LMFDB - Embedded newform 1339.2.a.e.1.7

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:	$21$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
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)

$f(q)$	$=$	$q-1.10076 q^{2} +0.263891 q^{3} -0.788329 q^{4} +3.16171 q^{5} -0.290481 q^{6} +1.73698 q^{7} +3.06928 q^{8} -2.93036 q^{9} +O(q^{10})$	\(q-1.10076 q^{2} +0.263891 q^{3} -0.788329 q^{4} +3.16171 q^{5} -0.290481 q^{6} +1.73698 q^{7} +3.06928 q^{8} -2.93036 q^{9} -3.48028 q^{10} -3.10474 q^{11} -0.208033 q^{12} -1.00000 q^{13} -1.91200 q^{14} +0.834349 q^{15} -1.80188 q^{16} -3.41611 q^{17} +3.22562 q^{18} -6.05193 q^{19} -2.49247 q^{20} +0.458375 q^{21} +3.41758 q^{22} -2.99631 q^{23} +0.809956 q^{24} +4.99643 q^{25} +1.10076 q^{26} -1.56497 q^{27} -1.36931 q^{28} -5.40188 q^{29} -0.918417 q^{30} -1.12545 q^{31} -4.15512 q^{32} -0.819315 q^{33} +3.76032 q^{34} +5.49184 q^{35} +2.31009 q^{36} -8.81605 q^{37} +6.66172 q^{38} -0.263891 q^{39} +9.70418 q^{40} +4.25911 q^{41} -0.504560 q^{42} +5.73199 q^{43} +2.44756 q^{44} -9.26496 q^{45} +3.29821 q^{46} -0.599133 q^{47} -0.475500 q^{48} -3.98289 q^{49} -5.49986 q^{50} -0.901483 q^{51} +0.788329 q^{52} +6.66172 q^{53} +1.72266 q^{54} -9.81631 q^{55} +5.33128 q^{56} -1.59705 q^{57} +5.94616 q^{58} +3.21729 q^{59} -0.657742 q^{60} +3.12684 q^{61} +1.23885 q^{62} -5.08999 q^{63} +8.17755 q^{64} -3.16171 q^{65} +0.901869 q^{66} -4.98173 q^{67} +2.69302 q^{68} -0.790700 q^{69} -6.04519 q^{70} +11.4683 q^{71} -8.99410 q^{72} -2.91540 q^{73} +9.70434 q^{74} +1.31851 q^{75} +4.77091 q^{76} -5.39289 q^{77} +0.290481 q^{78} +7.84215 q^{79} -5.69702 q^{80} +8.37810 q^{81} -4.68825 q^{82} -12.1494 q^{83} -0.361350 q^{84} -10.8008 q^{85} -6.30954 q^{86} -1.42551 q^{87} -9.52933 q^{88} -0.294986 q^{89} +10.1985 q^{90} -1.73698 q^{91} +2.36208 q^{92} -0.296997 q^{93} +0.659501 q^{94} -19.1345 q^{95} -1.09650 q^{96} -12.7010 q^{97} +4.38420 q^{98} +9.09802 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 21 q - q^{2} - 6 q^{3} + 15 q^{4} - 18 q^{5} - 8 q^{6} - 2 q^{7} + 6 q^{8} + 11 q^{9}+O(q^{10}) $ 21 * q - q^2 - 6 * q^3 + 15 * q^4 - 18 * q^5 - 8 * q^6 - 2 * q^7 + 6 * q^8 + 11 * q^9	\( 21 q - q^{2} - 6 q^{3} + 15 q^{4} - 18 q^{5} - 8 q^{6} - 2 q^{7} + 6 q^{8} + 11 q^{9} - 10 q^{10} - 5 q^{11} - 22 q^{12} - 21 q^{13} - 24 q^{14} - q^{15} - 5 q^{16} - 18 q^{17} + 10 q^{18} - 6 q^{19} - 30 q^{20} - 17 q^{21} - 8 q^{22} - 14 q^{23} - 13 q^{24} + 11 q^{25} + q^{26} - 33 q^{27} - 2 q^{28} - 36 q^{29} + 2 q^{30} + q^{31} - 9 q^{32} - 11 q^{33} - 25 q^{34} - 37 q^{35} + 3 q^{36} + 5 q^{37} - 13 q^{38} + 6 q^{39} - 2 q^{40} - 32 q^{41} + 7 q^{42} + 2 q^{43} + 4 q^{44} - 32 q^{45} - 5 q^{46} - 12 q^{47} - 18 q^{48} + 5 q^{49} - 2 q^{50} - 27 q^{51} - 15 q^{52} - 49 q^{53} - 31 q^{54} - 9 q^{55} - 53 q^{56} + 14 q^{57} - 2 q^{58} - 34 q^{59} + 25 q^{60} - 27 q^{61} - 12 q^{62} + 41 q^{63} - 70 q^{64} + 18 q^{65} - 15 q^{66} - 12 q^{67} - 20 q^{68} - 70 q^{69} + 70 q^{70} - 26 q^{71} - 14 q^{72} + 23 q^{73} + 20 q^{74} - 9 q^{75} - 36 q^{76} - 62 q^{77} + 8 q^{78} - 7 q^{79} - 41 q^{80} - 3 q^{81} - 33 q^{82} - q^{83} - 48 q^{84} + 10 q^{85} - 23 q^{86} - 7 q^{87} + 9 q^{88} - 16 q^{89} - 30 q^{90} + 2 q^{91} - 27 q^{92} + 2 q^{93} - 37 q^{94} - 35 q^{95} + 55 q^{96} - 30 q^{97} - 43 q^{98} - 42 q^{99}+O(q^{100}) \) 21 * q - q^2 - 6 * q^3 + 15 * q^4 - 18 * q^5 - 8 * q^6 - 2 * q^7 + 6 * q^8 + 11 * q^9 - 10 * q^10 - 5 * q^11 - 22 * q^12 - 21 * q^13 - 24 * q^14 - q^15 - 5 * q^16 - 18 * q^17 + 10 * q^18 - 6 * q^19 - 30 * q^20 - 17 * q^21 - 8 * q^22 - 14 * q^23 - 13 * q^24 + 11 * q^25 + q^26 - 33 * q^27 - 2 * q^28 - 36 * q^29 + 2 * q^30 + q^31 - 9 * q^32 - 11 * q^33 - 25 * q^34 - 37 * q^35 + 3 * q^36 + 5 * q^37 - 13 * q^38 + 6 * q^39 - 2 * q^40 - 32 * q^41 + 7 * q^42 + 2 * q^43 + 4 * q^44 - 32 * q^45 - 5 * q^46 - 12 * q^47 - 18 * q^48 + 5 * q^49 - 2 * q^50 - 27 * q^51 - 15 * q^52 - 49 * q^53 - 31 * q^54 - 9 * q^55 - 53 * q^56 + 14 * q^57 - 2 * q^58 - 34 * q^59 + 25 * q^60 - 27 * q^61 - 12 * q^62 + 41 * q^63 - 70 * q^64 + 18 * q^65 - 15 * q^66 - 12 * q^67 - 20 * q^68 - 70 * q^69 + 70 * q^70 - 26 * q^71 - 14 * q^72 + 23 * q^73 + 20 * q^74 - 9 * q^75 - 36 * q^76 - 62 * q^77 + 8 * q^78 - 7 * q^79 - 41 * q^80 - 3 * q^81 - 33 * q^82 - q^83 - 48 * q^84 + 10 * q^85 - 23 * q^86 - 7 * q^87 + 9 * q^88 - 16 * q^89 - 30 * q^90 + 2 * q^91 - 27 * q^92 + 2 * q^93 - 37 * q^94 - 35 * q^95 + 55 * q^96 - 30 * q^97 - 43 * q^98 - 42 * 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 $

$2$	−1.10076	−0.778354	−0.389177	−	0.921163i	$-0.627241\pi$
$2$	−1.10076	−0.778354	−0.389177	+	0.921163i	$0.627241\pi$
$3$	0.263891	0.152358	0.0761789	−	0.997094i	$-0.475728\pi$
$3$	0.263891	0.152358	0.0761789	+	0.997094i	$0.475728\pi$
$4$	−0.788329	−0.394165
$5$	3.16171	1.41396	0.706980	−	0.707233i	$-0.250057\pi$
$5$	3.16171	1.41396	0.706980	+	0.707233i	$0.250057\pi$
$6$	−0.290481	−0.118588
$7$	1.73698	0.656518	0.328259	−	0.944588i	$-0.393538\pi$
$7$	1.73698	0.656518	0.328259	+	0.944588i	$0.393538\pi$
$8$	3.06928	1.08515
$9$	−2.93036	−0.976787
$10$	−3.48028	−1.10056
$11$	−3.10474	−0.936116	−0.468058	−	0.883698i	$-0.655046\pi$
$11$	−3.10474	−0.936116	−0.468058	+	0.883698i	$0.655046\pi$
$12$	−0.208033	−0.0600540
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	−1.91200	−0.511003
$15$	0.834349	0.215428
$16$	−1.80188	−0.450470
$17$	−3.41611	−0.828530	−0.414265	−	0.910156i	$-0.635961\pi$
$17$	−3.41611	−0.828530	−0.414265	+	0.910156i	$0.635961\pi$
$18$	3.22562	0.760286
$19$	−6.05193	−1.38841	−0.694204	−	0.719778i	$-0.744243\pi$
$19$	−6.05193	−1.38841	−0.694204	+	0.719778i	$0.744243\pi$
$20$	−2.49247	−0.557333
$21$	0.458375	0.100026
$22$	3.41758	0.728630
$23$	−2.99631	−0.624773	−0.312387	−	0.949955i	$-0.601128\pi$
$23$	−2.99631	−0.624773	−0.312387	+	0.949955i	$0.601128\pi$
$24$	0.809956	0.165332
$25$	4.99643	0.999285
$26$	1.10076	0.215877
$27$	−1.56497	−0.301179
$28$	−1.36931	−0.258776
$29$	−5.40188	−1.00310	−0.501552	−	0.865128i	$-0.667237\pi$
$29$	−5.40188	−1.00310	−0.501552	+	0.865128i	$0.667237\pi$
$30$	−0.918417	−0.167679
$31$	−1.12545	−0.202137	−0.101069	−	0.994879i	$-0.532226\pi$
$31$	−1.12545	−0.202137	−0.101069	+	0.994879i	$0.532226\pi$
$32$	−4.15512	−0.734529
$33$	−0.819315	−0.142624
$34$	3.76032	0.644889
$35$	5.49184	0.928290
$36$	2.31009	0.385015
$37$	−8.81605	−1.44935	−0.724674	−	0.689091i	$-0.758010\pi$
$37$	−8.81605	−1.44935	−0.724674	+	0.689091i	$0.758010\pi$
$38$	6.66172	1.08067
$39$	−0.263891	−0.0422564
$40$	9.70418	1.53437
$41$	4.25911	0.665161	0.332581	−	0.943075i	$-0.392081\pi$
$41$	4.25911	0.665161	0.332581	+	0.943075i	$0.392081\pi$
$42$	−0.504560	−0.0778553
$43$	5.73199	0.874121	0.437060	−	0.899432i	$-0.356020\pi$
$43$	5.73199	0.874121	0.437060	+	0.899432i	$0.356020\pi$
$44$	2.44756	0.368984
$45$	−9.26496	−1.38114
$46$	3.29821	0.486295
$47$	−0.599133	−0.0873925	−0.0436963	−	0.999045i	$-0.513913\pi$
$47$	−0.599133	−0.0873925	−0.0436963	+	0.999045i	$0.513913\pi$
$48$	−0.475500	−0.0686325
$49$	−3.98289	−0.568985
$50$	−5.49986	−0.777798
$51$	−0.901483	−0.126233
$52$	0.788329	0.109322
$53$	6.66172	0.915057	0.457528	−	0.889195i	$-0.348735\pi$
$53$	6.66172	0.915057	0.457528	+	0.889195i	$0.348735\pi$
$54$	1.72266	0.234424
$55$	−9.81631	−1.32363
$56$	5.33128	0.712423
$57$	−1.59705	−0.211535
$58$	5.94616	0.780770
$59$	3.21729	0.418856	0.209428	−	0.977824i	$-0.432840\pi$
$59$	3.21729	0.418856	0.209428	+	0.977824i	$0.432840\pi$
$60$	−0.657742	−0.0849141
$61$	3.12684	0.400350	0.200175	−	0.979760i	$-0.435849\pi$
$61$	3.12684	0.400350	0.200175	+	0.979760i	$0.435849\pi$
$62$	1.23885	0.157335
$63$	−5.08999	−0.641278
$64$	8.17755	1.02219
$65$	−3.16171	−0.392162
$66$	0.901869	0.111012
$67$	−4.98173	−0.608615	−0.304307	−	0.952574i	$-0.598425\pi$
$67$	−4.98173	−0.608615	−0.304307	+	0.952574i	$0.598425\pi$
$68$	2.69302	0.326577
$69$	−0.790700	−0.0951891
$70$	−6.04519	−0.722539
$71$	11.4683	1.36104	0.680520	−	0.732729i	$-0.261754\pi$
$71$	11.4683	1.36104	0.680520	+	0.732729i	$0.261754\pi$
$72$	−8.99410	−1.05996
$73$	−2.91540	−0.341222	−0.170611	−	0.985338i	$-0.554574\pi$
$73$	−2.91540	−0.341222	−0.170611	+	0.985338i	$0.554574\pi$
$74$	9.70434	1.12811
$75$	1.31851	0.152249
$76$	4.77091	0.547261
$77$	−5.39289	−0.614577
$78$	0.290481	0.0328905
$79$	7.84215	0.882311	0.441155	−	0.897431i	$-0.354569\pi$
$79$	7.84215	0.882311	0.441155	+	0.897431i	$0.354569\pi$
$80$	−5.69702	−0.636946
$81$	8.37810	0.930900
$82$	−4.68825	−0.517731
$83$	−12.1494	−1.33357	−0.666785	−	0.745250i	$-0.732330\pi$
$83$	−12.1494	−1.33357	−0.666785	+	0.745250i	$0.732330\pi$
$84$	−0.361350	−0.0394265
$85$	−10.8008	−1.17151
$86$	−6.30954	−0.680376
$87$	−1.42551	−0.152831
$88$	−9.52933	−1.01583
$89$	−0.294986	−0.0312685	−0.0156342	−	0.999878i	$-0.504977\pi$
$89$	−0.294986	−0.0312685	−0.0156342	+	0.999878i	$0.504977\pi$
$90$	10.1985	1.07502
$91$	−1.73698	−0.182085
$92$	2.36208	0.246264
$93$	−0.296997	−0.0307972
$94$	0.659501	0.0680223
$95$	−19.1345	−1.96315
$96$	−1.09650	−0.111911
$97$	−12.7010	−1.28959	−0.644793	−	0.764357i	$-0.723057\pi$
$97$	−12.7010	−1.28959	−0.644793	+	0.764357i	$0.723057\pi$
$98$	4.38420	0.442871
$99$	9.09802	0.914386
$100$	−3.93883	−0.393883
$101$	−12.7552	−1.26919	−0.634594	−	0.772846i	$-0.718833\pi$
$101$	−12.7552	−1.26919	−0.634594	+	0.772846i	$0.718833\pi$
$102$	0.992316	0.0982539
$103$	−1.00000	−0.0985329
$103$	−1.00000	−0.0985329
$104$	−3.06928	−0.300968
$105$	1.44925	0.141432
$106$	−7.33294	−0.712238
$107$	−8.10549	−0.783587	−0.391794	−	0.920053i	$-0.628145\pi$
$107$	−8.10549	−0.783587	−0.391794	+	0.920053i	$0.628145\pi$
$108$	1.23371	0.118714
$109$	−13.5860	−1.30130	−0.650649	−	0.759379i	$-0.725503\pi$
$109$	−13.5860	−1.30130	−0.650649	+	0.759379i	$0.725503\pi$
$110$	10.8054	1.03025
$111$	−2.32648	−0.220820
$112$	−3.12983	−0.295741
$113$	−7.29668	−0.686414	−0.343207	−	0.939260i	$-0.611513\pi$
$113$	−7.29668	−0.686414	−0.343207	+	0.939260i	$0.611513\pi$
$114$	1.75797	0.164649
$115$	−9.47346	−0.883405
$116$	4.25846	0.395388
$117$	2.93036	0.270912
$118$	−3.54146	−0.326018
$119$	−5.93373	−0.543944
$120$	2.56085	0.233772
$121$	−1.36056	−0.123687
$122$	−3.44189	−0.311614
$123$	1.12394	0.101342
$124$	0.887228	0.0796754
$125$	−0.0112975	−0.00101048
$126$	5.60285	0.499141
$127$	14.8518	1.31788	0.658941	−	0.752195i	$-0.271005\pi$
$127$	14.8518	1.31788	0.658941	+	0.752195i	$0.271005\pi$
$128$	−0.691261	−0.0610994
$129$	1.51262	0.133179
$130$	3.48028	0.305241
$131$	8.73575	0.763246	0.381623	−	0.924318i	$-0.375365\pi$
$131$	8.73575	0.763246	0.381623	+	0.924318i	$0.375365\pi$
$132$	0.645890	0.0562175
$133$	−10.5121	−0.911514
$134$	5.48368	0.473718
$135$	−4.94799	−0.425855
$136$	−10.4850	−0.899082
$137$	0.552278	0.0471843	0.0235922	−	0.999722i	$-0.492490\pi$
$137$	0.552278	0.0471843	0.0235922	+	0.999722i	$0.492490\pi$
$138$	0.870370	0.0740908
$139$	6.56271	0.556642	0.278321	−	0.960488i	$-0.410222\pi$
$139$	6.56271	0.556642	0.278321	+	0.960488i	$0.410222\pi$
$140$	−4.32938	−0.365899
$141$	−0.158106	−0.0133149
$142$	−12.6239	−1.05937
$143$	3.10474	0.259632
$144$	5.28015	0.440013
$145$	−17.0792	−1.41835
$146$	3.20916	0.265592
$147$	−1.05105	−0.0866892
$148$	6.94995	0.571282
$149$	12.4780	1.02224	0.511119	−	0.859510i	$-0.329231\pi$
$149$	12.4780	1.02224	0.511119	+	0.859510i	$0.329231\pi$
$150$	−1.45137	−0.118504
$151$	−5.24916	−0.427171	−0.213585	−	0.976924i	$-0.568514\pi$
$151$	−5.24916	−0.427171	−0.213585	+	0.976924i	$0.568514\pi$
$152$	−18.5751	−1.50664
$153$	10.0105	0.809297
$154$	5.93627	0.478358
$155$	−3.55836	−0.285814
$156$	0.208033	0.0166560
$157$	17.7642	1.41774	0.708871	−	0.705339i	$-0.249205\pi$
$157$	17.7642	1.41774	0.708871	+	0.705339i	$0.249205\pi$
$158$	−8.63232	−0.686750
$159$	1.75797	0.139416
$160$	−13.1373	−1.03860
$161$	−5.20453	−0.410175
$162$	−9.22227	−0.724570
$163$	−12.3527	−0.967536	−0.483768	−	0.875196i	$-0.660732\pi$
$163$	−12.3527	−0.967536	−0.483768	+	0.875196i	$0.660732\pi$
$164$	−3.35758	−0.262183
$165$	−2.59044	−0.201665
$166$	13.3736	1.03799
$167$	−0.467727	−0.0361938	−0.0180969	−	0.999836i	$-0.505761\pi$
$167$	−0.467727	−0.0361938	−0.0180969	+	0.999836i	$0.505761\pi$
$168$	1.40688	0.108543
$169$	1.00000	0.0769231
$170$	11.8890	0.911848
$171$	17.7343	1.35618
$172$	−4.51870	−0.344548
$173$	−14.8040	−1.12552	−0.562762	−	0.826619i	$-0.690261\pi$
$173$	−14.8040	−1.12552	−0.562762	+	0.826619i	$0.690261\pi$
$174$	1.56914	0.118956
$175$	8.67871	0.656049
$176$	5.59437	0.421692
$177$	0.849016	0.0638159
$178$	0.324709	0.0243380
$179$	−14.6034	−1.09151	−0.545753	−	0.837946i	$-0.683756\pi$
$179$	−14.6034	−1.09151	−0.545753	+	0.837946i	$0.683756\pi$
$180$	7.30384	0.544396
$181$	−5.19668	−0.386266	−0.193133	−	0.981173i	$-0.561865\pi$
$181$	−5.19668	−0.386266	−0.193133	+	0.981173i	$0.561865\pi$
$182$	1.91200	0.141727
$183$	0.825145	0.0609965
$184$	−9.19650	−0.677975
$185$	−27.8738	−2.04932
$186$	0.326923	0.0239711
$187$	10.6062	0.775599
$188$	0.472314	0.0344470
$189$	−2.71833	−0.197729
$190$	21.0624	1.52803
$191$	−0.118115	−0.00854649	−0.00427324	−	0.999991i	$-0.501360\pi$
$191$	−0.118115	−0.00854649	−0.00427324	+	0.999991i	$0.501360\pi$
$192$	2.15798	0.155739
$193$	20.5606	1.47998	0.739992	−	0.672616i	$-0.234829\pi$
$193$	20.5606	1.47998	0.739992	+	0.672616i	$0.234829\pi$
$194$	13.9807	1.00376
$195$	−0.834349	−0.0597489
$196$	3.13983	0.224274
$197$	14.1373	1.00724	0.503619	−	0.863926i	$-0.332002\pi$
$197$	14.1373	1.00724	0.503619	+	0.863926i	$0.332002\pi$
$198$	−10.0147	−0.711716
$199$	5.28017	0.374301	0.187151	−	0.982331i	$-0.440075\pi$
$199$	5.28017	0.374301	0.187151	+	0.982331i	$0.440075\pi$
$200$	15.3354	1.08438
$201$	−1.31463	−0.0927272
$202$	14.0404	0.987878
$203$	−9.38296	−0.658555
$204$	0.710666	0.0497565
$205$	13.4661	0.940512
$206$	1.10076	0.0766935
$207$	8.78026	0.610271
$208$	1.80188	0.124938
$209$	18.7897	1.29971
$210$	−1.59527	−0.110084
$211$	5.05426	0.347950	0.173975	−	0.984750i	$-0.444339\pi$
$211$	5.05426	0.347950	0.173975	+	0.984750i	$0.444339\pi$
$212$	−5.25163	−0.360683
$213$	3.02639	0.207365
$214$	8.92219	0.609908
$215$	18.1229	1.23597
$216$	−4.80333	−0.326825
$217$	−1.95489	−0.132707
$218$	14.9549	1.01287
$219$	−0.769350	−0.0519879
$220$	7.73849	0.521729
$221$	3.41611	0.229793
$222$	2.56089	0.171876
$223$	22.2386	1.48921	0.744603	−	0.667507i	$-0.232639\pi$
$223$	22.2386	1.48921	0.744603	+	0.667507i	$0.232639\pi$
$224$	−7.21738	−0.482231
$225$	−14.6413	−0.976089
$226$	8.03188	0.534273
$227$	−13.0798	−0.868134	−0.434067	−	0.900881i	$-0.642922\pi$
$227$	−13.0798	−0.868134	−0.434067	+	0.900881i	$0.642922\pi$
$228$	1.25900	0.0833795
$229$	14.0950	0.931423	0.465711	−	0.884937i	$-0.345799\pi$
$229$	14.0950	0.931423	0.465711	+	0.884937i	$0.345799\pi$
$230$	10.4280	0.687602
$231$	−1.42314	−0.0936355
$232$	−16.5799	−1.08852
$233$	−12.0460	−0.789160	−0.394580	−	0.918861i	$-0.629110\pi$
$233$	−12.0460	−0.789160	−0.394580	+	0.918861i	$0.629110\pi$
$234$	−3.22562	−0.210866
$235$	−1.89429	−0.123570
$236$	−2.53629	−0.165098
$237$	2.06947	0.134427
$238$	6.53161	0.423381
$239$	−3.56470	−0.230581	−0.115291	−	0.993332i	$-0.536780\pi$
$239$	−3.56470	−0.230581	−0.115291	+	0.993332i	$0.536780\pi$
$240$	−1.50339	−0.0970437
$241$	3.93476	0.253460	0.126730	−	0.991937i	$-0.459552\pi$
$241$	3.93476	0.253460	0.126730	+	0.991937i	$0.459552\pi$
$242$	1.49765	0.0962726
$243$	6.90582	0.443009
$244$	−2.46498	−0.157804
$245$	−12.5928	−0.804522
$246$	−1.23719	−0.0788803
$247$	6.05193	0.385075
$248$	−3.45433	−0.219350
$249$	−3.20612	−0.203180
$250$	0.0124358	0.000786509 0
$251$	−9.33714	−0.589355	−0.294678	−	0.955597i	$-0.595212\pi$
$251$	−9.33714	−0.589355	−0.294678	+	0.955597i	$0.595212\pi$
$252$	4.01259	0.252769
$253$	9.30277	0.584860
$254$	−16.3482	−1.02578
$255$	−2.85023	−0.178488
$256$	−15.5942	−0.974636
$257$	−24.5673	−1.53247	−0.766235	−	0.642561i	$-0.777872\pi$
$257$	−24.5673	−1.53247	−0.766235	+	0.642561i	$0.777872\pi$
$258$	−1.66503	−0.103660
$259$	−15.3133	−0.951523
$260$	2.49247	0.154576
$261$	15.8294	0.979818
$262$	−9.61596	−0.594076
$263$	23.3054	1.43707	0.718535	−	0.695490i	$-0.244813\pi$
$263$	23.3054	1.43707	0.718535	+	0.695490i	$0.244813\pi$
$264$	−2.51471	−0.154770
$265$	21.0624	1.29385
$266$	11.5713	0.709481
$267$	−0.0778443	−0.00476400
$268$	3.92724	0.239894
$269$	−12.1281	−0.739461	−0.369730	−	0.929139i	$-0.620550\pi$
$269$	−12.1281	−0.739461	−0.369730	+	0.929139i	$0.620550\pi$
$270$	5.44654	0.331466
$271$	−11.8504	−0.719858	−0.359929	−	0.932980i	$-0.617199\pi$
$271$	−11.8504	−0.719858	−0.359929	+	0.932980i	$0.617199\pi$
$272$	6.15542	0.373227
$273$	−0.458375	−0.0277421
$274$	−0.607925	−0.0367261
$275$	−15.5126	−0.935447
$276$	0.623332	0.0375202
$277$	17.2208	1.03470	0.517349	−	0.855774i	$-0.326919\pi$
$277$	17.2208	1.03470	0.517349	+	0.855774i	$0.326919\pi$
$278$	−7.22396	−0.433264
$279$	3.29799	0.197445
$280$	16.8560	1.00734
$281$	5.95915	0.355493	0.177747	−	0.984076i	$-0.443119\pi$
$281$	5.95915	0.355493	0.177747	+	0.984076i	$0.443119\pi$
$282$	0.174037	0.0103637
$283$	−13.7617	−0.818050	−0.409025	−	0.912523i	$-0.634131\pi$
$283$	−13.7617	−0.818050	−0.409025	+	0.912523i	$0.634131\pi$
$284$	−9.04083	−0.536474
$285$	−5.04942	−0.299102
$286$	−3.41758	−0.202085
$287$	7.39800	0.436690
$288$	12.1760	0.717479
$289$	−5.33016	−0.313539
$290$	18.8001	1.10398
$291$	−3.35167	−0.196478
$292$	2.29830	0.134498
$293$	−9.92690	−0.579935	−0.289968	−	0.957036i	$-0.593645\pi$
$293$	−9.92690	−0.579935	−0.289968	+	0.957036i	$0.593645\pi$
$294$	1.15695	0.0674749
$295$	10.1722	0.592246
$296$	−27.0589	−1.57277
$297$	4.85884	0.281938
$298$	−13.7353	−0.795663
$299$	2.99631	0.173281
$300$	−1.03942	−0.0600111
$301$	9.95637	0.573876
$302$	5.77806	0.332490
$303$	−3.36598	−0.193371
$304$	10.9048	0.625435
$305$	9.88616	0.566080
$306$	−11.0191	−0.629920
$307$	12.9853	0.741112	0.370556	−	0.928810i	$-0.379167\pi$
$307$	12.9853	0.741112	0.370556	+	0.928810i	$0.379167\pi$
$308$	4.25137	0.242244
$309$	−0.263891	−0.0150123
$310$	3.91690	0.222465
$311$	−7.48495	−0.424433	−0.212216	−	0.977223i	$-0.568068\pi$
$311$	−7.48495	−0.424433	−0.212216	+	0.977223i	$0.568068\pi$
$312$	−0.809956	−0.0458547
$313$	3.31889	0.187595	0.0937974	−	0.995591i	$-0.470099\pi$
$313$	3.31889	0.187595	0.0937974	+	0.995591i	$0.470099\pi$
$314$	−19.5542	−1.10350
$315$	−16.0931	−0.906742
$316$	−6.18220	−0.347776
$317$	−15.5534	−0.873568	−0.436784	−	0.899566i	$-0.643883\pi$
$317$	−15.5534	−0.873568	−0.436784	+	0.899566i	$0.643883\pi$
$318$	−1.93510	−0.108515
$319$	16.7714	0.939021
$320$	25.8551	1.44534
$321$	−2.13897	−0.119386
$322$	5.72894	0.319261
$323$	20.6741	1.15034
$324$	−6.60470	−0.366928
$325$	−4.99643	−0.277152
$326$	13.5973	0.753086
$327$	−3.58522	−0.198263
$328$	13.0724	0.721802
$329$	−1.04068	−0.0573747
$330$	2.85145	0.156967
$331$	20.4970	1.12662	0.563308	−	0.826247i	$-0.309529\pi$
$331$	20.4970	1.12662	0.563308	+	0.826247i	$0.309529\pi$
$332$	9.57773	0.525646
$333$	25.8342	1.41571
$334$	0.514855	0.0281716
$335$	−15.7508	−0.860558
$336$	−0.825935	−0.0450585
$337$	−25.0946	−1.36699	−0.683496	−	0.729955i	$-0.739541\pi$
$337$	−25.0946	−1.36699	−0.683496	+	0.729955i	$0.739541\pi$
$338$	−1.10076	−0.0598734
$339$	−1.92553	−0.104580
$340$	8.51457	0.461767
$341$	3.49425	0.189224
$342$	−19.5212	−1.05559
$343$	−19.0771	−1.03007
$344$	17.5931	0.948556
$345$	−2.49997	−0.134594
$346$	16.2956	0.876057
$347$	−11.5726	−0.621252	−0.310626	−	0.950532i	$-0.600539\pi$
$347$	−11.5726	−0.621252	−0.310626	+	0.950532i	$0.600539\pi$
$348$	1.12377	0.0602404
$349$	27.1634	1.45403	0.727013	−	0.686624i	$-0.240908\pi$
$349$	27.1634	1.45403	0.727013	+	0.686624i	$0.240908\pi$
$350$	−9.55316	−0.510638
$351$	1.56497	0.0835320
$352$	12.9006	0.687604
$353$	3.18596	0.169572	0.0847858	−	0.996399i	$-0.472979\pi$
$353$	3.18596	0.169572	0.0847858	+	0.996399i	$0.472979\pi$
$354$	−0.934562	−0.0496714
$355$	36.2596	1.92446
$356$	0.232546	0.0123249
$357$	−1.56586	−0.0828741
$358$	16.0748	0.849578
$359$	−13.4701	−0.710927	−0.355464	−	0.934690i	$-0.615677\pi$
$359$	−13.4701	−0.710927	−0.355464	+	0.934690i	$0.615677\pi$
$360$	−28.4367	−1.49875
$361$	17.6259	0.927676
$362$	5.72029	0.300652
$363$	−0.359040	−0.0188447
$364$	1.36931	0.0717716
$365$	−9.21767	−0.482475
$366$	−0.908286	−0.0474769
$367$	15.5035	0.809278	0.404639	−	0.914476i	$-0.367397\pi$
$367$	15.5035	0.809278	0.404639	+	0.914476i	$0.367397\pi$
$368$	5.39898	0.281441
$369$	−12.4807	−0.649721
$370$	30.6823	1.59510
$371$	11.5713	0.600751
$372$	0.234132	0.0121392
$373$	1.94680	0.100801	0.0504006	−	0.998729i	$-0.483950\pi$
$373$	1.94680	0.100801	0.0504006	+	0.998729i	$0.483950\pi$
$374$	−11.6748	−0.603691
$375$	−0.00298131	−0.000153954 0
$376$	−1.83891	−0.0948343
$377$	5.40188	0.278211
$378$	2.99222	0.153903
$379$	34.9954	1.79759	0.898796	−	0.438367i	$-0.144443\pi$
$379$	34.9954	1.79759	0.898796	+	0.438367i	$0.144443\pi$
$380$	15.0843	0.773806
$381$	3.91926	0.200790
$382$	0.130016	0.00665220
$383$	17.5373	0.896115	0.448057	−	0.894005i	$-0.352116\pi$
$383$	17.5373	0.896115	0.448057	+	0.894005i	$0.352116\pi$
$384$	−0.182418	−0.00930896
$385$	−17.0508	−0.868987
$386$	−22.6323	−1.15195
$387$	−16.7968	−0.853830
$388$	10.0125	0.508309
$389$	−14.1694	−0.718415	−0.359208	−	0.933258i	$-0.616953\pi$
$389$	−14.1694	−0.718415	−0.359208	+	0.933258i	$0.616953\pi$
$390$	0.918417	0.0465058
$391$	10.2357	0.517643
$392$	−12.2246	−0.617436
$393$	2.30529	0.116286
$394$	−15.5617	−0.783988
$395$	24.7946	1.24755
$396$	−7.17224	−0.360419
$397$	−18.6458	−0.935805	−0.467903	−	0.883780i	$-0.654990\pi$
$397$	−18.6458	−0.935805	−0.467903	+	0.883780i	$0.654990\pi$
$398$	−5.81219	−0.291339
$399$	−2.77405	−0.138876
$400$	−9.00295	−0.450148
$401$	2.12251	0.105993	0.0529964	−	0.998595i	$-0.483123\pi$
$401$	2.12251	0.105993	0.0529964	+	0.998595i	$0.483123\pi$
$402$	1.44710	0.0721746
$403$	1.12545	0.0560628
$404$	10.0553	0.500269
$405$	26.4892	1.31626
$406$	10.3284	0.512589
$407$	27.3716	1.35676
$408$	−2.76690	−0.136982
$409$	−33.6111	−1.66196	−0.830980	−	0.556302i	$-0.812220\pi$
$409$	−33.6111	−1.66196	−0.830980	+	0.556302i	$0.812220\pi$
$410$	−14.8229	−0.732051
$411$	0.145741	0.00718890
$412$	0.788329	0.0388382
$413$	5.58838	0.274986
$414$	−9.66496	−0.475007
$415$	−38.4129	−1.88562
$416$	4.15512	0.203722
$417$	1.73184	0.0848087
$418$	−20.6829	−1.01164
$419$	−25.6417	−1.25268	−0.626340	−	0.779550i	$-0.715448\pi$
$419$	−25.6417	−1.25268	−0.626340	+	0.779550i	$0.715448\pi$
$420$	−1.14249	−0.0557476
$421$	22.9830	1.12012	0.560062	−	0.828451i	$-0.310777\pi$
$421$	22.9830	1.12012	0.560062	+	0.828451i	$0.310777\pi$
$422$	−5.56352	−0.270828
$423$	1.75568	0.0853639
$424$	20.4467	0.992978
$425$	−17.0684	−0.827937
$426$	−3.33133	−0.161404
$427$	5.43126	0.262837
$428$	6.38979	0.308862
$429$	0.819315	0.0395569
$430$	−19.9490	−0.962024
$431$	34.1899	1.64687	0.823434	−	0.567412i	$-0.192055\pi$
$431$	34.1899	1.64687	0.823434	+	0.567412i	$0.192055\pi$
$432$	2.81989	0.135672
$433$	25.1996	1.21102	0.605508	−	0.795839i	$-0.292970\pi$
$433$	25.1996	1.21102	0.605508	+	0.795839i	$0.292970\pi$
$434$	2.15187	0.103293
$435$	−4.50705	−0.216096
$436$	10.7102	0.512926
$437$	18.1334	0.867440
$438$	0.846869	0.0404650
$439$	36.5948	1.74658	0.873288	−	0.487204i	$-0.161983\pi$
$439$	36.5948	1.74658	0.873288	+	0.487204i	$0.161983\pi$
$440$	−30.1290	−1.43634
$441$	11.6713	0.555777
$442$	−3.76032	−0.178860
$443$	15.2865	0.726282	0.363141	−	0.931734i	$-0.381704\pi$
$443$	15.2865	0.726282	0.363141	+	0.931734i	$0.381704\pi$
$444$	1.83403	0.0870393
$445$	−0.932662	−0.0442124
$446$	−24.4793	−1.15913
$447$	3.29284	0.155746
$448$	14.2043	0.671088
$449$	−6.83324	−0.322481	−0.161240	−	0.986915i	$-0.551549\pi$
$449$	−6.83324	−0.322481	−0.161240	+	0.986915i	$0.551549\pi$
$450$	16.1166	0.759743
$451$	−13.2234	−0.622668
$452$	5.75218	0.270560
$453$	−1.38521	−0.0650827
$454$	14.3977	0.675716
$455$	−5.49184	−0.257461
$456$	−4.90180	−0.229548
$457$	−42.0609	−1.96753	−0.983763	−	0.179471i	$-0.942562\pi$
$457$	−42.0609	−1.96753	−0.983763	+	0.179471i	$0.942562\pi$
$458$	−15.5152	−0.724977
$459$	5.34612	0.249536
$460$	7.46821	0.348207
$461$	−25.5860	−1.19166	−0.595829	−	0.803111i	$-0.703176\pi$
$461$	−25.5860	−1.19166	−0.595829	+	0.803111i	$0.703176\pi$
$462$	1.56653	0.0728816
$463$	−3.74986	−0.174271	−0.0871353	−	0.996196i	$-0.527771\pi$
$463$	−3.74986	−0.174271	−0.0871353	+	0.996196i	$0.527771\pi$
$464$	9.73352	0.451867
$465$	−0.939021	−0.0435460
$466$	13.2598	0.614246
$467$	−37.4941	−1.73502	−0.867511	−	0.497418i	$-0.834282\pi$
$467$	−37.4941	−1.73502	−0.867511	+	0.497418i	$0.834282\pi$
$468$	−2.31009	−0.106784
$469$	−8.65317	−0.399566
$470$	2.08515	0.0961809
$471$	4.68783	0.216004
$472$	9.87477	0.454523
$473$	−17.7964	−0.818278
$474$	−2.27799	−0.104632
$475$	−30.2380	−1.38742
$476$	4.67774	0.214404
$477$	−19.5212	−0.893816
$478$	3.92387	0.179474
$479$	3.76694	0.172116	0.0860580	−	0.996290i	$-0.472573\pi$
$479$	3.76694	0.172116	0.0860580	+	0.996290i	$0.472573\pi$
$480$	−3.46682	−0.158238
$481$	8.81605	0.401977
$482$	−4.33122	−0.197282
$483$	−1.37343	−0.0624933
$484$	1.07257	0.0487532
$485$	−40.1568	−1.82342
$486$	−7.60165	−0.344818
$487$	36.6134	1.65911	0.829555	−	0.558424i	$-0.188594\pi$
$487$	36.6134	1.65911	0.829555	+	0.558424i	$0.188594\pi$
$488$	9.59713	0.434442
$489$	−3.25977	−0.147412
$490$	13.8616	0.626203
$491$	14.8096	0.668347	0.334173	−	0.942512i	$-0.391543\pi$
$491$	14.8096	0.668347	0.334173	+	0.942512i	$0.391543\pi$
$492$	−0.886037	−0.0399456
$493$	18.4534	0.831101
$494$	−6.66172	−0.299725
$495$	28.7653	1.29291
$496$	2.02793	0.0910567
$497$	19.9203	0.893547
$498$	3.52917	0.158146
$499$	−18.8786	−0.845121	−0.422561	−	0.906335i	$-0.638869\pi$
$499$	−18.8786	−0.845121	−0.422561	+	0.906335i	$0.638869\pi$
$500$	0.00890613	0.000398294 0
$501$	−0.123429	−0.00551441
$502$	10.2779	0.458727
$503$	29.4126	1.31144	0.655721	−	0.755004i	$-0.272365\pi$
$503$	29.4126	1.31144	0.655721	+	0.755004i	$0.272365\pi$
$504$	−15.6226	−0.695885
$505$	−40.3282	−1.79458
$506$	−10.2401	−0.455228
$507$	0.263891	0.0117198
$508$	−11.7081	−0.519463
$509$	−12.8973	−0.571662	−0.285831	−	0.958280i	$-0.592270\pi$
$509$	−12.8973	−0.571662	−0.285831	+	0.958280i	$0.592270\pi$
$510$	3.13742	0.138927
$511$	−5.06401	−0.224019
$512$	18.5480	0.819712
$513$	9.47109	0.418159
$514$	27.0427	1.19280
$515$	−3.16171	−0.139322
$516$	−1.19245	−0.0524945
$517$	1.86015	0.0818095
$518$	16.8563	0.740622
$519$	−3.90664	−0.171482
$520$	−9.70418	−0.425556
$521$	−4.47800	−0.196185	−0.0980925	−	0.995177i	$-0.531274\pi$
$521$	−4.47800	−0.196185	−0.0980925	+	0.995177i	$0.531274\pi$
$522$	−17.4244	−0.762646
$523$	−17.7306	−0.775305	−0.387652	−	0.921806i	$-0.626714\pi$
$523$	−17.7306	−0.775305	−0.387652	+	0.921806i	$0.626714\pi$
$524$	−6.88665	−0.300845
$525$	2.29024	0.0999541
$526$	−25.6536	−1.11855
$527$	3.84468	0.167477
$528$	1.47631	0.0642480
$529$	−14.0221	−0.609658
$530$	−23.1847	−1.00708
$531$	−9.42783	−0.409133
$532$	8.28699	0.359287
$533$	−4.25911	−0.184483
$534$	0.0856879	0.00370808
$535$	−25.6272	−1.10796
$536$	−15.2903	−0.660441
$537$	−3.85370	−0.166299
$538$	13.3501	0.575563
$539$	12.3659	0.532635
$540$	3.90065	0.167857
$541$	−15.8718	−0.682383	−0.341192	−	0.939994i	$-0.610830\pi$
$541$	−15.8718	−0.682383	−0.341192	+	0.939994i	$0.610830\pi$
$542$	13.0444	0.560304
$543$	−1.37136	−0.0588506
$544$	14.1944	0.608579
$545$	−42.9549	−1.83998
$546$	0.504560	0.0215932
$547$	25.7950	1.10291	0.551456	−	0.834204i	$-0.314072\pi$
$547$	25.7950	1.10291	0.551456	+	0.834204i	$0.314072\pi$
$548$	−0.435377	−0.0185984
$549$	−9.16276	−0.391057
$550$	17.0757	0.728109
$551$	32.6918	1.39272
$552$	−2.42688	−0.103295
$553$	13.6217	0.579252
$554$	−18.9560	−0.805362
$555$	−7.35566	−0.312230
$556$	−5.17358	−0.219408
$557$	−6.53991	−0.277105	−0.138552	−	0.990355i	$-0.544245\pi$
$557$	−6.53991	−0.277105	−0.138552	+	0.990355i	$0.544245\pi$
$558$	−3.63029	−0.153682
$559$	−5.73199	−0.242437
$560$	−9.89563	−0.418166
$561$	2.79887	0.118169
$562$	−6.55959	−0.276699
$563$	−33.1174	−1.39573	−0.697865	−	0.716229i	$-0.745867\pi$
$563$	−33.1174	−1.39573	−0.697865	+	0.716229i	$0.745867\pi$
$564$	0.124640	0.00524827
$565$	−23.0700	−0.970562
$566$	15.1483	0.636732
$567$	14.5526	0.611152
$568$	35.1995	1.47694
$569$	−18.1526	−0.760998	−0.380499	−	0.924781i	$-0.624248\pi$
$569$	−18.1526	−0.760998	−0.380499	+	0.924781i	$0.624248\pi$
$570$	5.55819	0.232807
$571$	−41.1393	−1.72163	−0.860814	−	0.508920i	$-0.830045\pi$
$571$	−41.1393	−1.72163	−0.860814	+	0.508920i	$0.830045\pi$
$572$	−2.44756	−0.102338
$573$	−0.0311695	−0.00130212
$574$	−8.14342	−0.339900
$575$	−14.9708	−0.624327
$576$	−23.9632	−0.998465
$577$	−22.6468	−0.942798	−0.471399	−	0.881920i	$-0.656251\pi$
$577$	−22.6468	−0.942798	−0.471399	+	0.881920i	$0.656251\pi$
$578$	5.86722	0.244044
$579$	5.42576	0.225487
$580$	13.4640	0.559063
$581$	−21.1033	−0.875513
$582$	3.68938	0.152930
$583$	−20.6829	−0.856599
$584$	−8.94819	−0.370279
$585$	9.26496	0.383059
$586$	10.9271	0.451395
$587$	−6.64834	−0.274406	−0.137203	−	0.990543i	$-0.543811\pi$
$587$	−6.64834	−0.274406	−0.137203	+	0.990543i	$0.543811\pi$
$588$	0.828574	0.0341698
$589$	6.81117	0.280649
$590$	−11.1971	−0.460977
$591$	3.73070	0.153460
$592$	15.8854	0.652887
$593$	−2.58618	−0.106202	−0.0531009	−	0.998589i	$-0.516911\pi$
$593$	−2.58618	−0.106202	−0.0531009	+	0.998589i	$0.516911\pi$
$594$	−5.34841	−0.219448
$595$	−18.7608	−0.769116
$596$	−9.83678	−0.402930
$597$	1.39339	0.0570277
$598$	−3.29821	−0.134874
$599$	21.0609	0.860524	0.430262	−	0.902704i	$-0.358421\pi$
$599$	21.0609	0.860524	0.430262	+	0.902704i	$0.358421\pi$
$600$	4.04689	0.165213
$601$	−15.9284	−0.649735	−0.324867	−	0.945760i	$-0.605320\pi$
$601$	−15.9284	−0.649735	−0.324867	+	0.945760i	$0.605320\pi$
$602$	−10.9596	−0.446679
$603$	14.5983	0.594487
$604$	4.13807	0.168376
$605$	−4.30171	−0.174889
$606$	3.70514	0.150511
$607$	39.9268	1.62058	0.810289	−	0.586030i	$-0.199310\pi$
$607$	39.9268	1.62058	0.810289	+	0.586030i	$0.199310\pi$
$608$	25.1465	1.01983
$609$	−2.47608	−0.100336
$610$	−10.8823	−0.440611
$611$	0.599133	0.0242383
$612$	−7.89153	−0.318996
$613$	34.2985	1.38530	0.692652	−	0.721272i	$-0.256442\pi$
$613$	34.2985	1.38530	0.692652	+	0.721272i	$0.256442\pi$
$614$	−14.2937	−0.576848
$615$	3.55358	0.143294
$616$	−16.5523	−0.666910
$617$	−20.1861	−0.812660	−0.406330	−	0.913726i	$-0.633192\pi$
$617$	−20.1861	−0.812660	−0.406330	+	0.913726i	$0.633192\pi$
$618$	0.290481	0.0116849
$619$	11.0382	0.443663	0.221831	−	0.975085i	$-0.428797\pi$
$619$	11.0382	0.443663	0.221831	+	0.975085i	$0.428797\pi$
$620$	2.80516	0.112658
$621$	4.68914	0.188169
$622$	8.23913	0.330359
$623$	−0.512386	−0.0205283
$624$	0.475500	0.0190352
$625$	−25.0179	−1.00071
$626$	−3.65330	−0.146015
$627$	4.95844	0.198021
$628$	−14.0041	−0.558824
$629$	30.1166	1.20083
$630$	17.7146	0.705767
$631$	1.72110	0.0685158	0.0342579	−	0.999413i	$-0.489093\pi$
$631$	1.72110	0.0685158	0.0342579	+	0.999413i	$0.489093\pi$
$632$	24.0697	0.957443
$633$	1.33378	0.0530128
$634$	17.1206	0.679945
$635$	46.9570	1.86343
$636$	−1.38586	−0.0549529
$637$	3.98289	0.157808
$638$	−18.4613	−0.730891
$639$	−33.6064	−1.32945
$640$	−2.18557	−0.0863921
$641$	−27.1843	−1.07372	−0.536859	−	0.843672i	$-0.680389\pi$
$641$	−27.1843	−1.07372	−0.536859	+	0.843672i	$0.680389\pi$
$642$	2.35449	0.0929243
$643$	28.5059	1.12416	0.562082	−	0.827081i	$-0.310000\pi$
$643$	28.5059	1.12416	0.562082	+	0.827081i	$0.310000\pi$
$644$	4.10289	0.161676
$645$	4.78248	0.188310
$646$	−22.7572	−0.895370
$647$	−31.1183	−1.22338	−0.611692	−	0.791096i	$-0.709511\pi$
$647$	−31.1183	−1.22338	−0.611692	+	0.791096i	$0.709511\pi$
$648$	25.7147	1.01017
$649$	−9.98887	−0.392097
$650$	5.49986	0.215722
$651$	−0.515879	−0.0202189
$652$	9.73798	0.381369
$653$	47.7154	1.86725	0.933624	−	0.358254i	$-0.116628\pi$
$653$	47.7154	1.86725	0.933624	+	0.358254i	$0.116628\pi$
$654$	3.94646	0.154319
$655$	27.6199	1.07920
$656$	−7.67440	−0.299635
$657$	8.54319	0.333302
$658$	1.14554	0.0446579
$659$	−35.6330	−1.38806	−0.694031	−	0.719945i	$-0.744167\pi$
$659$	−35.6330	−1.38806	−0.694031	+	0.719945i	$0.744167\pi$
$660$	2.04212	0.0794894
$661$	−9.03681	−0.351491	−0.175746	−	0.984436i	$-0.556234\pi$
$661$	−9.03681	−0.351491	−0.175746	+	0.984436i	$0.556234\pi$
$662$	−22.5622	−0.876906
$663$	0.901483	0.0350107
$664$	−37.2899	−1.44713
$665$	−33.2362	−1.28885
$666$	−28.4372	−1.10192
$667$	16.1857	0.626712
$668$	0.368723	0.0142663
$669$	5.86857	0.226892
$670$	17.3378	0.669819
$671$	−9.70803	−0.374774
$672$	−1.90460	−0.0734717
$673$	0.927646	0.0357581	0.0178791	−	0.999840i	$-0.494309\pi$
$673$	0.927646	0.0357581	0.0178791	+	0.999840i	$0.494309\pi$
$674$	27.6231	1.06400
$675$	−7.81926	−0.300964
$676$	−0.788329	−0.0303204
$677$	0.952769	0.0366179	0.0183089	−	0.999832i	$-0.494172\pi$
$677$	0.952769	0.0366179	0.0183089	+	0.999832i	$0.494172\pi$
$678$	2.11954	0.0814006
$679$	−22.0613	−0.846636
$680$	−33.1506	−1.27127
$681$	−3.45164	−0.132267
$682$	−3.84632	−0.147283
$683$	14.3313	0.548372	0.274186	−	0.961677i	$-0.411592\pi$
$683$	14.3313	0.548372	0.274186	+	0.961677i	$0.411592\pi$
$684$	−13.9805	−0.534558
$685$	1.74615	0.0667168
$686$	20.9993	0.801756
$687$	3.71954	0.141909
$688$	−10.3284	−0.393765
$689$	−6.66172	−0.253791
$690$	2.75186	0.104762
$691$	14.8375	0.564443	0.282222	−	0.959349i	$-0.408929\pi$
$691$	14.8375	0.564443	0.282222	+	0.959349i	$0.408929\pi$
$692$	11.6704	0.443642
$693$	15.8031	0.600310
$694$	12.7387	0.483554
$695$	20.7494	0.787070
$696$	−4.37528	−0.165845
$697$	−14.5496	−0.551106
$698$	−29.9004	−1.13175
$699$	−3.17884	−0.120235
$700$	−6.84168	−0.258591
$701$	26.4039	0.997263	0.498631	−	0.866814i	$-0.333836\pi$
$701$	26.4039	0.997263	0.498631	+	0.866814i	$0.333836\pi$
$702$	−1.72266	−0.0650175
$703$	53.3541	2.01229
$704$	−25.3892	−0.956891
$705$	−0.499886	−0.0188268
$706$	−3.50698	−0.131987
$707$	−22.1555	−0.833244
$708$	−0.669304	−0.0251540
$709$	−47.3669	−1.77890	−0.889451	−	0.457031i	$-0.848913\pi$
$709$	−47.3669	−1.77890	−0.889451	+	0.457031i	$0.848913\pi$
$710$	−39.9131	−1.49791
$711$	−22.9803	−0.861830
$712$	−0.905395	−0.0339311
$713$	3.37221	0.126290
$714$	1.72364	0.0645054
$715$	9.81631	0.367109
$716$	11.5123	0.430233
$717$	−0.940693	−0.0351308
$718$	14.8274	0.553353
$719$	−49.4897	−1.84565	−0.922827	−	0.385214i	$-0.874128\pi$
$719$	−49.4897	−1.84565	−0.922827	+	0.385214i	$0.874128\pi$
$720$	16.6943	0.622161
$721$	−1.73698	−0.0646886
$722$	−19.4018	−0.722061
$723$	1.03835	0.0386166
$724$	4.09669	0.152252
$725$	−26.9901	−1.00239
$726$	0.395217	0.0146679
$727$	10.7448	0.398501	0.199251	−	0.979949i	$-0.436149\pi$
$727$	10.7448	0.398501	0.199251	+	0.979949i	$0.436149\pi$
$728$	−5.33128	−0.197591
$729$	−23.3119	−0.863404
$730$	10.1464	0.375536
$731$	−19.5811	−0.724235
$732$	−0.650486	−0.0240427
$733$	14.3346	0.529461	0.264730	−	0.964323i	$-0.414717\pi$
$733$	14.3346	0.529461	0.264730	+	0.964323i	$0.414717\pi$
$734$	−17.0657	−0.629905
$735$	−3.32312	−0.122575
$736$	12.4500	0.458914
$737$	15.4670	0.569734
$738$	13.7383	0.505713
$739$	19.4387	0.715064	0.357532	−	0.933901i	$-0.383618\pi$
$739$	19.4387	0.715064	0.357532	+	0.933901i	$0.383618\pi$
$740$	21.9737	0.807771
$741$	1.59705	0.0586692
$742$	−12.7372	−0.467597
$743$	−33.4487	−1.22711	−0.613557	−	0.789651i	$-0.710262\pi$
$743$	−33.4487	−1.22711	−0.613557	+	0.789651i	$0.710262\pi$
$744$	−0.911568	−0.0334197
$745$	39.4519	1.44540
$746$	−2.14295	−0.0784591
$747$	35.6022	1.30261
$748$	−8.36115	−0.305714
$749$	−14.0791	−0.514439
$750$	0.00328170	0.000119831 0
$751$	−32.7335	−1.19446	−0.597230	−	0.802070i	$-0.703732\pi$
$751$	−32.7335	−1.19446	−0.597230	+	0.802070i	$0.703732\pi$
$752$	1.07956	0.0393677
$753$	−2.46399	−0.0897928
$754$	−5.94616	−0.216547
$755$	−16.5963	−0.604002
$756$	2.14294	0.0779379
$757$	15.9592	0.580046	0.290023	−	0.957020i	$-0.406337\pi$
$757$	15.9592	0.580046	0.290023	+	0.957020i	$0.406337\pi$
$758$	−38.5215	−1.39916
$759$	2.45492	0.0891080
$760$	−58.7290	−2.13032
$761$	4.29239	0.155599	0.0777995	−	0.996969i	$-0.475211\pi$
$761$	4.29239	0.155599	0.0777995	+	0.996969i	$0.475211\pi$
$762$	−4.31416	−0.156285
$763$	−23.5986	−0.854325
$764$	0.0931134	0.00336872
$765$	31.6502	1.14431
$766$	−19.3044	−0.697495
$767$	−3.21729	−0.116170
$768$	−4.11517	−0.148493
$769$	−20.0911	−0.724505	−0.362252	−	0.932080i	$-0.617992\pi$
$769$	−20.0911	−0.724505	−0.362252	+	0.932080i	$0.617992\pi$
$770$	18.7688	0.676380
$771$	−6.48311	−0.233484
$772$	−16.2085	−0.583357
$773$	11.1253	0.400150	0.200075	−	0.979781i	$-0.435881\pi$
$773$	11.1253	0.400150	0.200075	+	0.979781i	$0.435881\pi$
$774$	18.4892	0.664582
$775$	−5.62325	−0.201993
$776$	−38.9828	−1.39940
$777$	−4.04105	−0.144972
$778$	15.5971	0.559181
$779$	−25.7758	−0.923515
$780$	0.657742	0.0235509
$781$	−35.6063	−1.27409
$782$	−11.2671	−0.402910
$783$	8.45378	0.302113
$784$	7.17668	0.256310
$785$	56.1654	2.00463
$786$	−2.53757	−0.0905121
$787$	46.1078	1.64357	0.821784	−	0.569800i	$-0.192979\pi$
$787$	46.1078	1.64357	0.821784	+	0.569800i	$0.192979\pi$
$788$	−11.1448	−0.397018
$789$	6.15009	0.218949
$790$	−27.2929	−0.971038
$791$	−12.6742	−0.450643
$792$	27.9244	0.992249
$793$	−3.12684	−0.111037
$794$	20.5245	0.728388
$795$	5.55819	0.197129
$796$	−4.16251	−0.147536
$797$	14.1437	0.500996	0.250498	−	0.968117i	$-0.419406\pi$
$797$	14.1437	0.500996	0.250498	+	0.968117i	$0.419406\pi$
$798$	3.05356	0.108095
$799$	2.04671	0.0724073
$800$	−20.7608	−0.734004
$801$	0.864417	0.0305427
$802$	−2.33637	−0.0825000
$803$	9.05159	0.319424
$804$	1.03637	0.0365498
$805$	−16.4552	−0.579971
$806$	−1.23885	−0.0436367
$807$	−3.20049	−0.112663
$808$	−39.1492	−1.37726
$809$	31.7587	1.11658	0.558288	−	0.829647i	$-0.311458\pi$
$809$	31.7587	1.11658	0.558288	+	0.829647i	$0.311458\pi$
$810$	−29.1582	−1.02451
$811$	1.21020	0.0424958	0.0212479	−	0.999774i	$-0.493236\pi$
$811$	1.21020	0.0424958	0.0212479	+	0.999774i	$0.493236\pi$
$812$	7.39687	0.259579
$813$	−3.12721	−0.109676
$814$	−30.1295	−1.05604
$815$	−39.0556	−1.36806
$816$	1.62436	0.0568641
$817$	−34.6896	−1.21364
$818$	36.9977	1.29359
$819$	5.08999	0.177859
$820$	−10.6157	−0.370717
$821$	−23.8763	−0.833288	−0.416644	−	0.909070i	$-0.636794\pi$
$821$	−23.8763	−0.833288	−0.416644	+	0.909070i	$0.636794\pi$
$822$	−0.160426	−0.00559551
$823$	−47.7247	−1.66358	−0.831789	−	0.555092i	$-0.812683\pi$
$823$	−47.7247	−1.66358	−0.831789	+	0.555092i	$0.812683\pi$
$824$	−3.06928	−0.106923
$825$	−4.09365	−0.142523
$826$	−6.15146	−0.214037
$827$	−30.7653	−1.06982	−0.534908	−	0.844910i	$-0.679654\pi$
$827$	−30.7653	−1.06982	−0.534908	+	0.844910i	$0.679654\pi$
$828$	−6.92174	−0.240547
$829$	55.2138	1.91766	0.958828	−	0.283989i	$-0.0916578\pi$
$829$	55.2138	1.91766	0.958828	+	0.283989i	$0.0916578\pi$
$830$	42.2834	1.46768
$831$	4.54443	0.157644
$832$	−8.17755	−0.283505
$833$	13.6060	0.471420
$834$	−1.90634	−0.0660112
$835$	−1.47882	−0.0511766
$836$	−14.8125	−0.512300
$837$	1.76130	0.0608795
$838$	28.2253	0.975028
$839$	20.9978	0.724926	0.362463	−	0.931998i	$-0.381936\pi$
$839$	20.9978	0.724926	0.362463	+	0.931998i	$0.381936\pi$
$840$	4.44815	0.153476
$841$	0.180258	0.00621580
$842$	−25.2988	−0.871853
$843$	1.57257	0.0541621
$844$	−3.98442	−0.137149
$845$	3.16171	0.108766
$846$	−1.93258	−0.0664433
$847$	−2.36327	−0.0812030
$848$	−12.0036	−0.412205
$849$	−3.63160	−0.124636
$850$	18.7882	0.644429
$851$	26.4156	0.905515
$852$	−2.38580	−0.0817360
$853$	48.0626	1.64563	0.822815	−	0.568309i	$-0.192402\pi$
$853$	48.0626	1.64563	0.822815	+	0.568309i	$0.192402\pi$
$854$	−5.97851	−0.204580
$855$	56.0709	1.91758
$856$	−24.8780	−0.850313
$857$	12.4189	0.424220	0.212110	−	0.977246i	$-0.431966\pi$
$857$	12.4189	0.424220	0.212110	+	0.977246i	$0.431966\pi$
$858$	−0.901869	−0.0307893
$859$	34.8447	1.18889	0.594443	−	0.804138i	$-0.297373\pi$
$859$	34.8447	1.18889	0.594443	+	0.804138i	$0.297373\pi$
$860$	−14.2868	−0.487177
$861$	1.95227	0.0665331
$862$	−37.6348	−1.28185
$863$	−33.7142	−1.14764	−0.573822	−	0.818980i	$-0.694540\pi$
$863$	−33.7142	−1.14764	−0.573822	+	0.818980i	$0.694540\pi$
$864$	6.50265	0.221225
$865$	−46.8059	−1.59145
$866$	−27.7387	−0.942599
$867$	−1.40658	−0.0477701
$868$	1.54110	0.0523083
$869$	−24.3479	−0.825945
$870$	4.96117	0.168200
$871$	4.98173	0.168799
$872$	−41.6991	−1.41211
$873$	37.2184	1.25965
$874$	−19.9606	−0.675176
$875$	−0.0196235	−0.000663396 0
$876$	0.606501	0.0204918
$877$	−23.3181	−0.787398	−0.393699	−	0.919239i	$-0.628805\pi$
$877$	−23.3181	−0.787398	−0.393699	+	0.919239i	$0.628805\pi$
$878$	−40.2821	−1.35945
$879$	−2.61962	−0.0883577
$880$	17.6878	0.596255
$881$	45.6232	1.53709	0.768543	−	0.639799i	$-0.220982\pi$
$881$	45.6232	1.53709	0.768543	+	0.639799i	$0.220982\pi$
$882$	−12.8473	−0.432591
$883$	−26.6097	−0.895487	−0.447744	−	0.894162i	$-0.647772\pi$
$883$	−26.6097	−0.895487	−0.447744	+	0.894162i	$0.647772\pi$
$884$	−2.69302	−0.0905762
$885$	2.68434	0.0902332
$886$	−16.8267	−0.565305
$887$	−29.7077	−0.997486	−0.498743	−	0.866750i	$-0.666205\pi$
$887$	−29.7077	−0.997486	−0.498743	+	0.866750i	$0.666205\pi$
$888$	−7.14061	−0.239623
$889$	25.7973	0.865213
$890$	1.02664	0.0344129
$891$	−26.0119	−0.871430
$892$	−17.5313	−0.586993
$893$	3.62591	0.121336
$894$	−3.62462	−0.121225
$895$	−46.1716	−1.54335
$896$	−1.20071	−0.0401128
$897$	0.790700	0.0264007
$898$	7.52175	0.251004
$899$	6.07956	0.202765
$900$	11.5422	0.384740
$901$	−22.7572	−0.758152
$902$	14.5558	0.484656
$903$	2.62740	0.0874344
$904$	−22.3955	−0.744864
$905$	−16.4304	−0.546165
$906$	1.52478	0.0506574
$907$	−18.2292	−0.605292	−0.302646	−	0.953103i	$-0.597870\pi$
$907$	−18.2292	−0.605292	−0.302646	+	0.953103i	$0.597870\pi$
$908$	10.3112	0.342188
$909$	37.3773	1.23973
$910$	6.04519	0.200396
$911$	24.8085	0.821943	0.410971	−	0.911648i	$-0.365190\pi$
$911$	24.8085	0.821943	0.410971	+	0.911648i	$0.365190\pi$
$912$	2.87769	0.0952899
$913$	37.7208	1.24838
$914$	46.2989	1.53143
$915$	2.60887	0.0862466
$916$	−11.1115	−0.367134
$917$	15.1739	0.501085
$918$	−5.88479	−0.194227
$919$	14.0389	0.463101	0.231551	−	0.972823i	$-0.425620\pi$
$919$	14.0389	0.463101	0.231551	+	0.972823i	$0.425620\pi$
$920$	−29.0767	−0.958631
$921$	3.42672	0.112914
$922$	28.1640	0.927532
$923$	−11.4683	−0.377485
$924$	1.12190	0.0369078
$925$	−44.0487	−1.44831
$926$	4.12769	0.135644
$927$	2.93036	0.0962457
$928$	22.4455	0.736808
$929$	23.8557	0.782679	0.391340	−	0.920246i	$-0.372012\pi$
$929$	23.8557	0.782679	0.391340	+	0.920246i	$0.372012\pi$
$930$	1.03364	0.0338942
$931$	24.1042	0.789983
$932$	9.49623	0.311059
$933$	−1.97521	−0.0646656
$934$	41.2720	1.35046
$935$	33.5336	1.09667
$936$	8.99410	0.293981
$937$	12.9285	0.422357	0.211178	−	0.977448i	$-0.432270\pi$
$937$	12.9285	0.422357	0.211178	+	0.977448i	$0.432270\pi$
$938$	9.52506	0.311004
$939$	0.875826	0.0285815
$940$	1.49332	0.0487068
$941$	34.4231	1.12216	0.561081	−	0.827761i	$-0.310386\pi$
$941$	34.4231	1.12216	0.561081	+	0.827761i	$0.310386\pi$
$942$	−5.16017	−0.168128
$943$	−12.7616	−0.415575
$944$	−5.79717	−0.188682
$945$	−8.59457	−0.279581
$946$	19.5895	0.636910
$947$	−19.6355	−0.638069	−0.319035	−	0.947743i	$-0.603359\pi$
$947$	−19.6355	−0.638069	−0.319035	+	0.947743i	$0.603359\pi$
$948$	−1.63143	−0.0529863
$949$	2.91540	0.0946380
$950$	33.2848	1.07990
$951$	−4.10442	−0.133095
$952$	−18.2123	−0.590263
$953$	−37.6119	−1.21837	−0.609185	−	0.793028i	$-0.708503\pi$
$953$	−37.6119	−1.21837	−0.609185	+	0.793028i	$0.708503\pi$
$954$	21.4882	0.695705
$955$	−0.373445	−0.0120844
$956$	2.81016	0.0908870
$957$	4.42584	0.143067
$958$	−4.14650	−0.133967
$959$	0.959298	0.0309773
$960$	6.82293	0.220209
$961$	−29.7334	−0.959140
$962$	−9.70434	−0.312881
$963$	23.7520	0.765398
$964$	−3.10188	−0.0999049
$965$	65.0067	2.09264
$966$	1.51182	0.0486419
$967$	38.7462	1.24599	0.622997	−	0.782224i	$-0.285915\pi$
$967$	38.7462	1.24599	0.622997	+	0.782224i	$0.285915\pi$
$968$	−4.17594	−0.134220
$969$	5.45571	0.175263
$970$	44.2029	1.41927
$971$	40.9505	1.31416	0.657082	−	0.753819i	$-0.271791\pi$
$971$	40.9505	1.31416	0.657082	+	0.753819i	$0.271791\pi$
$972$	−5.44406	−0.174618
$973$	11.3993	0.365445
$974$	−40.3025	−1.29138
$975$	−1.31851	−0.0422262
$976$	−5.63418	−0.180346
$977$	2.14262	0.0685483	0.0342742	−	0.999412i	$-0.489088\pi$
$977$	2.14262	0.0685483	0.0342742	+	0.999412i	$0.489088\pi$
$978$	3.58822	0.114738
$979$	0.915857	0.0292709
$980$	9.92724	0.317114
$981$	39.8117	1.27109
$982$	−16.3018	−0.520210
$983$	48.5276	1.54779	0.773895	−	0.633314i	$-0.218306\pi$
$983$	48.5276	1.54779	0.773895	+	0.633314i	$0.218306\pi$
$984$	3.44969	0.109972
$985$	44.6979	1.42419
$986$	−20.3128	−0.646891
$987$	−0.274627	−0.00874149
$988$	−4.77091	−0.151783
$989$	−17.1748	−0.546127
$990$	−31.6637	−1.00634
$991$	−9.91145	−0.314848	−0.157424	−	0.987531i	$-0.550319\pi$
$991$	−9.91145	−0.314848	−0.157424	+	0.987531i	$0.550319\pi$
$992$	4.67640	0.148476
$993$	5.40897	0.171649
$994$	−21.9274	−0.695496
$995$	16.6944	0.529247
$996$	2.52748	0.0800863
$997$	54.7385	1.73359	0.866793	−	0.498667i	$-0.166177\pi$
$997$	54.7385	1.73359	0.866793	+	0.498667i	$0.166177\pi$
$998$	20.7808	0.657804
$999$	13.7969	0.436513

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1339.2.a.e.1.7	✓	21

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1339.2.a.e.1.7	✓	21	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)

\(f(q)\)	\(=\)	\(q-1.10076 q^{2} +0.263891 q^{3} -0.788329 q^{4} +3.16171 q^{5} -0.290481 q^{6} +1.73698 q^{7} +3.06928 q^{8} -2.93036 q^{9} +O(q^{10})\)	\(q-1.10076 q^{2} +0.263891 q^{3} -0.788329 q^{4} +3.16171 q^{5} -0.290481 q^{6} +1.73698 q^{7} +3.06928 q^{8} -2.93036 q^{9} -3.48028 q^{10} -3.10474 q^{11} -0.208033 q^{12} -1.00000 q^{13} -1.91200 q^{14} +0.834349 q^{15} -1.80188 q^{16} -3.41611 q^{17} +3.22562 q^{18} -6.05193 q^{19} -2.49247 q^{20} +0.458375 q^{21} +3.41758 q^{22} -2.99631 q^{23} +0.809956 q^{24} +4.99643 q^{25} +1.10076 q^{26} -1.56497 q^{27} -1.36931 q^{28} -5.40188 q^{29} -0.918417 q^{30} -1.12545 q^{31} -4.15512 q^{32} -0.819315 q^{33} +3.76032 q^{34} +5.49184 q^{35} +2.31009 q^{36} -8.81605 q^{37} +6.66172 q^{38} -0.263891 q^{39} +9.70418 q^{40} +4.25911 q^{41} -0.504560 q^{42} +5.73199 q^{43} +2.44756 q^{44} -9.26496 q^{45} +3.29821 q^{46} -0.599133 q^{47} -0.475500 q^{48} -3.98289 q^{49} -5.49986 q^{50} -0.901483 q^{51} +0.788329 q^{52} +6.66172 q^{53} +1.72266 q^{54} -9.81631 q^{55} +5.33128 q^{56} -1.59705 q^{57} +5.94616 q^{58} +3.21729 q^{59} -0.657742 q^{60} +3.12684 q^{61} +1.23885 q^{62} -5.08999 q^{63} +8.17755 q^{64} -3.16171 q^{65} +0.901869 q^{66} -4.98173 q^{67} +2.69302 q^{68} -0.790700 q^{69} -6.04519 q^{70} +11.4683 q^{71} -8.99410 q^{72} -2.91540 q^{73} +9.70434 q^{74} +1.31851 q^{75} +4.77091 q^{76} -5.39289 q^{77} +0.290481 q^{78} +7.84215 q^{79} -5.69702 q^{80} +8.37810 q^{81} -4.68825 q^{82} -12.1494 q^{83} -0.361350 q^{84} -10.8008 q^{85} -6.30954 q^{86} -1.42551 q^{87} -9.52933 q^{88} -0.294986 q^{89} +10.1985 q^{90} -1.73698 q^{91} +2.36208 q^{92} -0.296997 q^{93} +0.659501 q^{94} -19.1345 q^{95} -1.09650 q^{96} -12.7010 q^{97} +4.38420 q^{98} +9.09802 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 21 q - q^{2} - 6 q^{3} + 15 q^{4} - 18 q^{5} - 8 q^{6} - 2 q^{7} + 6 q^{8} + 11 q^{9}+O(q^{10}) \) 21 * q - q^2 - 6 * q^3 + 15 * q^4 - 18 * q^5 - 8 * q^6 - 2 * q^7 + 6 * q^8 + 11 * q^9	\( 21 q - q^{2} - 6 q^{3} + 15 q^{4} - 18 q^{5} - 8 q^{6} - 2 q^{7} + 6 q^{8} + 11 q^{9} - 10 q^{10} - 5 q^{11} - 22 q^{12} - 21 q^{13} - 24 q^{14} - q^{15} - 5 q^{16} - 18 q^{17} + 10 q^{18} - 6 q^{19} - 30 q^{20} - 17 q^{21} - 8 q^{22} - 14 q^{23} - 13 q^{24} + 11 q^{25} + q^{26} - 33 q^{27} - 2 q^{28} - 36 q^{29} + 2 q^{30} + q^{31} - 9 q^{32} - 11 q^{33} - 25 q^{34} - 37 q^{35} + 3 q^{36} + 5 q^{37} - 13 q^{38} + 6 q^{39} - 2 q^{40} - 32 q^{41} + 7 q^{42} + 2 q^{43} + 4 q^{44} - 32 q^{45} - 5 q^{46} - 12 q^{47} - 18 q^{48} + 5 q^{49} - 2 q^{50} - 27 q^{51} - 15 q^{52} - 49 q^{53} - 31 q^{54} - 9 q^{55} - 53 q^{56} + 14 q^{57} - 2 q^{58} - 34 q^{59} + 25 q^{60} - 27 q^{61} - 12 q^{62} + 41 q^{63} - 70 q^{64} + 18 q^{65} - 15 q^{66} - 12 q^{67} - 20 q^{68} - 70 q^{69} + 70 q^{70} - 26 q^{71} - 14 q^{72} + 23 q^{73} + 20 q^{74} - 9 q^{75} - 36 q^{76} - 62 q^{77} + 8 q^{78} - 7 q^{79} - 41 q^{80} - 3 q^{81} - 33 q^{82} - q^{83} - 48 q^{84} + 10 q^{85} - 23 q^{86} - 7 q^{87} + 9 q^{88} - 16 q^{89} - 30 q^{90} + 2 q^{91} - 27 q^{92} + 2 q^{93} - 37 q^{94} - 35 q^{95} + 55 q^{96} - 30 q^{97} - 43 q^{98} - 42 q^{99}+O(q^{100}) \) 21 * q - q^2 - 6 * q^3 + 15 * q^4 - 18 * q^5 - 8 * q^6 - 2 * q^7 + 6 * q^8 + 11 * q^9 - 10 * q^10 - 5 * q^11 - 22 * q^12 - 21 * q^13 - 24 * q^14 - q^15 - 5 * q^16 - 18 * q^17 + 10 * q^18 - 6 * q^19 - 30 * q^20 - 17 * q^21 - 8 * q^22 - 14 * q^23 - 13 * q^24 + 11 * q^25 + q^26 - 33 * q^27 - 2 * q^28 - 36 * q^29 + 2 * q^30 + q^31 - 9 * q^32 - 11 * q^33 - 25 * q^34 - 37 * q^35 + 3 * q^36 + 5 * q^37 - 13 * q^38 + 6 * q^39 - 2 * q^40 - 32 * q^41 + 7 * q^42 + 2 * q^43 + 4 * q^44 - 32 * q^45 - 5 * q^46 - 12 * q^47 - 18 * q^48 + 5 * q^49 - 2 * q^50 - 27 * q^51 - 15 * q^52 - 49 * q^53 - 31 * q^54 - 9 * q^55 - 53 * q^56 + 14 * q^57 - 2 * q^58 - 34 * q^59 + 25 * q^60 - 27 * q^61 - 12 * q^62 + 41 * q^63 - 70 * q^64 + 18 * q^65 - 15 * q^66 - 12 * q^67 - 20 * q^68 - 70 * q^69 + 70 * q^70 - 26 * q^71 - 14 * q^72 + 23 * q^73 + 20 * q^74 - 9 * q^75 - 36 * q^76 - 62 * q^77 + 8 * q^78 - 7 * q^79 - 41 * q^80 - 3 * q^81 - 33 * q^82 - q^83 - 48 * q^84 + 10 * q^85 - 23 * q^86 - 7 * q^87 + 9 * q^88 - 16 * q^89 - 30 * q^90 + 2 * q^91 - 27 * q^92 + 2 * q^93 - 37 * q^94 - 35 * q^95 + 55 * q^96 - 30 * q^97 - 43 * q^98 - 42 * q^99

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