LMFDB - Embedded newform 1339.2.a.d.1.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1339,2,Mod(1,1339)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1339, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1339.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1339 = 13 \cdot 103 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1339.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:	$10.6919688306$
Analytic rank:	$1$
Dimension:	$19$
Coefficient field:	$\mathbb{Q}[x]/(x^{19} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{19} - 9 x^{18} + 13 x^{17} + 106 x^{16} - 351 x^{15} - 279 x^{14} + 2337 x^{13} - 1079 x^{12} + \cdots - 63 $ x^19 - 9x^18 + 13x^17 + 106x^16 - 351x^15 - 279x^14 + 2337x^13 - 1079x^12 - 6826x^11 + 7199x^10 + 9364x^9 - 14841x^8 - 5183x^7 + 14037x^6 - 108x^5 - 6099x^4 + 855x^3 + 1062x^2 - 135x - 63
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 3 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
Root			$-0.234290$ of defining polynomial
Character	$\chi$	$=$	1339.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.234290 q^{2} +1.28880 q^{3} -1.94511 q^{4} +2.43495 q^{5} +0.301953 q^{6} -2.52468 q^{7} -0.924301 q^{8} -1.33900 q^{9} +O(q^{10})$	\(q+0.234290 q^{2} +1.28880 q^{3} -1.94511 q^{4} +2.43495 q^{5} +0.301953 q^{6} -2.52468 q^{7} -0.924301 q^{8} -1.33900 q^{9} +0.570486 q^{10} -1.87833 q^{11} -2.50685 q^{12} +1.00000 q^{13} -0.591509 q^{14} +3.13816 q^{15} +3.67366 q^{16} -6.70137 q^{17} -0.313715 q^{18} +1.28617 q^{19} -4.73624 q^{20} -3.25380 q^{21} -0.440074 q^{22} -3.05667 q^{23} -1.19124 q^{24} +0.928983 q^{25} +0.234290 q^{26} -5.59209 q^{27} +4.91078 q^{28} -7.36530 q^{29} +0.735240 q^{30} +6.92773 q^{31} +2.70931 q^{32} -2.42078 q^{33} -1.57007 q^{34} -6.14748 q^{35} +2.60450 q^{36} -1.70801 q^{37} +0.301337 q^{38} +1.28880 q^{39} -2.25063 q^{40} -1.22577 q^{41} -0.762335 q^{42} -4.32013 q^{43} +3.65355 q^{44} -3.26040 q^{45} -0.716148 q^{46} -2.62920 q^{47} +4.73460 q^{48} -0.625975 q^{49} +0.217652 q^{50} -8.63671 q^{51} -1.94511 q^{52} +4.70761 q^{53} -1.31017 q^{54} -4.57363 q^{55} +2.33357 q^{56} +1.65761 q^{57} -1.72562 q^{58} -10.0840 q^{59} -6.10406 q^{60} +1.21800 q^{61} +1.62310 q^{62} +3.38055 q^{63} -6.71256 q^{64} +2.43495 q^{65} -0.567167 q^{66} -1.12824 q^{67} +13.0349 q^{68} -3.93942 q^{69} -1.44030 q^{70} -2.58893 q^{71} +1.23764 q^{72} +9.61558 q^{73} -0.400170 q^{74} +1.19727 q^{75} -2.50174 q^{76} +4.74218 q^{77} +0.301953 q^{78} +3.14111 q^{79} +8.94518 q^{80} -3.19007 q^{81} -0.287187 q^{82} +6.55920 q^{83} +6.32900 q^{84} -16.3175 q^{85} -1.01217 q^{86} -9.49238 q^{87} +1.73614 q^{88} -0.0690797 q^{89} -0.763881 q^{90} -2.52468 q^{91} +5.94555 q^{92} +8.92844 q^{93} -0.615996 q^{94} +3.13176 q^{95} +3.49175 q^{96} -9.15109 q^{97} -0.146660 q^{98} +2.51508 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 19 q - 9 q^{2} - 2 q^{3} + 17 q^{4} - 18 q^{5} - 8 q^{6} - 8 q^{7} - 24 q^{8} + 7 q^{9}+O(q^{10}) $ 19 * q - 9 * q^2 - 2 * q^3 + 17 * q^4 - 18 * q^5 - 8 * q^6 - 8 * q^7 - 24 * q^8 + 7 * q^9	\( 19 q - 9 q^{2} - 2 q^{3} + 17 q^{4} - 18 q^{5} - 8 q^{6} - 8 q^{7} - 24 q^{8} + 7 q^{9} - q^{11} + 18 q^{12} + 19 q^{13} - 8 q^{14} - 11 q^{15} + 21 q^{16} - 16 q^{17} - 10 q^{19} - 20 q^{20} - 33 q^{21} + 2 q^{22} - 14 q^{23} - 9 q^{24} + 23 q^{25} - 9 q^{26} + q^{27} + 10 q^{28} - 22 q^{29} + 28 q^{30} - 3 q^{31} - 47 q^{32} - 25 q^{33} - 35 q^{34} - 3 q^{35} - 33 q^{36} - 19 q^{37} - 15 q^{38} - 2 q^{39} + 8 q^{40} - 52 q^{41} - 3 q^{42} - 2 q^{43} - 54 q^{44} - 40 q^{45} + 33 q^{46} - 24 q^{47} - 8 q^{48} + 7 q^{49} - 10 q^{50} - 7 q^{51} + 17 q^{52} - 11 q^{53} - 23 q^{54} - 29 q^{55} - 17 q^{56} - 34 q^{57} + 18 q^{58} - 52 q^{59} - 71 q^{60} - 25 q^{61} - 22 q^{62} - 19 q^{63} + 48 q^{64} - 18 q^{65} + 9 q^{66} - 2 q^{67} + 18 q^{68} - 26 q^{69} - 12 q^{70} - 44 q^{71} - 6 q^{72} - 39 q^{73} - 4 q^{74} + 17 q^{75} - 10 q^{76} - 18 q^{77} - 8 q^{78} + q^{79} - 9 q^{80} - 13 q^{81} + 47 q^{82} - 27 q^{83} - 24 q^{84} - 26 q^{85} - q^{86} + 31 q^{87} + 19 q^{88} - 86 q^{89} + 48 q^{90} - 8 q^{91} - 19 q^{92} + 32 q^{93} + 45 q^{94} + 17 q^{95} - 25 q^{96} - 20 q^{97} + 39 q^{98} + 50 q^{99}+O(q^{100}) \) 19 * q - 9 * q^2 - 2 * q^3 + 17 * q^4 - 18 * q^5 - 8 * q^6 - 8 * q^7 - 24 * q^8 + 7 * q^9 - q^11 + 18 * q^12 + 19 * q^13 - 8 * q^14 - 11 * q^15 + 21 * q^16 - 16 * q^17 - 10 * q^19 - 20 * q^20 - 33 * q^21 + 2 * q^22 - 14 * q^23 - 9 * q^24 + 23 * q^25 - 9 * q^26 + q^27 + 10 * q^28 - 22 * q^29 + 28 * q^30 - 3 * q^31 - 47 * q^32 - 25 * q^33 - 35 * q^34 - 3 * q^35 - 33 * q^36 - 19 * q^37 - 15 * q^38 - 2 * q^39 + 8 * q^40 - 52 * q^41 - 3 * q^42 - 2 * q^43 - 54 * q^44 - 40 * q^45 + 33 * q^46 - 24 * q^47 - 8 * q^48 + 7 * q^49 - 10 * q^50 - 7 * q^51 + 17 * q^52 - 11 * q^53 - 23 * q^54 - 29 * q^55 - 17 * q^56 - 34 * q^57 + 18 * q^58 - 52 * q^59 - 71 * q^60 - 25 * q^61 - 22 * q^62 - 19 * q^63 + 48 * q^64 - 18 * q^65 + 9 * q^66 - 2 * q^67 + 18 * q^68 - 26 * q^69 - 12 * q^70 - 44 * q^71 - 6 * q^72 - 39 * q^73 - 4 * q^74 + 17 * q^75 - 10 * q^76 - 18 * q^77 - 8 * q^78 + q^79 - 9 * q^80 - 13 * q^81 + 47 * q^82 - 27 * q^83 - 24 * q^84 - 26 * q^85 - q^86 + 31 * q^87 + 19 * q^88 - 86 * q^89 + 48 * q^90 - 8 * q^91 - 19 * q^92 + 32 * q^93 + 45 * q^94 + 17 * q^95 - 25 * q^96 - 20 * q^97 + 39 * q^98 + 50 * 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 $

$2$	0.234290	0.165668	0.0828342	−	0.996563i	$-0.473603\pi$
$2$	0.234290	0.165668	0.0828342	+	0.996563i	$0.473603\pi$
$3$	1.28880	0.744088	0.372044	−	0.928215i	$-0.378657\pi$
$3$	1.28880	0.744088	0.372044	+	0.928215i	$0.378657\pi$
$4$	−1.94511	−0.972554
$5$	2.43495	1.08894	0.544471	−	0.838779i	$-0.316730\pi$
$5$	2.43495	1.08894	0.544471	+	0.838779i	$0.316730\pi$
$6$	0.301953	0.123272
$7$	−2.52468	−0.954241	−0.477120	−	0.878838i	$-0.658319\pi$
$7$	−2.52468	−0.954241	−0.477120	+	0.878838i	$0.658319\pi$
$8$	−0.924301	−0.326790
$9$	−1.33900	−0.446334
$10$	0.570486	0.180403
$11$	−1.87833	−0.566337	−0.283169	−	0.959070i	$-0.591386\pi$
$11$	−1.87833	−0.566337	−0.283169	+	0.959070i	$0.591386\pi$
$12$	−2.50685	−0.723665
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	−0.591509	−0.158087
$15$	3.13816	0.810269
$16$	3.67366	0.918415
$17$	−6.70137	−1.62532	−0.812661	−	0.582737i	$-0.801982\pi$
$17$	−6.70137	−1.62532	−0.812661	+	0.582737i	$0.801982\pi$
$18$	−0.313715	−0.0739434
$19$	1.28617	0.295067	0.147534	−	0.989057i	$-0.452867\pi$
$19$	1.28617	0.295067	0.147534	+	0.989057i	$0.452867\pi$
$20$	−4.73624	−1.05906
$21$	−3.25380	−0.710038
$22$	−0.440074	−0.0938241
$23$	−3.05667	−0.637359	−0.318679	−	0.947863i	$-0.603239\pi$
$23$	−3.05667	−0.637359	−0.318679	+	0.947863i	$0.603239\pi$
$24$	−1.19124	−0.243160
$25$	0.928983	0.185797
$26$	0.234290	0.0459481
$27$	−5.59209	−1.07620
$28$	4.91078	0.928050
$29$	−7.36530	−1.36770	−0.683851	−	0.729622i	$-0.739696\pi$
$29$	−7.36530	−1.36770	−0.683851	+	0.729622i	$0.739696\pi$
$30$	0.735240	0.134236
$31$	6.92773	1.24426	0.622129	−	0.782915i	$-0.286268\pi$
$31$	6.92773	1.24426	0.622129	+	0.782915i	$0.286268\pi$
$32$	2.70931	0.478942
$33$	−2.42078	−0.421404
$34$	−1.57007	−0.269264
$35$	−6.14748	−1.03911
$36$	2.60450	0.434084
$37$	−1.70801	−0.280795	−0.140397	−	0.990095i	$-0.544838\pi$
$37$	−1.70801	−0.280795	−0.140397	+	0.990095i	$0.544838\pi$
$38$	0.301337	0.0488833
$39$	1.28880	0.206373
$40$	−2.25063	−0.355855
$41$	−1.22577	−0.191434	−0.0957168	−	0.995409i	$-0.530514\pi$
$41$	−1.22577	−0.191434	−0.0957168	+	0.995409i	$0.530514\pi$
$42$	−0.762335	−0.117631
$43$	−4.32013	−0.658814	−0.329407	−	0.944188i	$-0.606849\pi$
$43$	−4.32013	−0.658814	−0.329407	+	0.944188i	$0.606849\pi$
$44$	3.65355	0.550793
$45$	−3.26040	−0.486032
$46$	−0.716148	−0.105590
$47$	−2.62920	−0.383508	−0.191754	−	0.981443i	$-0.561418\pi$
$47$	−2.62920	−0.383508	−0.191754	+	0.981443i	$0.561418\pi$
$48$	4.73460	0.683381
$49$	−0.625975	−0.0894251
$50$	0.217652	0.0307806
$51$	−8.63671	−1.20938
$52$	−1.94511	−0.269738
$53$	4.70761	0.646640	0.323320	−	0.946290i	$-0.395201\pi$
$53$	4.70761	0.646640	0.323320	+	0.946290i	$0.395201\pi$
$54$	−1.31017	−0.178292
$55$	−4.57363	−0.616709
$56$	2.33357	0.311836
$57$	1.65761	0.219556
$58$	−1.72562	−0.226585
$59$	−10.0840	−1.31283	−0.656413	−	0.754402i	$-0.727927\pi$
$59$	−10.0840	−1.31283	−0.656413	+	0.754402i	$0.727927\pi$
$60$	−6.10406	−0.788030
$61$	1.21800	0.155950	0.0779748	−	0.996955i	$-0.475155\pi$
$61$	1.21800	0.155950	0.0779748	+	0.996955i	$0.475155\pi$
$62$	1.62310	0.206134
$63$	3.38055	0.425910
$64$	−6.71256	−0.839070
$65$	2.43495	0.302018
$66$	−0.567167	−0.0698134
$67$	−1.12824	−0.137836	−0.0689182	−	0.997622i	$-0.521955\pi$
$67$	−1.12824	−0.137836	−0.0689182	+	0.997622i	$0.521955\pi$
$68$	13.0349	1.58071
$69$	−3.93942	−0.474251
$70$	−1.44030	−0.172148
$71$	−2.58893	−0.307249	−0.153624	−	0.988129i	$-0.549095\pi$
$71$	−2.58893	−0.307249	−0.153624	+	0.988129i	$0.549095\pi$
$72$	1.23764	0.145857
$73$	9.61558	1.12542	0.562709	−	0.826655i	$-0.309759\pi$
$73$	9.61558	1.12542	0.562709	+	0.826655i	$0.309759\pi$
$74$	−0.400170	−0.0465188
$75$	1.19727	0.138249
$76$	−2.50174	−0.286969
$77$	4.74218	0.540422
$78$	0.301953	0.0341894
$79$	3.14111	0.353402	0.176701	−	0.984265i	$-0.443457\pi$
$79$	3.14111	0.353402	0.176701	+	0.984265i	$0.443457\pi$
$80$	8.94518	1.00010
$81$	−3.19007	−0.354452
$82$	−0.287187	−0.0317145
$83$	6.55920	0.719966	0.359983	−	0.932959i	$-0.382783\pi$
$83$	6.55920	0.719966	0.359983	+	0.932959i	$0.382783\pi$
$84$	6.32900	0.690551
$85$	−16.3175	−1.76988
$86$	−1.01217	−0.109145
$87$	−9.49238	−1.01769
$88$	1.73614	0.185073
$89$	−0.0690797	−0.00732243	−0.00366121	−	0.999993i	$-0.501165\pi$
$89$	−0.0690797	−0.00732243	−0.00366121	+	0.999993i	$0.501165\pi$
$90$	−0.763881	−0.0805201
$91$	−2.52468	−0.264659
$92$	5.94555	0.619866
$93$	8.92844	0.925836
$94$	−0.615996	−0.0635352
$95$	3.13176	0.321311
$96$	3.49175	0.356375
$97$	−9.15109	−0.929153	−0.464576	−	0.885533i	$-0.653793\pi$
$97$	−9.15109	−0.929153	−0.464576	+	0.885533i	$0.653793\pi$
$98$	−0.146660	−0.0148149
$99$	2.51508	0.252775
$100$	−1.80697	−0.180697
$101$	6.82425	0.679038	0.339519	−	0.940599i	$-0.389736\pi$
$101$	6.82425	0.679038	0.339519	+	0.940599i	$0.389736\pi$
$102$	−2.02350	−0.200356
$103$	1.00000	0.0985329
$103$	1.00000	0.0985329
$104$	−0.924301	−0.0906352
$105$	−7.92285	−0.773191
$106$	1.10295	0.107128
$107$	8.83892	0.854491	0.427245	−	0.904136i	$-0.359484\pi$
$107$	8.83892	0.854491	0.427245	+	0.904136i	$0.359484\pi$
$108$	10.8772	1.04666
$109$	5.88841	0.564007	0.282004	−	0.959413i	$-0.409001\pi$
$109$	5.88841	0.564007	0.282004	+	0.959413i	$0.409001\pi$
$110$	−1.07156	−0.102169
$111$	−2.20127	−0.208936
$112$	−9.27483	−0.876389
$113$	−5.82383	−0.547860	−0.273930	−	0.961750i	$-0.588324\pi$
$113$	−5.82383	−0.547860	−0.273930	+	0.961750i	$0.588324\pi$
$114$	0.388362	0.0363735
$115$	−7.44283	−0.694047
$116$	14.3263	1.33016
$117$	−1.33900	−0.123791
$118$	−2.36259	−0.217494
$119$	16.9188	1.55095
$120$	−2.90060	−0.264788
$121$	−7.47189	−0.679262
$122$	0.285367	0.0258359
$123$	−1.57977	−0.142443
$124$	−13.4752	−1.21011
$125$	−9.91272	−0.886621
$126$	0.792032	0.0705598
$127$	6.05856	0.537610	0.268805	−	0.963195i	$-0.413371\pi$
$127$	6.05856	0.537610	0.268805	+	0.963195i	$0.413371\pi$
$128$	−6.99130	−0.617949
$129$	−5.56778	−0.490215
$130$	0.570486	0.0500349
$131$	−21.4600	−1.87497	−0.937483	−	0.348030i	$-0.886851\pi$
$131$	−21.4600	−1.87497	−0.937483	+	0.348030i	$0.886851\pi$
$132$	4.70869	0.409838
$133$	−3.24717	−0.281565
$134$	−0.264336	−0.0228351
$135$	−13.6165	−1.17192
$136$	6.19409	0.531138
$137$	15.4958	1.32390	0.661948	−	0.749550i	$-0.269730\pi$
$137$	15.4958	1.32390	0.661948	+	0.749550i	$0.269730\pi$
$138$	−0.922969	−0.0785684
$139$	−2.58658	−0.219390	−0.109695	−	0.993965i	$-0.534987\pi$
$139$	−2.58658	−0.219390	−0.109695	+	0.993965i	$0.534987\pi$
$140$	11.9575	1.01059
$141$	−3.38850	−0.285364
$142$	−0.606560	−0.0509014
$143$	−1.87833	−0.157074
$144$	−4.91904	−0.409920
$145$	−17.9341	−1.48935
$146$	2.25284	0.186446
$147$	−0.806756	−0.0665401
$148$	3.32226	0.273088
$149$	−15.5432	−1.27335	−0.636675	−	0.771132i	$-0.719691\pi$
$149$	−15.5432	−1.27335	−0.636675	+	0.771132i	$0.719691\pi$
$150$	0.280509	0.0229035
$151$	3.71104	0.302000	0.151000	−	0.988534i	$-0.451751\pi$
$151$	3.71104	0.302000	0.151000	+	0.988534i	$0.451751\pi$
$152$	−1.18881	−0.0964250
$153$	8.97315	0.725436
$154$	1.11105	0.0895308
$155$	16.8687	1.35493
$156$	−2.50685	−0.200709
$157$	−8.60472	−0.686731	−0.343366	−	0.939202i	$-0.611567\pi$
$157$	−8.60472	−0.686731	−0.343366	+	0.939202i	$0.611567\pi$
$158$	0.735932	0.0585476
$159$	6.06716	0.481157
$160$	6.59703	0.521541
$161$	7.71711	0.608194
$162$	−0.747403	−0.0587215
$163$	−6.47195	−0.506923	−0.253461	−	0.967346i	$-0.581569\pi$
$163$	−6.47195	−0.506923	−0.253461	+	0.967346i	$0.581569\pi$
$164$	2.38426	0.186179
$165$	−5.89449	−0.458885
$166$	1.53676	0.119276
$167$	12.6403	0.978133	0.489066	−	0.872247i	$-0.337338\pi$
$167$	12.6403	0.978133	0.489066	+	0.872247i	$0.337338\pi$
$168$	3.00750	0.232033
$169$	1.00000	0.0769231
$170$	−3.82304	−0.293214
$171$	−1.72218	−0.131699
$172$	8.40313	0.640732
$173$	19.1896	1.45896	0.729479	−	0.684004i	$-0.239763\pi$
$173$	19.1896	1.45896	0.729479	+	0.684004i	$0.239763\pi$
$174$	−2.22397	−0.168599
$175$	−2.34539	−0.177295
$176$	−6.90034	−0.520133
$177$	−12.9962	−0.976858
$178$	−0.0161847	−0.00121309
$179$	−5.63523	−0.421197	−0.210599	−	0.977573i	$-0.567541\pi$
$179$	−5.63523	−0.421197	−0.210599	+	0.977573i	$0.567541\pi$
$180$	6.34183	0.472692
$181$	7.15451	0.531790	0.265895	−	0.964002i	$-0.414333\pi$
$181$	7.15451	0.531790	0.265895	+	0.964002i	$0.414333\pi$
$182$	−0.591509	−0.0438456
$183$	1.56976	0.116040
$184$	2.82528	0.208282
$185$	−4.15891	−0.305769
$186$	2.09185	0.153382
$187$	12.5874	0.920480
$188$	5.11408	0.372982
$189$	14.1183	1.02695
$190$	0.733741	0.0532311
$191$	18.7834	1.35912	0.679560	−	0.733620i	$-0.262171\pi$
$191$	18.7834	1.35912	0.679560	+	0.733620i	$0.262171\pi$
$192$	−8.65113	−0.624341
$193$	−9.33584	−0.672009	−0.336004	−	0.941860i	$-0.609076\pi$
$193$	−9.33584	−0.672009	−0.336004	+	0.941860i	$0.609076\pi$
$194$	−2.14401	−0.153931
$195$	3.13816	0.224728
$196$	1.21759	0.0869707
$197$	−19.7553	−1.40751	−0.703754	−	0.710444i	$-0.748494\pi$
$197$	−19.7553	−1.40751	−0.703754	+	0.710444i	$0.748494\pi$
$198$	0.589260	0.0418769
$199$	−9.18752	−0.651286	−0.325643	−	0.945493i	$-0.605581\pi$
$199$	−9.18752	−0.651286	−0.325643	+	0.945493i	$0.605581\pi$
$200$	−0.858660	−0.0607164
$201$	−1.45407	−0.102562
$202$	1.59886	0.112495
$203$	18.5950	1.30512
$204$	16.7993	1.17619
$205$	−2.98470	−0.208460
$206$	0.234290	0.0163238
$207$	4.09288	0.284475
$208$	3.67366	0.254723
$209$	−2.41585	−0.167108
$210$	−1.85625	−0.128093
$211$	25.4937	1.75506	0.877528	−	0.479525i	$-0.159191\pi$
$211$	25.4937	1.75506	0.877528	+	0.479525i	$0.159191\pi$
$212$	−9.15681	−0.628893
$213$	−3.33660	−0.228620
$214$	2.07087	0.141562
$215$	−10.5193	−0.717411
$216$	5.16878	0.351691
$217$	−17.4903	−1.18732
$218$	1.37960	0.0934381
$219$	12.3925	0.837410
$220$	8.89621	0.599783
$221$	−6.70137	−0.450783
$222$	−0.515738	−0.0346140
$223$	20.5625	1.37697	0.688483	−	0.725252i	$-0.258277\pi$
$223$	20.5625	1.37697	0.688483	+	0.725252i	$0.258277\pi$
$224$	−6.84014	−0.457026
$225$	−1.24391	−0.0829273
$226$	−1.36447	−0.0907630
$227$	15.0251	0.997252	0.498626	−	0.866817i	$-0.333838\pi$
$227$	15.0251	0.997252	0.498626	+	0.866817i	$0.333838\pi$
$228$	−3.22423	−0.213530
$229$	−17.7033	−1.16986	−0.584932	−	0.811082i	$-0.698879\pi$
$229$	−17.7033	−1.16986	−0.584932	+	0.811082i	$0.698879\pi$
$230$	−1.74378	−0.114982
$231$	6.11171	0.402121
$232$	6.80775	0.446951
$233$	−18.3072	−1.19935	−0.599673	−	0.800245i	$-0.704703\pi$
$233$	−18.3072	−1.19935	−0.599673	+	0.800245i	$0.704703\pi$
$234$	−0.313715	−0.0205082
$235$	−6.40197	−0.417618
$236$	19.6145	1.27679
$237$	4.04825	0.262962
$238$	3.96392	0.256943
$239$	2.80361	0.181350	0.0906752	−	0.995881i	$-0.471097\pi$
$239$	2.80361	0.181350	0.0906752	+	0.995881i	$0.471097\pi$
$240$	11.5285	0.744163
$241$	−5.07650	−0.327006	−0.163503	−	0.986543i	$-0.552279\pi$
$241$	−5.07650	−0.327006	−0.163503	+	0.986543i	$0.552279\pi$
$242$	−1.75059	−0.112532
$243$	12.6649	0.812455
$244$	−2.36915	−0.151669
$245$	−1.52422	−0.0973788
$246$	−0.370126	−0.0235984
$247$	1.28617	0.0818369
$248$	−6.40331	−0.406611
$249$	8.45348	0.535718
$250$	−2.32246	−0.146885
$251$	−7.71567	−0.487009	−0.243505	−	0.969900i	$-0.578297\pi$
$251$	−7.71567	−0.487009	−0.243505	+	0.969900i	$0.578297\pi$
$252$	−6.57554	−0.414220
$253$	5.74142	0.360960
$254$	1.41946	0.0890650
$255$	−21.0300	−1.31695
$256$	11.7871	0.736695
$257$	−5.29788	−0.330473	−0.165236	−	0.986254i	$-0.552839\pi$
$257$	−5.29788	−0.330473	−0.165236	+	0.986254i	$0.552839\pi$
$258$	−1.30448	−0.0812132
$259$	4.31218	0.267946
$260$	−4.73624	−0.293729
$261$	9.86215	0.610452
$262$	−5.02787	−0.310623
$263$	−7.23113	−0.445891	−0.222945	−	0.974831i	$-0.571567\pi$
$263$	−7.23113	−0.445891	−0.222945	+	0.974831i	$0.571567\pi$
$264$	2.23753	0.137711
$265$	11.4628	0.704154
$266$	−0.760780	−0.0466464
$267$	−0.0890297	−0.00544853
$268$	2.19455	0.134053
$269$	−2.68540	−0.163732	−0.0818658	−	0.996643i	$-0.526088\pi$
$269$	−2.68540	−0.163732	−0.0818658	+	0.996643i	$0.526088\pi$
$270$	−3.19021	−0.194150
$271$	20.5357	1.24746	0.623728	−	0.781641i	$-0.285617\pi$
$271$	20.5357	1.24746	0.623728	+	0.781641i	$0.285617\pi$
$272$	−24.6186	−1.49272
$273$	−3.25380	−0.196929
$274$	3.63052	0.219328
$275$	−1.74493	−0.105223
$276$	7.66260	0.461234
$277$	15.0238	0.902690	0.451345	−	0.892350i	$-0.350944\pi$
$277$	15.0238	0.902690	0.451345	+	0.892350i	$0.350944\pi$
$278$	−0.606010	−0.0363461
$279$	−9.27624	−0.555354
$280$	5.68212	0.339572
$281$	−2.60264	−0.155261	−0.0776303	−	0.996982i	$-0.524735\pi$
$281$	−2.60264	−0.155261	−0.0776303	+	0.996982i	$0.524735\pi$
$282$	−0.793894	−0.0472757
$283$	0.589769	0.0350581	0.0175291	−	0.999846i	$-0.494420\pi$
$283$	0.589769	0.0350581	0.0175291	+	0.999846i	$0.494420\pi$
$284$	5.03574	0.298816
$285$	4.03620	0.239084
$286$	−0.440074	−0.0260221
$287$	3.09469	0.182674
$288$	−3.62776	−0.213768
$289$	27.9084	1.64167
$290$	−4.20180	−0.246738
$291$	−11.7939	−0.691371
$292$	−18.7033	−1.09453
$293$	17.0533	0.996265	0.498132	−	0.867101i	$-0.334019\pi$
$293$	17.0533	0.996265	0.498132	+	0.867101i	$0.334019\pi$
$294$	−0.189015	−0.0110236
$295$	−24.5541	−1.42959
$296$	1.57871	0.0917608
$297$	10.5038	0.609491
$298$	−3.64163	−0.210954
$299$	−3.05667	−0.176772
$300$	−2.32882	−0.134455
$301$	10.9070	0.628667
$302$	0.869462	0.0500319
$303$	8.79508	0.505264
$304$	4.72495	0.270994
$305$	2.96578	0.169820
$306$	2.10232	0.120182
$307$	7.75143	0.442398	0.221199	−	0.975229i	$-0.429003\pi$
$307$	7.75143	0.442398	0.221199	+	0.975229i	$0.429003\pi$
$308$	−9.22406	−0.525589
$309$	1.28880	0.0733171
$310$	3.95217	0.224468
$311$	12.6014	0.714561	0.357280	−	0.933997i	$-0.383704\pi$
$311$	12.6014	0.714561	0.357280	+	0.933997i	$0.383704\pi$
$312$	−1.19124	−0.0674405
$313$	−11.1206	−0.628574	−0.314287	−	0.949328i	$-0.601765\pi$
$313$	−11.1206	−0.628574	−0.314287	+	0.949328i	$0.601765\pi$
$314$	−2.01600	−0.113770
$315$	8.23148	0.463791
$316$	−6.10979	−0.343703
$317$	−25.9077	−1.45512	−0.727560	−	0.686044i	$-0.759346\pi$
$317$	−25.9077	−1.45512	−0.727560	+	0.686044i	$0.759346\pi$
$318$	1.42148	0.0797125
$319$	13.8344	0.774580
$320$	−16.3447	−0.913699
$321$	11.3916	0.635816
$322$	1.80805	0.100758
$323$	−8.61909	−0.479579
$324$	6.20503	0.344724
$325$	0.928983	0.0515307
$326$	−1.51632	−0.0839810
$327$	7.58896	0.419671
$328$	1.13298	0.0625585
$329$	6.63789	0.365959
$330$	−1.38102	−0.0760228
$331$	−8.35549	−0.459259	−0.229630	−	0.973278i	$-0.573751\pi$
$331$	−8.35549	−0.459259	−0.229630	+	0.973278i	$0.573751\pi$
$332$	−12.7584	−0.700206
$333$	2.28702	0.125328
$334$	2.96149	0.162046
$335$	−2.74721	−0.150096
$336$	−11.9534	−0.652110
$337$	−10.1676	−0.553863	−0.276931	−	0.960890i	$-0.589318\pi$
$337$	−10.1676	−0.553863	−0.276931	+	0.960890i	$0.589318\pi$
$338$	0.234290	0.0127437
$339$	−7.50574	−0.407656
$340$	31.7393	1.72131
$341$	−13.0126	−0.704669
$342$	−0.403491	−0.0218183
$343$	19.2532	1.03957
$344$	3.99310	0.215294
$345$	−9.59230	−0.516432
$346$	4.49594	0.241703
$347$	27.8231	1.49362	0.746812	−	0.665036i	$-0.231584\pi$
$347$	27.8231	1.49362	0.746812	+	0.665036i	$0.231584\pi$
$348$	18.4637	0.989758
$349$	31.5937	1.69117	0.845585	−	0.533840i	$-0.179252\pi$
$349$	31.5937	1.69117	0.845585	+	0.533840i	$0.179252\pi$
$350$	−0.549502	−0.0293721
$351$	−5.59209	−0.298484
$352$	−5.08896	−0.271243
$353$	−6.04336	−0.321656	−0.160828	−	0.986982i	$-0.551416\pi$
$353$	−6.04336	−0.321656	−0.160828	+	0.986982i	$0.551416\pi$
$354$	−3.04490	−0.161834
$355$	−6.30390	−0.334576
$356$	0.134367	0.00712146
$357$	21.8050	1.15404
$358$	−1.32028	−0.0697791
$359$	3.27049	0.172610	0.0863049	−	0.996269i	$-0.472494\pi$
$359$	3.27049	0.172610	0.0863049	+	0.996269i	$0.472494\pi$
$360$	3.01359	0.158830
$361$	−17.3458	−0.912935
$362$	1.67623	0.0881008
$363$	−9.62975	−0.505431
$364$	4.91078	0.257395
$365$	23.4135	1.22552
$366$	0.367780	0.0192242
$367$	−28.1375	−1.46877	−0.734383	−	0.678736i	$-0.762528\pi$
$367$	−28.1375	−1.46877	−0.734383	+	0.678736i	$0.762528\pi$
$368$	−11.2292	−0.585360
$369$	1.64131	0.0854433
$370$	−0.974393	−0.0506563
$371$	−11.8852	−0.617050
$372$	−17.3668	−0.900426
$373$	−22.0087	−1.13957	−0.569785	−	0.821794i	$-0.692973\pi$
$373$	−22.0087	−1.13957	−0.569785	+	0.821794i	$0.692973\pi$
$374$	2.94910	0.152494
$375$	−12.7755	−0.659724
$376$	2.43017	0.125327
$377$	−7.36530	−0.379332
$378$	3.30777	0.170134
$379$	15.7665	0.809872	0.404936	−	0.914345i	$-0.367294\pi$
$379$	15.7665	0.809872	0.404936	+	0.914345i	$0.367294\pi$
$380$	−6.09160	−0.312493
$381$	7.80825	0.400029
$382$	4.40078	0.225163
$383$	−16.2334	−0.829486	−0.414743	−	0.909938i	$-0.636129\pi$
$383$	−16.2334	−0.829486	−0.414743	+	0.909938i	$0.636129\pi$
$384$	−9.01037	−0.459808
$385$	11.5470	0.588488
$386$	−2.18730	−0.111331
$387$	5.78466	0.294051
$388$	17.7999	0.903651
$389$	27.5466	1.39667	0.698335	−	0.715771i	$-0.253925\pi$
$389$	27.5466	1.39667	0.698335	+	0.715771i	$0.253925\pi$
$390$	0.735240	0.0372303
$391$	20.4839	1.03591
$392$	0.578590	0.0292232
$393$	−27.6576	−1.39514
$394$	−4.62848	−0.233180
$395$	7.64844	0.384835
$396$	−4.89211	−0.245838
$397$	−8.53361	−0.428290	−0.214145	−	0.976802i	$-0.568696\pi$
$397$	−8.53361	−0.428290	−0.214145	+	0.976802i	$0.568696\pi$
$398$	−2.15255	−0.107897
$399$	−4.18494	−0.209509
$400$	3.41277	0.170638
$401$	−38.9778	−1.94646	−0.973229	−	0.229836i	$-0.926181\pi$
$401$	−38.9778	−1.94646	−0.973229	+	0.229836i	$0.926181\pi$
$402$	−0.340675	−0.0169913
$403$	6.92773	0.345095
$404$	−13.2739	−0.660401
$405$	−7.76767	−0.385978
$406$	4.35664	0.216217
$407$	3.20820	0.159024
$408$	7.98292	0.395214
$409$	5.42043	0.268023	0.134011	−	0.990980i	$-0.457214\pi$
$409$	5.42043	0.268023	0.134011	+	0.990980i	$0.457214\pi$
$410$	−0.699286	−0.0345353
$411$	19.9710	0.985095
$412$	−1.94511	−0.0958286
$413$	25.4589	1.25275
$414$	0.958923	0.0471285
$415$	15.9713	0.784002
$416$	2.70931	0.132835
$417$	−3.33357	−0.163246
$418$	−0.566010	−0.0276844
$419$	4.36803	0.213392	0.106696	−	0.994292i	$-0.465973\pi$
$419$	4.36803	0.213392	0.106696	+	0.994292i	$0.465973\pi$
$420$	15.4108	0.751970
$421$	−24.4210	−1.19021	−0.595103	−	0.803649i	$-0.702889\pi$
$421$	−24.4210	−1.19021	−0.595103	+	0.803649i	$0.702889\pi$
$422$	5.97292	0.290757
$423$	3.52050	0.171173
$424$	−4.35125	−0.211315
$425$	−6.22546	−0.301979
$426$	−0.781734	−0.0378751
$427$	−3.07508	−0.148813
$428$	−17.1927	−0.831038
$429$	−2.42078	−0.116877
$430$	−2.46457	−0.118852
$431$	−18.1349	−0.873527	−0.436764	−	0.899576i	$-0.643875\pi$
$431$	−18.1349	−0.873527	−0.436764	+	0.899576i	$0.643875\pi$
$432$	−20.5435	−0.988397
$433$	34.2650	1.64667	0.823335	−	0.567556i	$-0.192111\pi$
$433$	34.2650	1.64667	0.823335	+	0.567556i	$0.192111\pi$
$434$	−4.09782	−0.196702
$435$	−23.1135	−1.10821
$436$	−11.4536	−0.548527
$437$	−3.93139	−0.188064
$438$	2.90345	0.138732
$439$	−17.7704	−0.848135	−0.424067	−	0.905631i	$-0.639398\pi$
$439$	−17.7704	−0.848135	−0.424067	+	0.905631i	$0.639398\pi$
$440$	4.22742	0.201534
$441$	0.838182	0.0399134
$442$	−1.57007	−0.0746805
$443$	−15.2608	−0.725063	−0.362532	−	0.931971i	$-0.618088\pi$
$443$	−15.2608	−0.725063	−0.362532	+	0.931971i	$0.618088\pi$
$444$	4.28172	0.203201
$445$	−0.168206	−0.00797371
$446$	4.81760	0.228120
$447$	−20.0321	−0.947484
$448$	16.9471	0.800674
$449$	−20.5694	−0.970728	−0.485364	−	0.874312i	$-0.661313\pi$
$449$	−20.5694	−0.970728	−0.485364	+	0.874312i	$0.661313\pi$
$450$	−0.291436	−0.0137384
$451$	2.30240	0.108416
$452$	11.3280	0.532823
$453$	4.78278	0.224715
$454$	3.52024	0.165213
$455$	−6.14748	−0.288198
$456$	−1.53213	−0.0717486
$457$	7.23982	0.338665	0.169332	−	0.985559i	$-0.445839\pi$
$457$	7.23982	0.338665	0.169332	+	0.985559i	$0.445839\pi$
$458$	−4.14770	−0.193809
$459$	37.4747	1.74917
$460$	14.4771	0.674999
$461$	−13.4678	−0.627256	−0.313628	−	0.949546i	$-0.601545\pi$
$461$	−13.4678	−0.627256	−0.313628	+	0.949546i	$0.601545\pi$
$462$	1.43192	0.0666187
$463$	−19.7485	−0.917790	−0.458895	−	0.888491i	$-0.651755\pi$
$463$	−19.7485	−0.917790	−0.458895	+	0.888491i	$0.651755\pi$
$464$	−27.0576	−1.25612
$465$	21.7403	1.00818
$466$	−4.28921	−0.198694
$467$	27.8027	1.28656	0.643278	−	0.765633i	$-0.277574\pi$
$467$	27.8027	1.28656	0.643278	+	0.765633i	$0.277574\pi$
$468$	2.60450	0.120393
$469$	2.84845	0.131529
$470$	−1.49992	−0.0691861
$471$	−11.0897	−0.510988
$472$	9.32066	0.429018
$473$	8.11463	0.373111
$474$	0.948467	0.0435645
$475$	1.19483	0.0548225
$476$	−32.9090	−1.50838
$477$	−6.30350	−0.288617
$478$	0.656859	0.0300440
$479$	−10.2948	−0.470380	−0.235190	−	0.971949i	$-0.575571\pi$
$479$	−10.2948	−0.470380	−0.235190	+	0.971949i	$0.575571\pi$
$480$	8.50223	0.388072
$481$	−1.70801	−0.0778784
$482$	−1.18938	−0.0541746
$483$	9.94579	0.452549
$484$	14.5336	0.660619
$485$	−22.2825	−1.01179
$486$	2.96727	0.134598
$487$	11.0858	0.502344	0.251172	−	0.967942i	$-0.419184\pi$
$487$	11.0858	0.502344	0.251172	+	0.967942i	$0.419184\pi$
$488$	−1.12580	−0.0509627
$489$	−8.34104	−0.377195
$490$	−0.357110	−0.0161326
$491$	−20.3801	−0.919740	−0.459870	−	0.887986i	$-0.652104\pi$
$491$	−20.3801	−0.919740	−0.459870	+	0.887986i	$0.652104\pi$
$492$	3.07283	0.138534
$493$	49.3576	2.22296
$494$	0.301337	0.0135578
$495$	6.12410	0.275258
$496$	25.4501	1.14275
$497$	6.53622	0.293189
$498$	1.98057	0.0887515
$499$	−14.8811	−0.666167	−0.333084	−	0.942897i	$-0.608089\pi$
$499$	−14.8811	−0.666167	−0.333084	+	0.942897i	$0.608089\pi$
$500$	19.2813	0.862287
$501$	16.2907	0.727816
$502$	−1.80771	−0.0806820
$503$	11.8133	0.526729	0.263364	−	0.964696i	$-0.415168\pi$
$503$	11.8133	0.526729	0.263364	+	0.964696i	$0.415168\pi$
$504$	−3.12465	−0.139183
$505$	16.6167	0.739434
$506$	1.34516	0.0597996
$507$	1.28880	0.0572375
$508$	−11.7846	−0.522855
$509$	−3.66506	−0.162451	−0.0812255	−	0.996696i	$-0.525883\pi$
$509$	−3.66506	−0.162451	−0.0812255	+	0.996696i	$0.525883\pi$
$510$	−4.92712	−0.218177
$511$	−24.2763	−1.07392
$512$	16.7442	0.739997
$513$	−7.19237	−0.317551
$514$	−1.24124	−0.0547488
$515$	2.43495	0.107297
$516$	10.8299	0.476761
$517$	4.93850	0.217195
$518$	1.01030	0.0443901
$519$	24.7315	1.08559
$520$	−2.25063	−0.0986965
$521$	9.07289	0.397491	0.198745	−	0.980051i	$-0.436313\pi$
$521$	9.07289	0.397491	0.198745	+	0.980051i	$0.436313\pi$
$522$	2.31061	0.101133
$523$	10.2960	0.450212	0.225106	−	0.974334i	$-0.427727\pi$
$523$	10.2960	0.450212	0.225106	+	0.974334i	$0.427727\pi$
$524$	41.7420	1.82351
$525$	−3.02273	−0.131923
$526$	−1.69418	−0.0738700
$527$	−46.4253	−2.02232
$528$	−8.89314	−0.387024
$529$	−13.6568	−0.593774
$530$	2.68562	0.116656
$531$	13.5025	0.585959
$532$	6.31609	0.273837
$533$	−1.22577	−0.0530941
$534$	−0.0208588	−0.000902649 0
$535$	21.5223	0.930492
$536$	1.04283	0.0450436
$537$	−7.26268	−0.313408
$538$	−0.629163	−0.0271252
$539$	1.17579	0.0506447
$540$	26.4855	1.13975
$541$	38.0719	1.63684	0.818419	−	0.574622i	$-0.194851\pi$
$541$	38.0719	1.63684	0.818419	+	0.574622i	$0.194851\pi$
$542$	4.81132	0.206664
$543$	9.22071	0.395699
$544$	−18.1561	−0.778435
$545$	14.3380	0.614171
$546$	−0.762335	−0.0326249
$547$	12.9069	0.551858	0.275929	−	0.961178i	$-0.411015\pi$
$547$	12.9069	0.551858	0.275929	+	0.961178i	$0.411015\pi$
$548$	−30.1410	−1.28756
$549$	−1.63091	−0.0696055
$550$	−0.408821	−0.0174322
$551$	−9.47302	−0.403564
$552$	3.64121	0.154980
$553$	−7.93030	−0.337231
$554$	3.51992	0.149547
$555$	−5.35999	−0.227519
$556$	5.03117	0.213369
$557$	−39.1808	−1.66014	−0.830071	−	0.557657i	$-0.811700\pi$
$557$	−39.1808	−1.66014	−0.830071	+	0.557657i	$0.811700\pi$
$558$	−2.17334	−0.0920046
$559$	−4.32013	−0.182722
$560$	−22.5837	−0.954338
$561$	16.2226	0.684918
$562$	−0.609774	−0.0257218
$563$	34.3378	1.44716	0.723582	−	0.690238i	$-0.242494\pi$
$563$	34.3378	1.44716	0.723582	+	0.690238i	$0.242494\pi$
$564$	6.59101	0.277531
$565$	−14.1807	−0.596588
$566$	0.138177	0.00580802
$567$	8.05392	0.338233
$568$	2.39295	0.100406
$569$	−22.8092	−0.956213	−0.478107	−	0.878302i	$-0.658677\pi$
$569$	−22.8092	−0.956213	−0.478107	+	0.878302i	$0.658677\pi$
$570$	0.945643	0.0396086
$571$	−5.22026	−0.218461	−0.109231	−	0.994016i	$-0.534839\pi$
$571$	−5.22026	−0.218461	−0.109231	+	0.994016i	$0.534839\pi$
$572$	3.65355	0.152763
$573$	24.2080	1.01130
$574$	0.725056	0.0302632
$575$	−2.83959	−0.118419
$576$	8.98812	0.374505
$577$	−41.9993	−1.74845	−0.874226	−	0.485519i	$-0.838631\pi$
$577$	−41.9993	−1.74845	−0.874226	+	0.485519i	$0.838631\pi$
$578$	6.53867	0.271973
$579$	−12.0320	−0.500033
$580$	34.8838	1.44847
$581$	−16.5599	−0.687021
$582$	−2.76320	−0.114538
$583$	−8.84244	−0.366216
$584$	−8.88769	−0.367775
$585$	−3.26040	−0.134801
$586$	3.99543	0.165050
$587$	18.3688	0.758161	0.379080	−	0.925364i	$-0.376240\pi$
$587$	18.3688	0.758161	0.379080	+	0.925364i	$0.376240\pi$
$588$	1.56923	0.0647138
$589$	8.91023	0.367140
$590$	−5.75278	−0.236838
$591$	−25.4606	−1.04731
$592$	−6.27464	−0.257886
$593$	−8.15537	−0.334901	−0.167450	−	0.985880i	$-0.553553\pi$
$593$	−8.15537	−0.334901	−0.167450	+	0.985880i	$0.553553\pi$
$594$	2.46094	0.100973
$595$	41.1965	1.68889
$596$	30.2332	1.23840
$597$	−11.8409	−0.484614
$598$	−0.716148	−0.0292855
$599$	−30.0784	−1.22897	−0.614485	−	0.788929i	$-0.710636\pi$
$599$	−30.0784	−1.22897	−0.614485	+	0.788929i	$0.710636\pi$
$600$	−1.10664	−0.0451783
$601$	−14.6169	−0.596236	−0.298118	−	0.954529i	$-0.596359\pi$
$601$	−14.6169	−0.596236	−0.298118	+	0.954529i	$0.596359\pi$
$602$	2.55540	0.104150
$603$	1.51072	0.0615211
$604$	−7.21838	−0.293712
$605$	−18.1937	−0.739678
$606$	2.06060	0.0837062
$607$	10.0931	0.409664	0.204832	−	0.978797i	$-0.434335\pi$
$607$	10.0931	0.409664	0.204832	+	0.978797i	$0.434335\pi$
$608$	3.48462	0.141320
$609$	23.9652	0.971121
$610$	0.694854	0.0281338
$611$	−2.62920	−0.106366
$612$	−17.4537	−0.705526
$613$	4.13756	0.167115	0.0835573	−	0.996503i	$-0.473372\pi$
$613$	4.13756	0.167115	0.0835573	+	0.996503i	$0.473372\pi$
$614$	1.81609	0.0732913
$615$	−3.84667	−0.155113
$616$	−4.38320	−0.176604
$617$	−35.3055	−1.42135	−0.710673	−	0.703523i	$-0.751609\pi$
$617$	−35.3055	−1.42135	−0.710673	+	0.703523i	$0.751609\pi$
$618$	0.301953	0.0121463
$619$	35.0343	1.40815	0.704073	−	0.710128i	$-0.251363\pi$
$619$	35.0343	1.40815	0.704073	+	0.710128i	$0.251363\pi$
$620$	−32.8114	−1.31774
$621$	17.0932	0.685925
$622$	2.95239	0.118380
$623$	0.174404	0.00698736
$624$	4.73460	0.189536
$625$	−28.7819	−1.15128
$626$	−2.60545	−0.104135
$627$	−3.11354	−0.124343
$628$	16.7371	0.667883
$629$	11.4460	0.456381
$630$	1.92856	0.0768356
$631$	−28.8688	−1.14925	−0.574624	−	0.818418i	$-0.694852\pi$
$631$	−28.8688	−1.14925	−0.574624	+	0.818418i	$0.694852\pi$
$632$	−2.90333	−0.115488
$633$	32.8562	1.30592
$634$	−6.06992	−0.241067
$635$	14.7523	0.585427
$636$	−11.8013	−0.467951
$637$	−0.625975	−0.0248021
$638$	3.24128	0.128323
$639$	3.46657	0.137136
$640$	−17.0235	−0.672912
$641$	−12.1008	−0.477952	−0.238976	−	0.971025i	$-0.576812\pi$
$641$	−12.1008	−0.477952	−0.238976	+	0.971025i	$0.576812\pi$
$642$	2.66894	0.105335
$643$	3.30183	0.130211	0.0651057	−	0.997878i	$-0.479262\pi$
$643$	3.30183	0.130211	0.0651057	+	0.997878i	$0.479262\pi$
$644$	−15.0106	−0.591501
$645$	−13.5573	−0.533816
$646$	−2.01937	−0.0794511
$647$	−31.6678	−1.24499	−0.622496	−	0.782623i	$-0.713881\pi$
$647$	−31.6678	−1.24499	−0.622496	+	0.782623i	$0.713881\pi$
$648$	2.94859	0.115831
$649$	18.9411	0.743502
$650$	0.217652	0.00853701
$651$	−22.5415	−0.883471
$652$	12.5887	0.493010
$653$	−18.7174	−0.732468	−0.366234	−	0.930523i	$-0.619353\pi$
$653$	−18.7174	−0.732468	−0.366234	+	0.930523i	$0.619353\pi$
$654$	1.77802	0.0695261
$655$	−52.2540	−2.04173
$656$	−4.50307	−0.175815
$657$	−12.8753	−0.502312
$658$	1.55520	0.0606278
$659$	−42.2788	−1.64695	−0.823474	−	0.567353i	$-0.807967\pi$
$659$	−42.2788	−1.64695	−0.823474	+	0.567353i	$0.807967\pi$
$660$	11.4654	0.446291
$661$	−45.3197	−1.76273	−0.881366	−	0.472433i	$-0.843376\pi$
$661$	−45.3197	−1.76273	−0.881366	+	0.472433i	$0.843376\pi$
$662$	−1.95761	−0.0760847
$663$	−8.63671	−0.335422
$664$	−6.06268	−0.235277
$665$	−7.90669	−0.306608
$666$	0.535828	0.0207629
$667$	22.5133	0.871717
$668$	−24.5867	−0.951287
$669$	26.5009	1.02458
$670$	−0.643645	−0.0248662
$671$	−2.28781	−0.0883200
$672$	−8.81555	−0.340067
$673$	−33.7161	−1.29966	−0.649831	−	0.760079i	$-0.725160\pi$
$673$	−33.7161	−1.29966	−0.649831	+	0.760079i	$0.725160\pi$
$674$	−2.38217	−0.0917576
$675$	−5.19496	−0.199954
$676$	−1.94511	−0.0748118
$677$	−14.1408	−0.543475	−0.271737	−	0.962371i	$-0.587598\pi$
$677$	−14.1408	−0.543475	−0.271737	+	0.962371i	$0.587598\pi$
$678$	−1.75852	−0.0675356
$679$	23.1036	0.886635
$680$	15.0823	0.578379
$681$	19.3643	0.742043
$682$	−3.04872	−0.116741
$683$	−10.7295	−0.410551	−0.205276	−	0.978704i	$-0.565809\pi$
$683$	−10.7295	−0.410551	−0.205276	+	0.978704i	$0.565809\pi$
$684$	3.34983	0.128084
$685$	37.7315	1.44165
$686$	4.51083	0.172224
$687$	−22.8159	−0.870481
$688$	−15.8707	−0.605065
$689$	4.70761	0.179346
$690$	−2.24738	−0.0855564
$691$	−17.5371	−0.667144	−0.333572	−	0.942725i	$-0.608254\pi$
$691$	−17.5371	−0.667144	−0.333572	+	0.942725i	$0.608254\pi$
$692$	−37.3258	−1.41891
$693$	−6.34979	−0.241208
$694$	6.51869	0.247446
$695$	−6.29818	−0.238904
$696$	8.77382	0.332571
$697$	8.21436	0.311141
$698$	7.40210	0.280174
$699$	−23.5943	−0.892419
$700$	4.56203	0.172429
$701$	23.4489	0.885652	0.442826	−	0.896607i	$-0.353976\pi$
$701$	23.4489	0.885652	0.442826	+	0.896607i	$0.353976\pi$
$702$	−1.31017	−0.0494493
$703$	−2.19678	−0.0828533
$704$	12.6084	0.475196
$705$	−8.25084	−0.310745
$706$	−1.41590	−0.0532881
$707$	−17.2291	−0.647966
$708$	25.2791	0.950047
$709$	21.8837	0.821860	0.410930	−	0.911667i	$-0.365204\pi$
$709$	21.8837	0.821860	0.410930	+	0.911667i	$0.365204\pi$
$710$	−1.47694	−0.0554287
$711$	−4.20595	−0.157735
$712$	0.0638504	0.00239289
$713$	−21.1758	−0.793039
$714$	5.10869	0.191188
$715$	−4.57363	−0.171044
$716$	10.9611	0.409637
$717$	3.61328	0.134941
$718$	0.766244	0.0285960
$719$	13.2841	0.495415	0.247707	−	0.968835i	$-0.420323\pi$
$719$	13.2841	0.495415	0.247707	+	0.968835i	$0.420323\pi$
$720$	−11.9776	−0.446379
$721$	−2.52468	−0.0940241
$722$	−4.06395	−0.151244
$723$	−6.54258	−0.243321
$724$	−13.9163	−0.517195
$725$	−6.84224	−0.254114
$726$	−2.25616	−0.0837339
$727$	−46.1540	−1.71176	−0.855878	−	0.517178i	$-0.826983\pi$
$727$	−46.1540	−1.71176	−0.855878	+	0.517178i	$0.826983\pi$
$728$	2.33357	0.0864878
$729$	25.8927	0.958990
$730$	5.48555	0.203029
$731$	28.9508	1.07078
$732$	−3.05336	−0.112855
$733$	−33.9699	−1.25471	−0.627354	−	0.778734i	$-0.715862\pi$
$733$	−33.9699	−1.25471	−0.627354	+	0.778734i	$0.715862\pi$
$734$	−6.59235	−0.243328
$735$	−1.96441	−0.0724583
$736$	−8.28144	−0.305258
$737$	2.11920	0.0780619
$738$	0.384544	0.0141552
$739$	−36.1288	−1.32902	−0.664510	−	0.747280i	$-0.731359\pi$
$739$	−36.1288	−1.32902	−0.664510	+	0.747280i	$0.731359\pi$
$740$	8.08953	0.297377
$741$	1.65761	0.0608938
$742$	−2.78460	−0.102226
$743$	−30.9018	−1.13368	−0.566838	−	0.823829i	$-0.691833\pi$
$743$	−30.9018	−1.13368	−0.566838	+	0.823829i	$0.691833\pi$
$744$	−8.25257	−0.302554
$745$	−37.8470	−1.38661
$746$	−5.15644	−0.188791
$747$	−8.78278	−0.321345
$748$	−24.4838	−0.895216
$749$	−22.3155	−0.815390
$750$	−2.99318	−0.109295
$751$	−30.2833	−1.10505	−0.552527	−	0.833495i	$-0.686336\pi$
$751$	−30.2833	−1.10505	−0.552527	+	0.833495i	$0.686336\pi$
$752$	−9.65878	−0.352220
$753$	−9.94394	−0.362377
$754$	−1.72562	−0.0628434
$755$	9.03620	0.328861
$756$	−27.4615	−0.998767
$757$	−28.0388	−1.01909	−0.509543	−	0.860445i	$-0.670186\pi$
$757$	−28.0388	−1.01909	−0.509543	+	0.860445i	$0.670186\pi$
$758$	3.69395	0.134170
$759$	7.39953	0.268586
$760$	−2.89469	−0.105001
$761$	−9.21637	−0.334093	−0.167047	−	0.985949i	$-0.553423\pi$
$761$	−9.21637	−0.334093	−0.167047	+	0.985949i	$0.553423\pi$
$762$	1.82940	0.0662722
$763$	−14.8664	−0.538198
$764$	−36.5358	−1.32182
$765$	21.8492	0.789958
$766$	−3.80332	−0.137420
$767$	−10.0840	−0.364112
$768$	15.1912	0.548166
$769$	33.5990	1.21161	0.605806	−	0.795612i	$-0.292851\pi$
$769$	33.5990	1.21161	0.605806	+	0.795612i	$0.292851\pi$
$770$	2.70535	0.0974939
$771$	−6.82789	−0.245900
$772$	18.1592	0.653565
$773$	−46.8792	−1.68613	−0.843064	−	0.537813i	$-0.819251\pi$
$773$	−46.8792	−1.68613	−0.843064	+	0.537813i	$0.819251\pi$
$774$	1.35529	0.0487149
$775$	6.43574	0.231179
$776$	8.45836	0.303638
$777$	5.55752	0.199375
$778$	6.45391	0.231384
$779$	−1.57655	−0.0564858
$780$	−6.10406	−0.218560
$781$	4.86285	0.174006
$782$	4.79917	0.171618
$783$	41.1874	1.47192
$784$	−2.29962	−0.0821293
$785$	−20.9521	−0.747811
$786$	−6.47990	−0.231130
$787$	−25.1802	−0.897576	−0.448788	−	0.893638i	$-0.648144\pi$
$787$	−25.1802	−0.897576	−0.448788	+	0.893638i	$0.648144\pi$
$788$	38.4262	1.36888
$789$	−9.31946	−0.331782
$790$	1.79196	0.0637550
$791$	14.7033	0.522790
$792$	−2.32469	−0.0826044
$793$	1.21800	0.0432526
$794$	−1.99934	−0.0709541
$795$	14.7732	0.523952
$796$	17.8707	0.633411
$797$	22.4577	0.795493	0.397746	−	0.917495i	$-0.369792\pi$
$797$	22.4577	0.795493	0.397746	+	0.917495i	$0.369792\pi$
$798$	−0.980492	−0.0347090
$799$	17.6192	0.623324
$800$	2.51690	0.0889858
$801$	0.0924977	0.00326825
$802$	−9.13213	−0.322467
$803$	−18.0612	−0.637366
$804$	2.82833	0.0997475
$805$	18.7908	0.662288
$806$	1.62310	0.0571713
$807$	−3.46093	−0.121831
$808$	−6.30766	−0.221903
$809$	22.8101	0.801960	0.400980	−	0.916087i	$-0.368670\pi$
$809$	22.8101	0.801960	0.400980	+	0.916087i	$0.368670\pi$
$810$	−1.81989	−0.0639444
$811$	−23.2126	−0.815106	−0.407553	−	0.913182i	$-0.633618\pi$
$811$	−23.2126	−0.815106	−0.407553	+	0.913182i	$0.633618\pi$
$812$	−36.1694	−1.26930
$813$	26.4664	0.928217
$814$	0.751650	0.0263453
$815$	−15.7589	−0.552010
$816$	−31.7283	−1.11071
$817$	−5.55642	−0.194394
$818$	1.26995	0.0444029
$819$	3.38055	0.118126
$820$	5.80556	0.202739
$821$	−31.6879	−1.10591	−0.552957	−	0.833210i	$-0.686501\pi$
$821$	−31.6879	−1.10591	−0.552957	+	0.833210i	$0.686501\pi$
$822$	4.67900	0.163199
$823$	25.4058	0.885590	0.442795	−	0.896623i	$-0.353987\pi$
$823$	25.4058	0.885590	0.442795	+	0.896623i	$0.353987\pi$
$824$	−0.924301	−0.0321996
$825$	−2.24887	−0.0782955
$826$	5.96478	0.207541
$827$	−10.9700	−0.381463	−0.190732	−	0.981642i	$-0.561086\pi$
$827$	−10.9700	−0.381463	−0.190732	+	0.981642i	$0.561086\pi$
$828$	−7.96109	−0.276667
$829$	−6.00376	−0.208519	−0.104260	−	0.994550i	$-0.533247\pi$
$829$	−6.00376	−0.208519	−0.104260	+	0.994550i	$0.533247\pi$
$830$	3.74193	0.129884
$831$	19.3626	0.671680
$832$	−6.71256	−0.232716
$833$	4.19489	0.145344
$834$	−0.781024	−0.0270447
$835$	30.7784	1.06513
$836$	4.69908	0.162521
$837$	−38.7405	−1.33907
$838$	1.02339	0.0353524
$839$	34.2864	1.18370	0.591850	−	0.806048i	$-0.298398\pi$
$839$	34.2864	1.18370	0.591850	+	0.806048i	$0.298398\pi$
$840$	7.32310	0.252671
$841$	25.2476	0.870608
$842$	−5.72161	−0.197180
$843$	−3.35428	−0.115528
$844$	−49.5879	−1.70689
$845$	2.43495	0.0837648
$846$	0.824820	0.0283579
$847$	18.8641	0.648180
$848$	17.2942	0.593884
$849$	0.760093	0.0260863
$850$	−1.45857	−0.0500284
$851$	5.22081	0.178967
$852$	6.49005	0.222345
$853$	17.2768	0.591546	0.295773	−	0.955258i	$-0.404423\pi$
$853$	17.2768	0.591546	0.295773	+	0.955258i	$0.404423\pi$
$854$	−0.720461	−0.0246537
$855$	−4.19343	−0.143412
$856$	−8.16982	−0.279239
$857$	51.7560	1.76795	0.883976	−	0.467532i	$-0.154857\pi$
$857$	51.7560	1.76795	0.883976	+	0.467532i	$0.154857\pi$
$858$	−0.567167	−0.0193627
$859$	−18.9487	−0.646520	−0.323260	−	0.946310i	$-0.604779\pi$
$859$	−18.9487	−0.646520	−0.323260	+	0.946310i	$0.604779\pi$
$860$	20.4612	0.697721
$861$	3.98843	0.135925
$862$	−4.24883	−0.144716
$863$	0.234262	0.00797436	0.00398718	−	0.999992i	$-0.498731\pi$
$863$	0.234262	0.00797436	0.00398718	+	0.999992i	$0.498731\pi$
$864$	−15.1507	−0.515437
$865$	46.7257	1.58872
$866$	8.02796	0.272801
$867$	35.9683	1.22155
$868$	34.0206	1.15473
$869$	−5.90003	−0.200145
$870$	−5.41527	−0.183595
$871$	−1.12824	−0.0382290
$872$	−5.44266	−0.184312
$873$	12.2533	0.414712
$874$	−0.921087	−0.0311562
$875$	25.0265	0.846050
$876$	−24.1048	−0.814426
$877$	15.8446	0.535034	0.267517	−	0.963553i	$-0.413797\pi$
$877$	15.8446	0.535034	0.267517	+	0.963553i	$0.413797\pi$
$878$	−4.16343	−0.140509
$879$	21.9783	0.741308
$880$	−16.8020	−0.566395
$881$	−14.6188	−0.492521	−0.246261	−	0.969204i	$-0.579202\pi$
$881$	−14.6188	−0.492521	−0.246261	+	0.969204i	$0.579202\pi$
$882$	0.196378	0.00661239
$883$	49.9341	1.68042	0.840208	−	0.542264i	$-0.182433\pi$
$883$	49.9341	1.68042	0.840208	+	0.542264i	$0.182433\pi$
$884$	13.0349	0.438411
$885$	−31.6452	−1.06374
$886$	−3.57546	−0.120120
$887$	49.0527	1.64703	0.823514	−	0.567296i	$-0.192010\pi$
$887$	49.0527	1.64703	0.823514	+	0.567296i	$0.192010\pi$
$888$	2.03464	0.0682781
$889$	−15.2959	−0.513009
$890$	−0.0394089	−0.00132099
$891$	5.99200	0.200740
$892$	−39.9963	−1.33917
$893$	−3.38159	−0.113161
$894$	−4.69332	−0.156968
$895$	−13.7215	−0.458660
$896$	17.6508	0.589672
$897$	−3.93942	−0.131534
$898$	−4.81920	−0.160819
$899$	−51.0248	−1.70177
$900$	2.41954	0.0806513
$901$	−31.5475	−1.05100
$902$	0.539431	0.0179611
$903$	14.0569	0.467783
$904$	5.38297	0.179035
$905$	17.4209	0.579089
$906$	1.12056	0.0372281
$907$	−26.1163	−0.867176	−0.433588	−	0.901111i	$-0.642753\pi$
$907$	−26.1163	−0.867176	−0.433588	+	0.901111i	$0.642753\pi$
$908$	−29.2255	−0.969881
$909$	−9.13768	−0.303078
$910$	−1.44030	−0.0477453
$911$	−19.1302	−0.633811	−0.316905	−	0.948457i	$-0.602644\pi$
$911$	−19.1302	−0.633811	−0.316905	+	0.948457i	$0.602644\pi$
$912$	6.08950	0.201643
$913$	−12.3203	−0.407743
$914$	1.69622	0.0561060
$915$	3.82229	0.126361
$916$	34.4347	1.13776
$917$	54.1796	1.78917
$918$	8.77996	0.289782
$919$	0.921336	0.0303921	0.0151960	−	0.999885i	$-0.495163\pi$
$919$	0.921336	0.0303921	0.0151960	+	0.999885i	$0.495163\pi$
$920$	6.87942	0.226808
$921$	9.99002	0.329182
$922$	−3.15537	−0.103917
$923$	−2.58893	−0.0852155
$924$	−11.8879	−0.391084
$925$	−1.58671	−0.0521707
$926$	−4.62688	−0.152049
$927$	−1.33900	−0.0439786
$928$	−19.9549	−0.655050
$929$	39.5446	1.29742	0.648708	−	0.761037i	$-0.275310\pi$
$929$	39.5446	1.29742	0.648708	+	0.761037i	$0.275310\pi$
$930$	5.09355	0.167024
$931$	−0.805110	−0.0263864
$932$	35.6096	1.16643
$933$	16.2407	0.531696
$934$	6.51391	0.213142
$935$	30.6496	1.00235
$936$	1.23764	0.0404535
$937$	30.0137	0.980506	0.490253	−	0.871580i	$-0.336904\pi$
$937$	30.0137	0.980506	0.490253	+	0.871580i	$0.336904\pi$
$938$	0.667364	0.0217902
$939$	−14.3322	−0.467714
$940$	12.4525	0.406156
$941$	−34.1168	−1.11218	−0.556088	−	0.831123i	$-0.687698\pi$
$941$	−34.1168	−1.11218	−0.556088	+	0.831123i	$0.687698\pi$
$942$	−2.59822	−0.0846546
$943$	3.74678	0.122012
$944$	−37.0452	−1.20572
$945$	34.3773	1.11829
$946$	1.90118	0.0618127
$947$	−24.8327	−0.806954	−0.403477	−	0.914990i	$-0.632199\pi$
$947$	−24.8327	−0.806954	−0.403477	+	0.914990i	$0.632199\pi$
$948$	−7.87429	−0.255745
$949$	9.61558	0.312135
$950$	0.279937	0.00908235
$951$	−33.3897	−1.08274
$952$	−15.6381	−0.506834
$953$	−26.6853	−0.864423	−0.432211	−	0.901772i	$-0.642267\pi$
$953$	−26.6853	−0.864423	−0.432211	+	0.901772i	$0.642267\pi$
$954$	−1.47685	−0.0478148
$955$	45.7367	1.48000
$956$	−5.45332	−0.176373
$957$	17.8298	0.576356
$958$	−2.41196	−0.0779270
$959$	−39.1220	−1.26332
$960$	−21.0651	−0.679872
$961$	16.9935	0.548177
$962$	−0.400170	−0.0129020
$963$	−11.8353	−0.381388
$964$	9.87435	0.318031
$965$	−22.7323	−0.731779
$966$	2.33020	0.0749731
$967$	−59.5547	−1.91515	−0.957575	−	0.288184i	$-0.906948\pi$
$967$	−59.5547	−1.91515	−0.957575	+	0.288184i	$0.906948\pi$
$968$	6.90627	0.221976
$969$	−11.1083	−0.356849
$970$	−5.22057	−0.167622
$971$	23.3771	0.750207	0.375104	−	0.926983i	$-0.377607\pi$
$971$	23.3771	0.750207	0.375104	+	0.926983i	$0.377607\pi$
$972$	−24.6346	−0.790157
$973$	6.53028	0.209351
$974$	2.59729	0.0832225
$975$	1.19727	0.0383433
$976$	4.47454	0.143226
$977$	30.1796	0.965530	0.482765	−	0.875750i	$-0.339633\pi$
$977$	30.1796	0.965530	0.482765	+	0.875750i	$0.339633\pi$
$978$	−1.95423	−0.0624892
$979$	0.129754	0.00414696
$980$	2.96477	0.0947061
$981$	−7.88458	−0.251735
$982$	−4.77486	−0.152372
$983$	−1.98885	−0.0634346	−0.0317173	−	0.999497i	$-0.510098\pi$
$983$	−1.98885	−0.0634346	−0.0317173	+	0.999497i	$0.510098\pi$
$984$	1.46019	0.0465490
$985$	−48.1032	−1.53270
$986$	11.5640	0.368273
$987$	8.55490	0.272305
$988$	−2.50174	−0.0795908
$989$	13.2052	0.419901
$990$	1.43482	0.0456015
$991$	48.0189	1.52537	0.762686	−	0.646769i	$-0.223880\pi$
$991$	48.0189	1.52537	0.762686	+	0.646769i	$0.223880\pi$
$992$	18.7693	0.595927
$993$	−10.7685	−0.341729
$994$	1.53137	0.0485722
$995$	−22.3712	−0.709213
$996$	−16.4429	−0.521014
$997$	52.1504	1.65162	0.825810	−	0.563948i	$-0.190718\pi$
$997$	52.1504	1.65162	0.825810	+	0.563948i	$0.190718\pi$
$998$	−3.48649	−0.110363
$999$	9.55133	0.302191

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1339.2.a.d.1.12	✓	19

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1339.2.a.d.1.12	✓	19	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:	\( N \)	\(=\)	\( 1339 = 13 \cdot 103 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1339.a (trivial)

\(f(q)\)	\(=\)	\(q+0.234290 q^{2} +1.28880 q^{3} -1.94511 q^{4} +2.43495 q^{5} +0.301953 q^{6} -2.52468 q^{7} -0.924301 q^{8} -1.33900 q^{9} +O(q^{10})\)	\(q+0.234290 q^{2} +1.28880 q^{3} -1.94511 q^{4} +2.43495 q^{5} +0.301953 q^{6} -2.52468 q^{7} -0.924301 q^{8} -1.33900 q^{9} +0.570486 q^{10} -1.87833 q^{11} -2.50685 q^{12} +1.00000 q^{13} -0.591509 q^{14} +3.13816 q^{15} +3.67366 q^{16} -6.70137 q^{17} -0.313715 q^{18} +1.28617 q^{19} -4.73624 q^{20} -3.25380 q^{21} -0.440074 q^{22} -3.05667 q^{23} -1.19124 q^{24} +0.928983 q^{25} +0.234290 q^{26} -5.59209 q^{27} +4.91078 q^{28} -7.36530 q^{29} +0.735240 q^{30} +6.92773 q^{31} +2.70931 q^{32} -2.42078 q^{33} -1.57007 q^{34} -6.14748 q^{35} +2.60450 q^{36} -1.70801 q^{37} +0.301337 q^{38} +1.28880 q^{39} -2.25063 q^{40} -1.22577 q^{41} -0.762335 q^{42} -4.32013 q^{43} +3.65355 q^{44} -3.26040 q^{45} -0.716148 q^{46} -2.62920 q^{47} +4.73460 q^{48} -0.625975 q^{49} +0.217652 q^{50} -8.63671 q^{51} -1.94511 q^{52} +4.70761 q^{53} -1.31017 q^{54} -4.57363 q^{55} +2.33357 q^{56} +1.65761 q^{57} -1.72562 q^{58} -10.0840 q^{59} -6.10406 q^{60} +1.21800 q^{61} +1.62310 q^{62} +3.38055 q^{63} -6.71256 q^{64} +2.43495 q^{65} -0.567167 q^{66} -1.12824 q^{67} +13.0349 q^{68} -3.93942 q^{69} -1.44030 q^{70} -2.58893 q^{71} +1.23764 q^{72} +9.61558 q^{73} -0.400170 q^{74} +1.19727 q^{75} -2.50174 q^{76} +4.74218 q^{77} +0.301953 q^{78} +3.14111 q^{79} +8.94518 q^{80} -3.19007 q^{81} -0.287187 q^{82} +6.55920 q^{83} +6.32900 q^{84} -16.3175 q^{85} -1.01217 q^{86} -9.49238 q^{87} +1.73614 q^{88} -0.0690797 q^{89} -0.763881 q^{90} -2.52468 q^{91} +5.94555 q^{92} +8.92844 q^{93} -0.615996 q^{94} +3.13176 q^{95} +3.49175 q^{96} -9.15109 q^{97} -0.146660 q^{98} +2.51508 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 19 q - 9 q^{2} - 2 q^{3} + 17 q^{4} - 18 q^{5} - 8 q^{6} - 8 q^{7} - 24 q^{8} + 7 q^{9}+O(q^{10}) \) 19 * q - 9 * q^2 - 2 * q^3 + 17 * q^4 - 18 * q^5 - 8 * q^6 - 8 * q^7 - 24 * q^8 + 7 * q^9	\( 19 q - 9 q^{2} - 2 q^{3} + 17 q^{4} - 18 q^{5} - 8 q^{6} - 8 q^{7} - 24 q^{8} + 7 q^{9} - q^{11} + 18 q^{12} + 19 q^{13} - 8 q^{14} - 11 q^{15} + 21 q^{16} - 16 q^{17} - 10 q^{19} - 20 q^{20} - 33 q^{21} + 2 q^{22} - 14 q^{23} - 9 q^{24} + 23 q^{25} - 9 q^{26} + q^{27} + 10 q^{28} - 22 q^{29} + 28 q^{30} - 3 q^{31} - 47 q^{32} - 25 q^{33} - 35 q^{34} - 3 q^{35} - 33 q^{36} - 19 q^{37} - 15 q^{38} - 2 q^{39} + 8 q^{40} - 52 q^{41} - 3 q^{42} - 2 q^{43} - 54 q^{44} - 40 q^{45} + 33 q^{46} - 24 q^{47} - 8 q^{48} + 7 q^{49} - 10 q^{50} - 7 q^{51} + 17 q^{52} - 11 q^{53} - 23 q^{54} - 29 q^{55} - 17 q^{56} - 34 q^{57} + 18 q^{58} - 52 q^{59} - 71 q^{60} - 25 q^{61} - 22 q^{62} - 19 q^{63} + 48 q^{64} - 18 q^{65} + 9 q^{66} - 2 q^{67} + 18 q^{68} - 26 q^{69} - 12 q^{70} - 44 q^{71} - 6 q^{72} - 39 q^{73} - 4 q^{74} + 17 q^{75} - 10 q^{76} - 18 q^{77} - 8 q^{78} + q^{79} - 9 q^{80} - 13 q^{81} + 47 q^{82} - 27 q^{83} - 24 q^{84} - 26 q^{85} - q^{86} + 31 q^{87} + 19 q^{88} - 86 q^{89} + 48 q^{90} - 8 q^{91} - 19 q^{92} + 32 q^{93} + 45 q^{94} + 17 q^{95} - 25 q^{96} - 20 q^{97} + 39 q^{98} + 50 q^{99}+O(q^{100}) \) 19 * q - 9 * q^2 - 2 * q^3 + 17 * q^4 - 18 * q^5 - 8 * q^6 - 8 * q^7 - 24 * q^8 + 7 * q^9 - q^11 + 18 * q^12 + 19 * q^13 - 8 * q^14 - 11 * q^15 + 21 * q^16 - 16 * q^17 - 10 * q^19 - 20 * q^20 - 33 * q^21 + 2 * q^22 - 14 * q^23 - 9 * q^24 + 23 * q^25 - 9 * q^26 + q^27 + 10 * q^28 - 22 * q^29 + 28 * q^30 - 3 * q^31 - 47 * q^32 - 25 * q^33 - 35 * q^34 - 3 * q^35 - 33 * q^36 - 19 * q^37 - 15 * q^38 - 2 * q^39 + 8 * q^40 - 52 * q^41 - 3 * q^42 - 2 * q^43 - 54 * q^44 - 40 * q^45 + 33 * q^46 - 24 * q^47 - 8 * q^48 + 7 * q^49 - 10 * q^50 - 7 * q^51 + 17 * q^52 - 11 * q^53 - 23 * q^54 - 29 * q^55 - 17 * q^56 - 34 * q^57 + 18 * q^58 - 52 * q^59 - 71 * q^60 - 25 * q^61 - 22 * q^62 - 19 * q^63 + 48 * q^64 - 18 * q^65 + 9 * q^66 - 2 * q^67 + 18 * q^68 - 26 * q^69 - 12 * q^70 - 44 * q^71 - 6 * q^72 - 39 * q^73 - 4 * q^74 + 17 * q^75 - 10 * q^76 - 18 * q^77 - 8 * q^78 + q^79 - 9 * q^80 - 13 * q^81 + 47 * q^82 - 27 * q^83 - 24 * q^84 - 26 * q^85 - q^86 + 31 * q^87 + 19 * q^88 - 86 * q^89 + 48 * q^90 - 8 * q^91 - 19 * q^92 + 32 * q^93 + 45 * q^94 + 17 * q^95 - 25 * q^96 - 20 * q^97 + 39 * q^98 + 50 * q^99

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