -0.152339 q^{9} +O(q^{10})\)	\(q-2.41976 q^{2} -1.68750 q^{3} +3.85525 q^{4} -0.342140 q^{5} +4.08335 q^{6} +2.33028 q^{7} -4.48927 q^{8} -0.152339 q^{9} +0.827898 q^{10} +1.44102 q^{11} -6.50574 q^{12} -1.29517 q^{13} -5.63872 q^{14} +0.577362 q^{15} +3.15247 q^{16} +0.986005 q^{17} +0.368624 q^{18} -0.0757367 q^{19} -1.31904 q^{20} -3.93235 q^{21} -3.48692 q^{22} +2.87604 q^{23} +7.57565 q^{24} -4.88294 q^{25} +3.13401 q^{26} +5.31958 q^{27} +8.98381 q^{28} +0.748536 q^{29} -1.39708 q^{30} +5.91336 q^{31} +1.35032 q^{32} -2.43172 q^{33} -2.38590 q^{34} -0.797282 q^{35} -0.587306 q^{36} -3.84784 q^{37} +0.183265 q^{38} +2.18560 q^{39} +1.53596 q^{40} -4.65625 q^{41} +9.51535 q^{42} -8.59010 q^{43} +5.55548 q^{44} +0.0521213 q^{45} -6.95934 q^{46} +3.47831 q^{47} -5.31979 q^{48} -1.56981 q^{49} +11.8156 q^{50} -1.66388 q^{51} -4.99321 q^{52} -0.153372 q^{53} -12.8721 q^{54} -0.493030 q^{55} -10.4612 q^{56} +0.127806 q^{57} -1.81128 q^{58} -2.50685 q^{59} +2.22588 q^{60} -3.96388 q^{61} -14.3089 q^{62} -0.354992 q^{63} -9.57239 q^{64} +0.443130 q^{65} +5.88418 q^{66} +1.64267 q^{67} +3.80130 q^{68} -4.85333 q^{69} +1.92923 q^{70} +8.90909 q^{71} +0.683891 q^{72} -5.75235 q^{73} +9.31086 q^{74} +8.23997 q^{75} -0.291984 q^{76} +3.35797 q^{77} -5.28864 q^{78} -9.14863 q^{79} -1.07859 q^{80} -8.51978 q^{81} +11.2670 q^{82} +3.02346 q^{83} -15.1602 q^{84} -0.337352 q^{85} +20.7860 q^{86} -1.26316 q^{87} -6.46911 q^{88} -7.91527 q^{89} -0.126121 q^{90} -3.01811 q^{91} +11.0879 q^{92} -9.97881 q^{93} -8.41669 q^{94} +0.0259126 q^{95} -2.27866 q^{96} -12.9207 q^{97} +3.79856 q^{98} -0.219523 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 149 q - 6 q^{2} - 28 q^{3} + 116 q^{4} - 19 q^{5} - 31 q^{6} - 47 q^{7} - 15 q^{8} + 115 q^{9}+O(q^{10}) \) 149 * q - 6 * q^2 - 28 * q^3 + 116 * q^4 - 19 * q^5 - 31 * q^6 - 47 * q^7 - 15 * q^8 + 115 * q^9	\( 149 q - 6 q^{2} - 28 q^{3} + 116 q^{4} - 19 q^{5} - 31 q^{6} - 47 q^{7} - 15 q^{8} + 115 q^{9} - 48 q^{10} - 31 q^{11} - 61 q^{12} - 54 q^{13} - 44 q^{14} - 65 q^{15} + 58 q^{16} - 26 q^{17} - 23 q^{18} - 86 q^{19} - 52 q^{20} - 30 q^{21} - 56 q^{22} - 63 q^{23} - 90 q^{24} + 92 q^{25} - 38 q^{26} - 103 q^{27} - 77 q^{28} - 51 q^{29} - 22 q^{30} - 256 q^{31} - 21 q^{32} - 36 q^{33} - 124 q^{34} - 50 q^{35} + 45 q^{36} - 42 q^{37} - 14 q^{38} - 119 q^{39} - 131 q^{40} - 55 q^{41} - 5 q^{42} - 55 q^{43} - 54 q^{44} - 68 q^{45} - 59 q^{46} - 82 q^{47} - 89 q^{48} + 30 q^{49} + 13 q^{50} - 83 q^{51} - 126 q^{52} - 23 q^{53} - 83 q^{54} - 244 q^{55} - 94 q^{56} - 14 q^{57} - 60 q^{58} - 93 q^{59} - 70 q^{60} - 139 q^{61} - 10 q^{62} - 120 q^{63} - 45 q^{64} - 37 q^{65} - 28 q^{66} - 110 q^{67} - 27 q^{68} - 79 q^{69} - 78 q^{70} - 123 q^{71} - 74 q^{73} - 25 q^{74} - 146 q^{75} - 192 q^{76} - q^{77} + 26 q^{78} - 273 q^{79} - 51 q^{80} + 41 q^{81} - 120 q^{82} - 27 q^{83} - 14 q^{84} - 71 q^{85} + 3 q^{86} - 98 q^{87} - 143 q^{88} - 70 q^{89} - 24 q^{90} - 256 q^{91} - 47 q^{92} + 16 q^{93} - 151 q^{94} - 83 q^{95} - 137 q^{96} - 108 q^{97} + 17 q^{98} - 131 q^{99}+O(q^{100}) \) 149 * q - 6 * q^2 - 28 * q^3 + 116 * q^4 - 19 * q^5 - 31 * q^6 - 47 * q^7 - 15 * q^8 + 115 * q^9 - 48 * q^10 - 31 * q^11 - 61 * q^12 - 54 * q^13 - 44 * q^14 - 65 * q^15 + 58 * q^16 - 26 * q^17 - 23 * q^18 - 86 * q^19 - 52 * q^20 - 30 * q^21 - 56 * q^22 - 63 * q^23 - 90 * q^24 + 92 * q^25 - 38 * q^26 - 103 * q^27 - 77 * q^28 - 51 * q^29 - 22 * q^30 - 256 * q^31 - 21 * q^32 - 36 * q^33 - 124 * q^34 - 50 * q^35 + 45 * q^36 - 42 * q^37 - 14 * q^38 - 119 * q^39 - 131 * q^40 - 55 * q^41 - 5 * q^42 - 55 * q^43 - 54 * q^44 - 68 * q^45 - 59 * q^46 - 82 * q^47 - 89 * q^48 + 30 * q^49 + 13 * q^50 - 83 * q^51 - 126 * q^52 - 23 * q^53 - 83 * q^54 - 244 * q^55 - 94 * q^56 - 14 * q^57 - 60 * q^58 - 93 * q^59 - 70 * q^60 - 139 * q^61 - 10 * q^62 - 120 * q^63 - 45 * q^64 - 37 * q^65 - 28 * q^66 - 110 * q^67 - 27 * q^68 - 79 * q^69 - 78 * q^70 - 123 * q^71 - 74 * q^73 - 25 * q^74 - 146 * q^75 - 192 * q^76 - q^77 + 26 * q^78 - 273 * q^79 - 51 * q^80 + 41 * q^81 - 120 * q^82 - 27 * q^83 - 14 * q^84 - 71 * q^85 + 3 * q^86 - 98 * q^87 - 143 * q^88 - 70 * q^89 - 24 * q^90 - 256 * q^91 - 47 * q^92 + 16 * q^93 - 151 * q^94 - 83 * q^95 - 137 * q^96 - 108 * q^97 + 17 * q^98 - 131 * q^99

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