LMFDB - Embedded newform 403.2.a.d.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [403,2,Mod(1,403)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(403, 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("403.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 403 = 13 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	403.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:	$3.21797120146$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3x^{7} - 7x^{6} + 19x^{5} + 21x^{4} - 31x^{3} - 29x^{2} - x + 1 $ x^8 - 3x^7 - 7x^6 + 19x^5 + 21x^4 - 31x^3 - 29x^2 - x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$1.93310$ of defining polynomial
Character	$\chi$	$=$	403.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.933096 q^{2} +1.42306 q^{3} -1.12933 q^{4} -3.66529 q^{5} +1.32785 q^{6} -3.96149 q^{7} -2.91997 q^{8} -0.974910 q^{9} +O(q^{10})$	\(q+0.933096 q^{2} +1.42306 q^{3} -1.12933 q^{4} -3.66529 q^{5} +1.32785 q^{6} -3.96149 q^{7} -2.91997 q^{8} -0.974910 q^{9} -3.42006 q^{10} +4.36293 q^{11} -1.60710 q^{12} -1.00000 q^{13} -3.69645 q^{14} -5.21591 q^{15} -0.465945 q^{16} +3.00507 q^{17} -0.909684 q^{18} -7.36472 q^{19} +4.13933 q^{20} -5.63743 q^{21} +4.07103 q^{22} +7.15784 q^{23} -4.15528 q^{24} +8.43433 q^{25} -0.933096 q^{26} -5.65652 q^{27} +4.47384 q^{28} -8.93598 q^{29} -4.86694 q^{30} -1.00000 q^{31} +5.40516 q^{32} +6.20870 q^{33} +2.80402 q^{34} +14.5200 q^{35} +1.10100 q^{36} +3.43749 q^{37} -6.87199 q^{38} -1.42306 q^{39} +10.7025 q^{40} -0.259319 q^{41} -5.26026 q^{42} +0.991709 q^{43} -4.92720 q^{44} +3.57332 q^{45} +6.67895 q^{46} -8.92173 q^{47} -0.663066 q^{48} +8.69341 q^{49} +7.87003 q^{50} +4.27638 q^{51} +1.12933 q^{52} -6.09433 q^{53} -5.27808 q^{54} -15.9914 q^{55} +11.5674 q^{56} -10.4804 q^{57} -8.33812 q^{58} -8.53935 q^{59} +5.89049 q^{60} -2.95055 q^{61} -0.933096 q^{62} +3.86210 q^{63} +5.97543 q^{64} +3.66529 q^{65} +5.79331 q^{66} +2.75239 q^{67} -3.39372 q^{68} +10.1860 q^{69} +13.5486 q^{70} -10.3705 q^{71} +2.84670 q^{72} +1.28405 q^{73} +3.20751 q^{74} +12.0025 q^{75} +8.31722 q^{76} -17.2837 q^{77} -1.32785 q^{78} +4.40043 q^{79} +1.70782 q^{80} -5.12482 q^{81} -0.241970 q^{82} -11.2464 q^{83} +6.36653 q^{84} -11.0144 q^{85} +0.925360 q^{86} -12.7164 q^{87} -12.7396 q^{88} +3.69483 q^{89} +3.33425 q^{90} +3.96149 q^{91} -8.08358 q^{92} -1.42306 q^{93} -8.32483 q^{94} +26.9938 q^{95} +7.69185 q^{96} +7.38065 q^{97} +8.11179 q^{98} -4.25346 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 5 q^{2} - 3 q^{3} + 9 q^{4} - 15 q^{5} - 4 q^{7} - 15 q^{8} + 9 q^{9}+O(q^{10}) $ 8 * q - 5 * q^2 - 3 * q^3 + 9 * q^4 - 15 * q^5 - 4 * q^7 - 15 * q^8 + 9 * q^9	\( 8 q - 5 q^{2} - 3 q^{3} + 9 q^{4} - 15 q^{5} - 4 q^{7} - 15 q^{8} + 9 q^{9} + 9 q^{10} - 5 q^{11} - 9 q^{12} - 8 q^{13} + 2 q^{14} + 2 q^{15} + 3 q^{16} - 11 q^{17} - 30 q^{18} - 9 q^{19} - 31 q^{20} - 16 q^{21} - 2 q^{22} + 13 q^{24} + 19 q^{25} + 5 q^{26} - 9 q^{27} + 16 q^{28} - 12 q^{29} - 7 q^{30} - 8 q^{31} - 25 q^{32} - 14 q^{33} + 22 q^{34} + 7 q^{35} + 37 q^{36} - 9 q^{37} - 9 q^{38} + 3 q^{39} + 55 q^{40} - 25 q^{41} - 3 q^{42} + 7 q^{43} - 26 q^{44} - 45 q^{45} + 5 q^{46} - 17 q^{47} - 9 q^{48} - 11 q^{50} - 10 q^{51} - 9 q^{52} - 15 q^{53} + 54 q^{54} + 7 q^{55} - 14 q^{56} - 7 q^{57} - 5 q^{58} - 15 q^{59} + 61 q^{60} + 11 q^{61} + 5 q^{62} - 21 q^{63} + 47 q^{64} + 15 q^{65} + 83 q^{66} + 18 q^{67} - 16 q^{68} - 15 q^{69} - 24 q^{70} - 7 q^{71} - 21 q^{72} + 24 q^{73} + 48 q^{74} - 17 q^{75} - 3 q^{76} - 49 q^{77} + 33 q^{79} - 16 q^{80} + 20 q^{81} - q^{82} - 13 q^{83} - 6 q^{84} + q^{85} + 19 q^{86} + 18 q^{87} + 37 q^{88} - 23 q^{89} + 117 q^{90} + 4 q^{91} + 22 q^{92} + 3 q^{93} + 10 q^{94} + 43 q^{95} + 46 q^{96} - 17 q^{97} + 52 q^{98} + 51 q^{99}+O(q^{100}) \) 8 * q - 5 * q^2 - 3 * q^3 + 9 * q^4 - 15 * q^5 - 4 * q^7 - 15 * q^8 + 9 * q^9 + 9 * q^10 - 5 * q^11 - 9 * q^12 - 8 * q^13 + 2 * q^14 + 2 * q^15 + 3 * q^16 - 11 * q^17 - 30 * q^18 - 9 * q^19 - 31 * q^20 - 16 * q^21 - 2 * q^22 + 13 * q^24 + 19 * q^25 + 5 * q^26 - 9 * q^27 + 16 * q^28 - 12 * q^29 - 7 * q^30 - 8 * q^31 - 25 * q^32 - 14 * q^33 + 22 * q^34 + 7 * q^35 + 37 * q^36 - 9 * q^37 - 9 * q^38 + 3 * q^39 + 55 * q^40 - 25 * q^41 - 3 * q^42 + 7 * q^43 - 26 * q^44 - 45 * q^45 + 5 * q^46 - 17 * q^47 - 9 * q^48 - 11 * q^50 - 10 * q^51 - 9 * q^52 - 15 * q^53 + 54 * q^54 + 7 * q^55 - 14 * q^56 - 7 * q^57 - 5 * q^58 - 15 * q^59 + 61 * q^60 + 11 * q^61 + 5 * q^62 - 21 * q^63 + 47 * q^64 + 15 * q^65 + 83 * q^66 + 18 * q^67 - 16 * q^68 - 15 * q^69 - 24 * q^70 - 7 * q^71 - 21 * q^72 + 24 * q^73 + 48 * q^74 - 17 * q^75 - 3 * q^76 - 49 * q^77 + 33 * q^79 - 16 * q^80 + 20 * q^81 - q^82 - 13 * q^83 - 6 * q^84 + q^85 + 19 * q^86 + 18 * q^87 + 37 * q^88 - 23 * q^89 + 117 * q^90 + 4 * q^91 + 22 * q^92 + 3 * q^93 + 10 * q^94 + 43 * q^95 + 46 * q^96 - 17 * q^97 + 52 * q^98 + 51 * 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 $

$2$	0.933096	0.659798	0.329899	−	0.944016i	$-0.392985\pi$
$2$	0.933096	0.659798	0.329899	+	0.944016i	$0.392985\pi$
$3$	1.42306	0.821602	0.410801	−	0.911725i	$-0.365249\pi$
$3$	1.42306	0.821602	0.410801	+	0.911725i	$0.365249\pi$
$4$	−1.12933	−0.564666
$5$	−3.66529	−1.63917	−0.819583	−	0.572960i	$-0.805795\pi$
$5$	−3.66529	−1.63917	−0.819583	+	0.572960i	$0.805795\pi$
$6$	1.32785	0.542092
$7$	−3.96149	−1.49730	−0.748652	−	0.662964i	$-0.769298\pi$
$7$	−3.96149	−1.49730	−0.748652	+	0.662964i	$0.769298\pi$
$8$	−2.91997	−1.03236
$9$	−0.974910	−0.324970
$10$	−3.42006	−1.08152
$11$	4.36293	1.31547	0.657737	−	0.753248i	$-0.271514\pi$
$11$	4.36293	1.31547	0.657737	+	0.753248i	$0.271514\pi$
$12$	−1.60710	−0.463931
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	−3.69645	−0.987918
$15$	−5.21591	−1.34674
$16$	−0.465945	−0.116486
$17$	3.00507	0.728836	0.364418	−	0.931235i	$-0.381268\pi$
$17$	3.00507	0.728836	0.364418	+	0.931235i	$0.381268\pi$
$18$	−0.909684	−0.214415
$19$	−7.36472	−1.68958	−0.844792	−	0.535095i	$-0.820276\pi$
$19$	−7.36472	−1.68958	−0.844792	+	0.535095i	$0.820276\pi$
$20$	4.13933	0.925581
$21$	−5.63743	−1.23019
$22$	4.07103	0.867947
$23$	7.15784	1.49251	0.746256	−	0.665659i	$-0.231849\pi$
$23$	7.15784	1.49251	0.746256	+	0.665659i	$0.231849\pi$
$24$	−4.15528	−0.848193
$25$	8.43433	1.68687
$26$	−0.933096	−0.182995
$27$	−5.65652	−1.08860
$28$	4.47384	0.845476
$29$	−8.93598	−1.65937	−0.829685	−	0.558232i	$-0.811480\pi$
$29$	−8.93598	−1.65937	−0.829685	+	0.558232i	$0.811480\pi$
$30$	−4.86694	−0.888579
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	5.40516	0.955507
$33$	6.20870	1.08080
$34$	2.80402	0.480885
$35$	14.5200	2.45433
$36$	1.10100	0.183499
$37$	3.43749	0.565120	0.282560	−	0.959250i	$-0.408816\pi$
$37$	3.43749	0.565120	0.282560	+	0.959250i	$0.408816\pi$
$38$	−6.87199	−1.11478
$39$	−1.42306	−0.227871
$40$	10.7025	1.69222
$41$	−0.259319	−0.0404989	−0.0202494	−	0.999795i	$-0.506446\pi$
$41$	−0.259319	−0.0404989	−0.0202494	+	0.999795i	$0.506446\pi$
$42$	−5.26026	−0.811676
$43$	0.991709	0.151234	0.0756171	−	0.997137i	$-0.475907\pi$
$43$	0.991709	0.151234	0.0756171	+	0.997137i	$0.475907\pi$
$44$	−4.92720	−0.742803
$45$	3.57332	0.532680
$46$	6.67895	0.984758
$47$	−8.92173	−1.30137	−0.650684	−	0.759349i	$-0.725518\pi$
$47$	−8.92173	−1.30137	−0.650684	+	0.759349i	$0.725518\pi$
$48$	−0.663066	−0.0957054
$49$	8.69341	1.24192
$50$	7.87003	1.11299
$51$	4.27638	0.598814
$52$	1.12933	0.156610
$53$	−6.09433	−0.837120	−0.418560	−	0.908189i	$-0.637465\pi$
$53$	−6.09433	−0.837120	−0.418560	+	0.908189i	$0.637465\pi$
$54$	−5.27808	−0.718255
$55$	−15.9914	−2.15628
$56$	11.5674	1.54576
$57$	−10.4804	−1.38817
$58$	−8.33812	−1.09485
$59$	−8.53935	−1.11173	−0.555864	−	0.831273i	$-0.687612\pi$
$59$	−8.53935	−1.11173	−0.555864	+	0.831273i	$0.687612\pi$
$60$	5.89049	0.760460
$61$	−2.95055	−0.377780	−0.188890	−	0.981998i	$-0.560489\pi$
$61$	−2.95055	−0.377780	−0.188890	+	0.981998i	$0.560489\pi$
$62$	−0.933096	−0.118503
$63$	3.86210	0.486578
$64$	5.97543	0.746928
$65$	3.66529	0.454623
$66$	5.79331	0.713107
$67$	2.75239	0.336258	0.168129	−	0.985765i	$-0.446228\pi$
$67$	2.75239	0.336258	0.168129	+	0.985765i	$0.446228\pi$
$68$	−3.39372	−0.411549
$69$	10.1860	1.22625
$70$	13.5486	1.61936
$71$	−10.3705	−1.23075	−0.615376	−	0.788233i	$-0.710996\pi$
$71$	−10.3705	−1.23075	−0.615376	+	0.788233i	$0.710996\pi$
$72$	2.84670	0.335487
$73$	1.28405	0.150287	0.0751434	−	0.997173i	$-0.476059\pi$
$73$	1.28405	0.150287	0.0751434	+	0.997173i	$0.476059\pi$
$74$	3.20751	0.372865
$75$	12.0025	1.38593
$76$	8.31722	0.954051
$77$	−17.2837	−1.96966
$78$	−1.32785	−0.150349
$79$	4.40043	0.495087	0.247543	−	0.968877i	$-0.420377\pi$
$79$	4.40043	0.495087	0.247543	+	0.968877i	$0.420377\pi$
$80$	1.70782	0.190940
$81$	−5.12482	−0.569425
$82$	−0.241970	−0.0267211
$83$	−11.2464	−1.23446	−0.617228	−	0.786785i	$-0.711744\pi$
$83$	−11.2464	−1.23446	−0.617228	+	0.786785i	$0.711744\pi$
$84$	6.36653	0.694645
$85$	−11.0144	−1.19468
$86$	0.925360	0.0997841
$87$	−12.7164	−1.36334
$88$	−12.7396	−1.35805
$89$	3.69483	0.391652	0.195826	−	0.980639i	$-0.437261\pi$
$89$	3.69483	0.391652	0.195826	+	0.980639i	$0.437261\pi$
$90$	3.33425	0.351461
$91$	3.96149	0.415277
$92$	−8.08358	−0.842771
$93$	−1.42306	−0.147564
$94$	−8.32483	−0.858640
$95$	26.9938	2.76951
$96$	7.69185	0.785046
$97$	7.38065	0.749391	0.374696	−	0.927148i	$-0.377747\pi$
$97$	7.38065	0.749391	0.374696	+	0.927148i	$0.377747\pi$
$98$	8.11179	0.819414
$99$	−4.25346	−0.427489
$100$	−9.52515	−0.952515
$101$	4.45365	0.443155	0.221577	−	0.975143i	$-0.428879\pi$
$101$	4.45365	0.443155	0.221577	+	0.975143i	$0.428879\pi$
$102$	3.99028	0.395096
$103$	−9.44393	−0.930538	−0.465269	−	0.885169i	$-0.654042\pi$
$103$	−9.44393	−0.930538	−0.465269	+	0.885169i	$0.654042\pi$
$104$	2.91997	0.286326
$105$	20.6628	2.01648
$106$	−5.68659	−0.552331
$107$	−1.42974	−0.138218	−0.0691091	−	0.997609i	$-0.522016\pi$
$107$	−1.42974	−0.138218	−0.0691091	+	0.997609i	$0.522016\pi$
$108$	6.38809	0.614694
$109$	1.92198	0.184092	0.0920461	−	0.995755i	$-0.470659\pi$
$109$	1.92198	0.184092	0.0920461	+	0.995755i	$0.470659\pi$
$110$	−14.9215	−1.42271
$111$	4.89175	0.464304
$112$	1.84584	0.174415
$113$	−3.67214	−0.345445	−0.172723	−	0.984970i	$-0.555256\pi$
$113$	−3.67214	−0.345445	−0.172723	+	0.984970i	$0.555256\pi$
$114$	−9.77924	−0.915910
$115$	−26.2355	−2.44648
$116$	10.0917	0.936989
$117$	0.974910	0.0901304
$118$	−7.96803	−0.733516
$119$	−11.9046	−1.09129
$120$	15.2303	1.39033
$121$	8.03518	0.730471
$122$	−2.75315	−0.249259
$123$	−0.369026	−0.0332740
$124$	1.12933	0.101417
$125$	−12.5878	−1.12589
$126$	3.60371	0.321044
$127$	4.57665	0.406112	0.203056	−	0.979167i	$-0.434913\pi$
$127$	4.57665	0.406112	0.203056	+	0.979167i	$0.434913\pi$
$128$	−5.23468	−0.462685
$129$	1.41126	0.124254
$130$	3.42006	0.299959
$131$	19.9891	1.74646	0.873230	−	0.487309i	$-0.162022\pi$
$131$	19.9891	1.74646	0.873230	+	0.487309i	$0.162022\pi$
$132$	−7.01168	−0.610289
$133$	29.1753	2.52982
$134$	2.56824	0.221862
$135$	20.7328	1.78439
$136$	−8.77470	−0.752425
$137$	−7.56535	−0.646352	−0.323176	−	0.946339i	$-0.604751\pi$
$137$	−7.56535	−0.646352	−0.323176	+	0.946339i	$0.604751\pi$
$138$	9.50453	0.809079
$139$	11.2700	0.955907	0.477953	−	0.878385i	$-0.341379\pi$
$139$	11.2700	0.955907	0.477953	+	0.878385i	$0.341379\pi$
$140$	−16.3979	−1.38588
$141$	−12.6961	−1.06921
$142$	−9.67668	−0.812049
$143$	−4.36293	−0.364847
$144$	0.454254	0.0378545
$145$	32.7529	2.71998
$146$	1.19814	0.0991590
$147$	12.3712	1.02036
$148$	−3.88207	−0.319104
$149$	3.68395	0.301801	0.150900	−	0.988549i	$-0.451783\pi$
$149$	3.68395	0.301801	0.150900	+	0.988549i	$0.451783\pi$
$150$	11.1995	0.914436
$151$	16.4980	1.34259	0.671293	−	0.741192i	$-0.265739\pi$
$151$	16.4980	1.34259	0.671293	+	0.741192i	$0.265739\pi$
$152$	21.5048	1.74427
$153$	−2.92967	−0.236850
$154$	−16.1274	−1.29958
$155$	3.66529	0.294403
$156$	1.60710	0.128671
$157$	10.8873	0.868898	0.434449	−	0.900696i	$-0.356943\pi$
$157$	10.8873	0.868898	0.434449	+	0.900696i	$0.356943\pi$
$158$	4.10602	0.326657
$159$	−8.67257	−0.687780
$160$	−19.8115	−1.56623
$161$	−28.3557	−2.23474
$162$	−4.78195	−0.375706
$163$	−0.966419	−0.0756958	−0.0378479	−	0.999284i	$-0.512050\pi$
$163$	−0.966419	−0.0756958	−0.0378479	+	0.999284i	$0.512050\pi$
$164$	0.292858	0.0228683
$165$	−22.7567	−1.77160
$166$	−10.4940	−0.814492
$167$	17.8864	1.38409	0.692045	−	0.721854i	$-0.256710\pi$
$167$	17.8864	1.38409	0.692045	+	0.721854i	$0.256710\pi$
$168$	16.4611	1.27000
$169$	1.00000	0.0769231
$170$	−10.2775	−0.788251
$171$	7.17994	0.549064
$172$	−1.11997	−0.0853968
$173$	−16.1876	−1.23072	−0.615362	−	0.788245i	$-0.710990\pi$
$173$	−16.1876	−1.23072	−0.615362	+	0.788245i	$0.710990\pi$
$174$	−11.8656	−0.899531
$175$	−33.4125	−2.52575
$176$	−2.03289	−0.153235
$177$	−12.1520	−0.913398
$178$	3.44763	0.258411
$179$	10.3778	0.775670	0.387835	−	0.921729i	$-0.373223\pi$
$179$	10.3778	0.775670	0.387835	+	0.921729i	$0.373223\pi$
$180$	−4.03547	−0.300786
$181$	−24.6080	−1.82910	−0.914549	−	0.404475i	$-0.867454\pi$
$181$	−24.6080	−1.82910	−0.914549	+	0.404475i	$0.867454\pi$
$182$	3.69645	0.273999
$183$	−4.19881	−0.310385
$184$	−20.9007	−1.54082
$185$	−12.5994	−0.926326
$186$	−1.32785	−0.0973626
$187$	13.1109	0.958765
$188$	10.0756	0.734838
$189$	22.4083	1.62996
$190$	25.1878	1.82732
$191$	−19.6692	−1.42321	−0.711607	−	0.702578i	$-0.752032\pi$
$191$	−19.6692	−1.42321	−0.711607	+	0.702578i	$0.752032\pi$
$192$	8.50337	0.613678
$193$	−15.8625	−1.14181	−0.570903	−	0.821017i	$-0.693407\pi$
$193$	−15.8625	−1.14181	−0.570903	+	0.821017i	$0.693407\pi$
$194$	6.88685	0.494447
$195$	5.21591	0.373519
$196$	−9.81775	−0.701268
$197$	−1.92478	−0.137135	−0.0685676	−	0.997646i	$-0.521843\pi$
$197$	−1.92478	−0.137135	−0.0685676	+	0.997646i	$0.521843\pi$
$198$	−3.96889	−0.282057
$199$	−9.55856	−0.677588	−0.338794	−	0.940860i	$-0.610019\pi$
$199$	−9.55856	−0.677588	−0.338794	+	0.940860i	$0.610019\pi$
$200$	−24.6280	−1.74146
$201$	3.91680	0.276270
$202$	4.15568	0.292393
$203$	35.3998	2.48458
$204$	−4.82946	−0.338130
$205$	0.950480	0.0663844
$206$	−8.81209	−0.613967
$207$	−6.97825	−0.485022
$208$	0.465945	0.0323075
$209$	−32.1318	−2.22260
$210$	19.2804	1.33047
$211$	−0.828736	−0.0570525	−0.0285263	−	0.999593i	$-0.509081\pi$
$211$	−0.828736	−0.0570525	−0.0285263	+	0.999593i	$0.509081\pi$
$212$	6.88252	0.472693
$213$	−14.7578	−1.01119
$214$	−1.33409	−0.0911962
$215$	−3.63490	−0.247898
$216$	16.5169	1.12383
$217$	3.96149	0.268924
$218$	1.79339	0.121464
$219$	1.82728	0.123476
$220$	18.0596	1.21758
$221$	−3.00507	−0.202143
$222$	4.56447	0.306347
$223$	−23.3844	−1.56594	−0.782968	−	0.622061i	$-0.786295\pi$
$223$	−23.3844	−1.56594	−0.782968	+	0.622061i	$0.786295\pi$
$224$	−21.4125	−1.43068
$225$	−8.22271	−0.548180
$226$	−3.42646	−0.227924
$227$	−21.7301	−1.44228	−0.721140	−	0.692790i	$-0.756381\pi$
$227$	−21.7301	−1.44228	−0.721140	+	0.692790i	$0.756381\pi$
$228$	11.8359	0.783850
$229$	−0.429856	−0.0284057	−0.0142028	−	0.999899i	$-0.504521\pi$
$229$	−0.429856	−0.0284057	−0.0142028	+	0.999899i	$0.504521\pi$
$230$	−24.4803	−1.61418
$231$	−24.5957	−1.61828
$232$	26.0928	1.71307
$233$	27.2836	1.78741	0.893706	−	0.448654i	$-0.148096\pi$
$233$	27.2836	1.78741	0.893706	+	0.448654i	$0.148096\pi$
$234$	0.909684	0.0594679
$235$	32.7007	2.13316
$236$	9.64376	0.627755
$237$	6.26206	0.406764
$238$	−11.1081	−0.720031
$239$	14.5543	0.941438	0.470719	−	0.882283i	$-0.343995\pi$
$239$	14.5543	0.941438	0.470719	+	0.882283i	$0.343995\pi$
$240$	2.43033	0.156877
$241$	5.28840	0.340656	0.170328	−	0.985387i	$-0.445517\pi$
$241$	5.28840	0.340656	0.170328	+	0.985387i	$0.445517\pi$
$242$	7.49759	0.481963
$243$	9.67665	0.620758
$244$	3.33216	0.213319
$245$	−31.8639	−2.03571
$246$	−0.344337	−0.0219541
$247$	7.36472	0.468606
$248$	2.91997	0.185418
$249$	−16.0043	−1.01423
$250$	−11.7456	−0.742858
$251$	−8.53799	−0.538913	−0.269457	−	0.963013i	$-0.586844\pi$
$251$	−8.53799	−0.538913	−0.269457	+	0.963013i	$0.586844\pi$
$252$	−4.36159	−0.274754
$253$	31.2292	1.96336
$254$	4.27046	0.267952
$255$	−15.6742	−0.981555
$256$	−16.8353	−1.05221
$257$	−23.1723	−1.44545	−0.722723	−	0.691138i	$-0.757110\pi$
$257$	−23.1723	−1.44545	−0.722723	+	0.691138i	$0.757110\pi$
$258$	1.31684	0.0819828
$259$	−13.6176	−0.846156
$260$	−4.13933	−0.256710
$261$	8.71177	0.539245
$262$	18.6518	1.15231
$263$	−23.6575	−1.45878	−0.729392	−	0.684096i	$-0.760197\pi$
$263$	−23.6575	−1.45878	−0.729392	+	0.684096i	$0.760197\pi$
$264$	−18.1292	−1.11578
$265$	22.3375	1.37218
$266$	27.2233	1.66917
$267$	5.25796	0.321782
$268$	−3.10836	−0.189873
$269$	−8.14827	−0.496809	−0.248404	−	0.968656i	$-0.579906\pi$
$269$	−8.14827	−0.496809	−0.248404	+	0.968656i	$0.579906\pi$
$270$	19.3457	1.17734
$271$	16.4892	1.00165	0.500824	−	0.865549i	$-0.333031\pi$
$271$	16.4892	1.00165	0.500824	+	0.865549i	$0.333031\pi$
$272$	−1.40020	−0.0848995
$273$	5.63743	0.341193
$274$	−7.05920	−0.426462
$275$	36.7984	2.21903
$276$	−11.5034	−0.692423
$277$	−22.8067	−1.37032	−0.685160	−	0.728393i	$-0.740268\pi$
$277$	−22.8067	−1.37032	−0.685160	+	0.728393i	$0.740268\pi$
$278$	10.5160	0.630706
$279$	0.974910	0.0583663
$280$	−42.3979	−2.53376
$281$	−14.3315	−0.854946	−0.427473	−	0.904028i	$-0.640596\pi$
$281$	−14.3315	−0.854946	−0.427473	+	0.904028i	$0.640596\pi$
$282$	−11.8467	−0.705461
$283$	20.2231	1.20214	0.601069	−	0.799197i	$-0.294742\pi$
$283$	20.2231	1.20214	0.601069	+	0.799197i	$0.294742\pi$
$284$	11.7117	0.694964
$285$	38.4137	2.27543
$286$	−4.07103	−0.240725
$287$	1.02729	0.0606391
$288$	−5.26954	−0.310511
$289$	−7.96956	−0.468797
$290$	30.5616	1.79464
$291$	10.5031	0.615702
$292$	−1.45012	−0.0848618
$293$	15.5642	0.909272	0.454636	−	0.890677i	$-0.349769\pi$
$293$	15.5642	0.909272	0.454636	+	0.890677i	$0.349769\pi$
$294$	11.5435	0.673233
$295$	31.2991	1.82231
$296$	−10.0374	−0.583410
$297$	−24.6790	−1.43202
$298$	3.43748	0.199128
$299$	−7.15784	−0.413949
$300$	−13.5548	−0.782589
$301$	−3.92865	−0.226443
$302$	15.3942	0.885836
$303$	6.33780	0.364097
$304$	3.43156	0.196813
$305$	10.8146	0.619244
$306$	−2.73366	−0.156273
$307$	−8.37006	−0.477705	−0.238852	−	0.971056i	$-0.576771\pi$
$307$	−8.37006	−0.477705	−0.238852	+	0.971056i	$0.576771\pi$
$308$	19.5191	1.11220
$309$	−13.4392	−0.764532
$310$	3.42006	0.194247
$311$	12.0451	0.683015	0.341507	−	0.939879i	$-0.389063\pi$
$311$	12.0451	0.683015	0.341507	+	0.939879i	$0.389063\pi$
$312$	4.15528	0.235246
$313$	11.9355	0.674632	0.337316	−	0.941392i	$-0.390481\pi$
$313$	11.9355	0.674632	0.337316	+	0.941392i	$0.390481\pi$
$314$	10.1589	0.573298
$315$	−14.1557	−0.797583
$316$	−4.96954	−0.279559
$317$	−26.8353	−1.50722	−0.753609	−	0.657323i	$-0.771689\pi$
$317$	−26.8353	−1.50722	−0.753609	+	0.657323i	$0.771689\pi$
$318$	−8.09234	−0.453796
$319$	−38.9871	−2.18286
$320$	−21.9016	−1.22434
$321$	−2.03460	−0.113560
$322$	−26.4586	−1.47448
$323$	−22.1315	−1.23143
$324$	5.78763	0.321535
$325$	−8.43433	−0.467852
$326$	−0.901761	−0.0499439
$327$	2.73508	0.151250
$328$	0.757204	0.0418096
$329$	35.3433	1.94854
$330$	−21.2342	−1.16890
$331$	17.8840	0.982995	0.491497	−	0.870879i	$-0.336450\pi$
$331$	17.8840	0.982995	0.491497	+	0.870879i	$0.336450\pi$
$332$	12.7009	0.697055
$333$	−3.35124	−0.183647
$334$	16.6897	0.913221
$335$	−10.0883	−0.551182
$336$	2.62673	0.143300
$337$	7.70516	0.419727	0.209863	−	0.977731i	$-0.432698\pi$
$337$	7.70516	0.419727	0.209863	+	0.977731i	$0.432698\pi$
$338$	0.933096	0.0507537
$339$	−5.22566	−0.283819
$340$	12.4390	0.674597
$341$	−4.36293	−0.236266
$342$	6.69957	0.362271
$343$	−6.70845	−0.362222
$344$	−2.89576	−0.156129
$345$	−37.3347	−2.01003
$346$	−15.1046	−0.812029
$347$	−9.72809	−0.522231	−0.261116	−	0.965307i	$-0.584090\pi$
$347$	−9.72809	−0.522231	−0.261116	+	0.965307i	$0.584090\pi$
$348$	14.3610	0.769833
$349$	9.55130	0.511269	0.255635	−	0.966773i	$-0.417716\pi$
$349$	9.55130	0.511269	0.255635	+	0.966773i	$0.417716\pi$
$350$	−31.1771	−1.66648
$351$	5.65652	0.301923
$352$	23.5824	1.25694
$353$	−3.68455	−0.196109	−0.0980545	−	0.995181i	$-0.531262\pi$
$353$	−3.68455	−0.196109	−0.0980545	+	0.995181i	$0.531262\pi$
$354$	−11.3390	−0.602659
$355$	38.0109	2.01741
$356$	−4.17269	−0.221152
$357$	−16.9409	−0.896605
$358$	9.68345	0.511786
$359$	20.2659	1.06959	0.534796	−	0.844981i	$-0.320389\pi$
$359$	20.2659	1.06959	0.534796	+	0.844981i	$0.320389\pi$
$360$	−10.4340	−0.549919
$361$	35.2392	1.85469
$362$	−22.9616	−1.20684
$363$	11.4345	0.600156
$364$	−4.47384	−0.234493
$365$	−4.70641	−0.246345
$366$	−3.91789	−0.204791
$367$	10.2659	0.535875	0.267937	−	0.963436i	$-0.413658\pi$
$367$	10.2659	0.535875	0.267937	+	0.963436i	$0.413658\pi$
$368$	−3.33516	−0.173857
$369$	0.252813	0.0131609
$370$	−11.7564	−0.611188
$371$	24.1426	1.25342
$372$	1.60710	0.0833244
$373$	−28.6917	−1.48560	−0.742799	−	0.669514i	$-0.766502\pi$
$373$	−28.6917	−1.48560	−0.742799	+	0.669514i	$0.766502\pi$
$374$	12.2337	0.632592
$375$	−17.9131	−0.925030
$376$	26.0511	1.34349
$377$	8.93598	0.460226
$378$	20.9091	1.07545
$379$	36.1299	1.85587	0.927934	−	0.372746i	$-0.121584\pi$
$379$	36.1299	1.85587	0.927934	+	0.372746i	$0.121584\pi$
$380$	−30.4850	−1.56385
$381$	6.51284	0.333663
$382$	−18.3533	−0.939035
$383$	−21.3211	−1.08946	−0.544729	−	0.838612i	$-0.683367\pi$
$383$	−21.3211	−1.08946	−0.544729	+	0.838612i	$0.683367\pi$
$384$	−7.44925	−0.380143
$385$	63.3498	3.22860
$386$	−14.8012	−0.753362
$387$	−0.966827	−0.0491466
$388$	−8.33520	−0.423156
$389$	15.3335	0.777438	0.388719	−	0.921356i	$-0.372918\pi$
$389$	15.3335	0.777438	0.388719	+	0.921356i	$0.372918\pi$
$390$	4.86694	0.246447
$391$	21.5098	1.08780
$392$	−25.3845	−1.28211
$393$	28.4457	1.43489
$394$	−1.79601	−0.0904816
$395$	−16.1288	−0.811529
$396$	4.80357	0.241389
$397$	−4.08605	−0.205073	−0.102536	−	0.994729i	$-0.532696\pi$
$397$	−4.08605	−0.205073	−0.102536	+	0.994729i	$0.532696\pi$
$398$	−8.91905	−0.447072
$399$	41.5181	2.07850
$400$	−3.92993	−0.196497
$401$	−28.7267	−1.43454	−0.717270	−	0.696795i	$-0.754609\pi$
$401$	−28.7267	−1.43454	−0.717270	+	0.696795i	$0.754609\pi$
$402$	3.65475	0.182283
$403$	1.00000	0.0498135
$404$	−5.02965	−0.250235
$405$	18.7839	0.933382
$406$	33.0314	1.63932
$407$	14.9975	0.743401
$408$	−12.4869	−0.618194
$409$	−8.82693	−0.436464	−0.218232	−	0.975897i	$-0.570029\pi$
$409$	−8.82693	−0.436464	−0.218232	+	0.975897i	$0.570029\pi$
$410$	0.886889	0.0438003
$411$	−10.7659	−0.531044
$412$	10.6653	0.525443
$413$	33.8285	1.66459
$414$	−6.51137	−0.320017
$415$	41.2214	2.02348
$416$	−5.40516	−0.265010
$417$	16.0378	0.785375
$418$	−29.9820	−1.46647
$419$	−25.5923	−1.25027	−0.625133	−	0.780519i	$-0.714955\pi$
$419$	−25.5923	−1.25027	−0.625133	+	0.780519i	$0.714955\pi$
$420$	−23.3351	−1.13864
$421$	−13.2992	−0.648165	−0.324082	−	0.946029i	$-0.605056\pi$
$421$	−13.2992	−0.648165	−0.324082	+	0.946029i	$0.605056\pi$
$422$	−0.773290	−0.0376432
$423$	8.69788	0.422905
$424$	17.7952	0.864213
$425$	25.3457	1.22945
$426$	−13.7705	−0.667181
$427$	11.6886	0.565651
$428$	1.61465	0.0780471
$429$	−6.20870	−0.299759
$430$	−3.39171	−0.163563
$431$	8.59692	0.414099	0.207050	−	0.978330i	$-0.433614\pi$
$431$	8.59692	0.414099	0.207050	+	0.978330i	$0.433614\pi$
$432$	2.63563	0.126807
$433$	−4.08683	−0.196400	−0.0982002	−	0.995167i	$-0.531309\pi$
$433$	−4.08683	−0.196400	−0.0982002	+	0.995167i	$0.531309\pi$
$434$	3.69645	0.177435
$435$	46.6093	2.23474
$436$	−2.17055	−0.103951
$437$	−52.7155	−2.52173
$438$	1.70502	0.0814692
$439$	34.6191	1.65228	0.826140	−	0.563464i	$-0.190532\pi$
$439$	34.6191	1.65228	0.826140	+	0.563464i	$0.190532\pi$
$440$	46.6943	2.22607
$441$	−8.47529	−0.403585
$442$	−2.80402	−0.133374
$443$	40.0171	1.90127	0.950635	−	0.310310i	$-0.100433\pi$
$443$	40.0171	1.90127	0.950635	+	0.310310i	$0.100433\pi$
$444$	−5.52441	−0.262177
$445$	−13.5426	−0.641982
$446$	−21.8199	−1.03320
$447$	5.24247	0.247960
$448$	−23.6716	−1.11838
$449$	3.71533	0.175337	0.0876686	−	0.996150i	$-0.472058\pi$
$449$	3.71533	0.175337	0.0876686	+	0.996150i	$0.472058\pi$
$450$	−7.67257	−0.361689
$451$	−1.13139	−0.0532752
$452$	4.14706	0.195061
$453$	23.4776	1.10307
$454$	−20.2763	−0.951614
$455$	−14.5200	−0.680708
$456$	30.6025	1.43309
$457$	−40.7792	−1.90757	−0.953785	−	0.300490i	$-0.902850\pi$
$457$	−40.7792	−1.90757	−0.953785	+	0.300490i	$0.902850\pi$
$458$	−0.401097	−0.0187420
$459$	−16.9982	−0.793410
$460$	29.6286	1.38144
$461$	−30.0007	−1.39727	−0.698635	−	0.715478i	$-0.746209\pi$
$461$	−30.0007	−1.39727	−0.698635	+	0.715478i	$0.746209\pi$
$462$	−22.9502	−1.06774
$463$	29.1739	1.35583	0.677914	−	0.735142i	$-0.262884\pi$
$463$	29.1739	1.35583	0.677914	+	0.735142i	$0.262884\pi$
$464$	4.16368	0.193294
$465$	5.21591	0.241882
$466$	25.4583	1.17933
$467$	−28.3103	−1.31004	−0.655022	−	0.755610i	$-0.727341\pi$
$467$	−28.3103	−1.31004	−0.655022	+	0.755610i	$0.727341\pi$
$468$	−1.10100	−0.0508936
$469$	−10.9036	−0.503480
$470$	30.5129	1.40745
$471$	15.4932	0.713889
$472$	24.9346	1.14771
$473$	4.32676	0.198945
$474$	5.84310	0.268382
$475$	−62.1165	−2.85010
$476$	13.4442	0.616214
$477$	5.94142	0.272039
$478$	13.5805	0.621159
$479$	7.03857	0.321601	0.160800	−	0.986987i	$-0.448592\pi$
$479$	7.03857	0.321601	0.160800	+	0.986987i	$0.448592\pi$
$480$	−28.1928	−1.28682
$481$	−3.43749	−0.156736
$482$	4.93459	0.224764
$483$	−40.3518	−1.83607
$484$	−9.07438	−0.412472
$485$	−27.0522	−1.22838
$486$	9.02925	0.409575
$487$	−15.4960	−0.702190	−0.351095	−	0.936340i	$-0.614191\pi$
$487$	−15.4960	−0.702190	−0.351095	+	0.936340i	$0.614191\pi$
$488$	8.61552	0.390006
$489$	−1.37527	−0.0621918
$490$	−29.7320	−1.34316
$491$	15.2454	0.688014	0.344007	−	0.938967i	$-0.388216\pi$
$491$	15.2454	0.688014	0.344007	+	0.938967i	$0.388216\pi$
$492$	0.416753	0.0187887
$493$	−26.8532	−1.20941
$494$	6.87199	0.309186
$495$	15.5902	0.700726
$496$	0.465945	0.0209216
$497$	41.0827	1.84281
$498$	−14.9335	−0.669188
$499$	−5.17163	−0.231514	−0.115757	−	0.993278i	$-0.536929\pi$
$499$	−5.17163	−0.231514	−0.115757	+	0.993278i	$0.536929\pi$
$500$	14.2158	0.635750
$501$	25.4534	1.13717
$502$	−7.96677	−0.355574
$503$	−7.14730	−0.318682	−0.159341	−	0.987224i	$-0.550937\pi$
$503$	−7.14730	−0.318682	−0.159341	+	0.987224i	$0.550937\pi$
$504$	−11.2772	−0.502326
$505$	−16.3239	−0.726405
$506$	29.1398	1.29542
$507$	1.42306	0.0632002
$508$	−5.16856	−0.229318
$509$	−8.71061	−0.386091	−0.193045	−	0.981190i	$-0.561836\pi$
$509$	−8.71061	−0.386091	−0.193045	+	0.981190i	$0.561836\pi$
$510$	−14.6255	−0.647628
$511$	−5.08675	−0.225025
$512$	−5.23960	−0.231560
$513$	41.6587	1.83928
$514$	−21.6219	−0.953703
$515$	34.6147	1.52531
$516$	−1.59378	−0.0701622
$517$	−38.9249	−1.71191
$518$	−12.7065	−0.558293
$519$	−23.0359	−1.01116
$520$	−10.7025	−0.469336
$521$	−30.0669	−1.31726	−0.658628	−	0.752469i	$-0.728863\pi$
$521$	−30.0669	−1.31726	−0.658628	+	0.752469i	$0.728863\pi$
$522$	8.12892	0.355793
$523$	0.563068	0.0246213	0.0123106	−	0.999924i	$-0.496081\pi$
$523$	0.563068	0.0246213	0.0123106	+	0.999924i	$0.496081\pi$
$524$	−22.5744	−0.986166
$525$	−47.5479	−2.07516
$526$	−22.0747	−0.962503
$527$	−3.00507	−0.130903
$528$	−2.89291	−0.125898
$529$	28.2347	1.22759
$530$	20.8430	0.905361
$531$	8.32509	0.361278
$532$	−32.9486	−1.42850
$533$	0.259319	0.0112324
$534$	4.90618	0.212311
$535$	5.24041	0.226563
$536$	−8.03688	−0.347140
$537$	14.7681	0.637292
$538$	−7.60312	−0.327794
$539$	37.9288	1.63371
$540$	−23.4142	−1.00759
$541$	6.51391	0.280055	0.140027	−	0.990148i	$-0.455281\pi$
$541$	6.51391	0.280055	0.140027	+	0.990148i	$0.455281\pi$
$542$	15.3860	0.660885
$543$	−35.0186	−1.50279
$544$	16.2429	0.696408
$545$	−7.04460	−0.301758
$546$	5.26026	0.225118
$547$	−28.0124	−1.19772	−0.598862	−	0.800853i	$-0.704380\pi$
$547$	−28.0124	−1.19772	−0.598862	+	0.800853i	$0.704380\pi$
$548$	8.54380	0.364973
$549$	2.87652	0.122767
$550$	34.3364	1.46411
$551$	65.8110	2.80364
$552$	−29.7428	−1.26594
$553$	−17.4323	−0.741295
$554$	−21.2808	−0.904135
$555$	−17.9297	−0.761071
$556$	−12.7275	−0.539768
$557$	2.21416	0.0938169	0.0469084	−	0.998899i	$-0.485063\pi$
$557$	2.21416	0.0938169	0.0469084	+	0.998899i	$0.485063\pi$
$558$	0.909684	0.0385100
$559$	−0.991709	−0.0419448
$560$	−6.76552	−0.285896
$561$	18.6576	0.787723
$562$	−13.3727	−0.564092
$563$	−43.4262	−1.83020	−0.915098	−	0.403232i	$-0.867887\pi$
$563$	−43.4262	−1.83020	−0.915098	+	0.403232i	$0.867887\pi$
$564$	14.3381	0.603745
$565$	13.4594	0.566242
$566$	18.8701	0.793169
$567$	20.3019	0.852601
$568$	30.2815	1.27059
$569$	5.43735	0.227946	0.113973	−	0.993484i	$-0.463642\pi$
$569$	5.43735	0.227946	0.113973	+	0.993484i	$0.463642\pi$
$570$	35.8437	1.50133
$571$	1.74433	0.0729979	0.0364990	−	0.999334i	$-0.488379\pi$
$571$	1.74433	0.0729979	0.0364990	+	0.999334i	$0.488379\pi$
$572$	4.92720	0.206017
$573$	−27.9904	−1.16932
$574$	0.958562	0.0400096
$575$	60.3716	2.51767
$576$	−5.82550	−0.242729
$577$	14.4784	0.602744	0.301372	−	0.953507i	$-0.402555\pi$
$577$	14.4784	0.602744	0.301372	+	0.953507i	$0.402555\pi$
$578$	−7.43636	−0.309312
$579$	−22.5732	−0.938111
$580$	−36.9889	−1.53588
$581$	44.5526	1.84835
$582$	9.80038	0.406239
$583$	−26.5891	−1.10121
$584$	−3.74938	−0.155151
$585$	−3.57332	−0.147739
$586$	14.5229	0.599937
$587$	−11.9448	−0.493015	−0.246508	−	0.969141i	$-0.579283\pi$
$587$	−11.9448	−0.493015	−0.246508	+	0.969141i	$0.579283\pi$
$588$	−13.9712	−0.576163
$589$	7.36472	0.303458
$590$	29.2051	1.20236
$591$	−2.73908	−0.112671
$592$	−1.60168	−0.0658288
$593$	5.50147	0.225918	0.112959	−	0.993600i	$-0.463967\pi$
$593$	5.50147	0.225918	0.112959	+	0.993600i	$0.463967\pi$
$594$	−23.0279	−0.944846
$595$	43.6336	1.78880
$596$	−4.16040	−0.170417
$597$	−13.6024	−0.556708
$598$	−6.67895	−0.273123
$599$	41.3879	1.69106	0.845531	−	0.533926i	$-0.179284\pi$
$599$	41.3879	1.69106	0.845531	+	0.533926i	$0.179284\pi$
$600$	−35.0470	−1.43079
$601$	−36.2163	−1.47729	−0.738646	−	0.674093i	$-0.764535\pi$
$601$	−36.2163	−1.47729	−0.738646	+	0.674093i	$0.764535\pi$
$602$	−3.66580	−0.149407
$603$	−2.68333	−0.109274
$604$	−18.6317	−0.758113
$605$	−29.4512	−1.19736
$606$	5.91378	0.240231
$607$	27.7153	1.12493	0.562465	−	0.826821i	$-0.309853\pi$
$607$	27.7153	1.12493	0.562465	+	0.826821i	$0.309853\pi$
$608$	−39.8075	−1.61441
$609$	50.3759	2.04134
$610$	10.0911	0.408576
$611$	8.92173	0.360934
$612$	3.30857	0.133741
$613$	−9.24961	−0.373588	−0.186794	−	0.982399i	$-0.559810\pi$
$613$	−9.24961	−0.373588	−0.186794	+	0.982399i	$0.559810\pi$
$614$	−7.81007	−0.315189
$615$	1.35259	0.0545416
$616$	50.4679	2.03341
$617$	−29.1884	−1.17508	−0.587541	−	0.809195i	$-0.699904\pi$
$617$	−29.1884	−1.17508	−0.587541	+	0.809195i	$0.699904\pi$
$618$	−12.5401	−0.504437
$619$	−28.1648	−1.13204	−0.566019	−	0.824392i	$-0.691517\pi$
$619$	−28.1648	−1.13204	−0.566019	+	0.824392i	$0.691517\pi$
$620$	−4.13933	−0.166239
$621$	−40.4885	−1.62475
$622$	11.2392	0.450652
$623$	−14.6370	−0.586421
$624$	0.663066	0.0265439
$625$	3.96622	0.158649
$626$	11.1369	0.445121
$627$	−45.7254	−1.82610
$628$	−12.2953	−0.490637
$629$	10.3299	0.411880
$630$	−13.2086	−0.526244
$631$	21.8309	0.869074	0.434537	−	0.900654i	$-0.356912\pi$
$631$	21.8309	0.869074	0.434537	+	0.900654i	$0.356912\pi$
$632$	−12.8491	−0.511110
$633$	−1.17934	−0.0468745
$634$	−25.0399	−0.994460
$635$	−16.7748	−0.665686
$636$	9.79421	0.388366
$637$	−8.69341	−0.344446
$638$	−36.3787	−1.44025
$639$	10.1103	0.399958
$640$	19.1866	0.758417
$641$	14.0275	0.554051	0.277026	−	0.960863i	$-0.410651\pi$
$641$	14.0275	0.554051	0.277026	+	0.960863i	$0.410651\pi$
$642$	−1.89848	−0.0749270
$643$	34.5606	1.36294	0.681468	−	0.731848i	$-0.261342\pi$
$643$	34.5606	1.36294	0.681468	+	0.731848i	$0.261342\pi$
$644$	32.0230	1.26188
$645$	−5.17266	−0.203673
$646$	−20.6508	−0.812496
$647$	31.6370	1.24378	0.621890	−	0.783105i	$-0.286365\pi$
$647$	31.6370	1.24378	0.621890	+	0.783105i	$0.286365\pi$
$648$	14.9643	0.587854
$649$	−37.2566	−1.46245
$650$	−7.87003	−0.308688
$651$	5.63743	0.220948
$652$	1.09141	0.0427428
$653$	17.6243	0.689691	0.344845	−	0.938659i	$-0.387931\pi$
$653$	17.6243	0.689691	0.344845	+	0.938659i	$0.387931\pi$
$654$	2.55210	0.0997948
$655$	−73.2659	−2.86274
$656$	0.120829	0.00471757
$657$	−1.25183	−0.0488387
$658$	32.9787	1.28564
$659$	−19.5738	−0.762486	−0.381243	−	0.924475i	$-0.624504\pi$
$659$	−19.5738	−0.762486	−0.381243	+	0.924475i	$0.624504\pi$
$660$	25.6998	1.00036
$661$	−25.3246	−0.985013	−0.492507	−	0.870309i	$-0.663919\pi$
$661$	−25.3246	−0.985013	−0.492507	+	0.870309i	$0.663919\pi$
$662$	16.6875	0.648578
$663$	−4.27638	−0.166081
$664$	32.8392	1.27441
$665$	−106.936	−4.14679
$666$	−3.12703	−0.121170
$667$	−63.9623	−2.47663
$668$	−20.1997	−0.781549
$669$	−33.2774	−1.28658
$670$	−9.41334	−0.363669
$671$	−12.8731	−0.496959
$672$	−30.4712	−1.17545
$673$	−6.96545	−0.268499	−0.134249	−	0.990948i	$-0.542862\pi$
$673$	−6.96545	−0.268499	−0.134249	+	0.990948i	$0.542862\pi$
$674$	7.18965	0.276935
$675$	−47.7089	−1.83632
$676$	−1.12933	−0.0434358
$677$	−6.73704	−0.258926	−0.129463	−	0.991584i	$-0.541325\pi$
$677$	−6.73704	−0.258926	−0.129463	+	0.991584i	$0.541325\pi$
$678$	−4.87604	−0.187263
$679$	−29.2384	−1.12207
$680$	32.1618	1.23335
$681$	−30.9232	−1.18498
$682$	−4.07103	−0.155888
$683$	5.31103	0.203221	0.101611	−	0.994824i	$-0.467600\pi$
$683$	5.31103	0.203221	0.101611	+	0.994824i	$0.467600\pi$
$684$	−8.10854	−0.310038
$685$	27.7292	1.05948
$686$	−6.25962	−0.238994
$687$	−0.611709	−0.0233381
$688$	−0.462082	−0.0176167
$689$	6.09433	0.232175
$690$	−34.8368	−1.32621
$691$	−24.0844	−0.916216	−0.458108	−	0.888897i	$-0.651473\pi$
$691$	−24.0844	−0.916216	−0.458108	+	0.888897i	$0.651473\pi$
$692$	18.2812	0.694948
$693$	16.8501	0.640081
$694$	−9.07725	−0.344568
$695$	−41.3077	−1.56689
$696$	37.1315	1.40746
$697$	−0.779273	−0.0295171
$698$	8.91228	0.337335
$699$	38.8262	1.46854
$700$	37.7338	1.42620
$701$	−2.88912	−0.109120	−0.0545602	−	0.998510i	$-0.517376\pi$
$701$	−2.88912	−0.109120	−0.0545602	+	0.998510i	$0.517376\pi$
$702$	5.27808	0.199208
$703$	−25.3162	−0.954818
$704$	26.0704	0.982564
$705$	46.5349	1.75261
$706$	−3.43804	−0.129392
$707$	−17.6431	−0.663537
$708$	13.7236	0.515765
$709$	15.7441	0.591283	0.295641	−	0.955299i	$-0.404467\pi$
$709$	15.7441	0.591283	0.295641	+	0.955299i	$0.404467\pi$
$710$	35.4678	1.33108
$711$	−4.29002	−0.160888
$712$	−10.7888	−0.404327
$713$	−7.15784	−0.268063
$714$	−15.8074	−0.591579
$715$	15.9914	0.598044
$716$	−11.7199	−0.437995
$717$	20.7116	0.773488
$718$	18.9100	0.705715
$719$	15.7275	0.586536	0.293268	−	0.956030i	$-0.405257\pi$
$719$	15.7275	0.586536	0.293268	+	0.956030i	$0.405257\pi$
$720$	−1.66497	−0.0620499
$721$	37.4120	1.39330
$722$	32.8815	1.22372
$723$	7.52570	0.279884
$724$	27.7906	1.03283
$725$	−75.3689	−2.79913
$726$	10.6695	0.395982
$727$	26.8866	0.997167	0.498583	−	0.866842i	$-0.333854\pi$
$727$	26.8866	0.997167	0.498583	+	0.866842i	$0.333854\pi$
$728$	−11.5674	−0.428717
$729$	29.1449	1.07944
$730$	−4.39153	−0.162538
$731$	2.98015	0.110225
$732$	4.74185	0.175264
$733$	−33.4511	−1.23554	−0.617772	−	0.786357i	$-0.711965\pi$
$733$	−33.4511	−1.23554	−0.617772	+	0.786357i	$0.711965\pi$
$734$	9.57906	0.353569
$735$	−45.3441	−1.67254
$736$	38.6893	1.42611
$737$	12.0085	0.442338
$738$	0.235899	0.00868355
$739$	−17.7871	−0.654309	−0.327154	−	0.944971i	$-0.606090\pi$
$739$	−17.7871	−0.654309	−0.327154	+	0.944971i	$0.606090\pi$
$740$	14.2289	0.523065
$741$	10.4804	0.385008
$742$	22.5274	0.827006
$743$	30.8605	1.13216	0.566081	−	0.824350i	$-0.308459\pi$
$743$	30.8605	1.13216	0.566081	+	0.824350i	$0.308459\pi$
$744$	4.15528	0.152340
$745$	−13.5027	−0.494702
$746$	−26.7721	−0.980196
$747$	10.9642	0.401161
$748$	−14.8066	−0.541382
$749$	5.66391	0.206955
$750$	−16.7147	−0.610334
$751$	2.13650	0.0779621	0.0389810	−	0.999240i	$-0.487589\pi$
$751$	2.13650	0.0779621	0.0389810	+	0.999240i	$0.487589\pi$
$752$	4.15704	0.151591
$753$	−12.1500	−0.442772
$754$	8.33812	0.303657
$755$	−60.4698	−2.20072
$756$	−25.3064	−0.920384
$757$	42.4487	1.54282	0.771411	−	0.636337i	$-0.219551\pi$
$757$	42.4487	1.54282	0.771411	+	0.636337i	$0.219551\pi$
$758$	33.7126	1.22450
$759$	44.4409	1.61310
$760$	−78.8211	−2.85914
$761$	42.0460	1.52417	0.762084	−	0.647478i	$-0.224176\pi$
$761$	42.0460	1.52417	0.762084	+	0.647478i	$0.224176\pi$
$762$	6.07710	0.220150
$763$	−7.61390	−0.275642
$764$	22.2131	0.803641
$765$	10.7381	0.388236
$766$	−19.8946	−0.718823
$767$	8.53935	0.308338
$768$	−23.9576	−0.864495
$769$	42.6180	1.53684	0.768421	−	0.639944i	$-0.221043\pi$
$769$	42.6180	1.53684	0.768421	+	0.639944i	$0.221043\pi$
$770$	59.1114	2.13023
$771$	−32.9754	−1.18758
$772$	17.9140	0.644739
$773$	28.1921	1.01400	0.506999	−	0.861947i	$-0.330755\pi$
$773$	28.1921	1.01400	0.506999	+	0.861947i	$0.330755\pi$
$774$	−0.902142	−0.0324268
$775$	−8.43433	−0.302970
$776$	−21.5513	−0.773645
$777$	−19.3786	−0.695204
$778$	14.3076	0.512952
$779$	1.90982	0.0684263
$780$	−5.89049	−0.210914
$781$	−45.2458	−1.61902
$782$	20.0707	0.717727
$783$	50.5465	1.80639
$784$	−4.05065	−0.144666
$785$	−39.9049	−1.42427
$786$	26.5425	0.946741
$787$	2.75942	0.0983626	0.0491813	−	0.998790i	$-0.484339\pi$
$787$	2.75942	0.0983626	0.0491813	+	0.998790i	$0.484339\pi$
$788$	2.17372	0.0774356
$789$	−33.6659	−1.19854
$790$	−15.0497	−0.535446
$791$	14.5471	0.517237
$792$	12.4200	0.441325
$793$	2.95055	0.104777
$794$	−3.81268	−0.135307
$795$	31.7875	1.12739
$796$	10.7948	0.382611
$797$	5.02580	0.178023	0.0890114	−	0.996031i	$-0.471629\pi$
$797$	5.02580	0.178023	0.0890114	+	0.996031i	$0.471629\pi$
$798$	38.7404	1.37139
$799$	−26.8104	−0.948484
$800$	45.5889	1.61181
$801$	−3.60213	−0.127275
$802$	−26.8047	−0.946508
$803$	5.60223	0.197698
$804$	−4.42337	−0.156000
$805$	103.932	3.66312
$806$	0.933096	0.0328669
$807$	−11.5954	−0.408179
$808$	−13.0045	−0.457497
$809$	32.7234	1.15049	0.575247	−	0.817980i	$-0.304906\pi$
$809$	32.7234	1.15049	0.575247	+	0.817980i	$0.304906\pi$
$810$	17.5272	0.615844
$811$	53.2821	1.87099	0.935495	−	0.353341i	$-0.114954\pi$
$811$	53.2821	1.87099	0.935495	+	0.353341i	$0.114954\pi$
$812$	−39.9781	−1.40296
$813$	23.4651	0.822956
$814$	13.9941	0.490495
$815$	3.54220	0.124078
$816$	−1.99256	−0.0697536
$817$	−7.30366	−0.255523
$818$	−8.23638	−0.287978
$819$	−3.86210	−0.134953
$820$	−1.07341	−0.0374850
$821$	8.53950	0.298031	0.149015	−	0.988835i	$-0.452390\pi$
$821$	8.53950	0.298031	0.149015	+	0.988835i	$0.452390\pi$
$822$	−10.0456	−0.350382
$823$	−33.1221	−1.15456	−0.577281	−	0.816545i	$-0.695886\pi$
$823$	−33.1221	−1.15456	−0.577281	+	0.816545i	$0.695886\pi$
$824$	27.5760	0.960654
$825$	52.3662	1.82316
$826$	31.5653	1.09830
$827$	−45.2853	−1.57473	−0.787363	−	0.616490i	$-0.788554\pi$
$827$	−45.2853	−1.57473	−0.787363	+	0.616490i	$0.788554\pi$
$828$	7.88076	0.273875
$829$	−16.6087	−0.576845	−0.288422	−	0.957503i	$-0.593131\pi$
$829$	−16.6087	−0.576845	−0.288422	+	0.957503i	$0.593131\pi$
$830$	38.4635	1.33509
$831$	−32.4552	−1.12586
$832$	−5.97543	−0.207161
$833$	26.1243	0.905154
$834$	14.9648	0.518189
$835$	−65.5588	−2.26875
$836$	36.2875	1.25503
$837$	5.65652	0.195518
$838$	−23.8801	−0.824923
$839$	8.05238	0.277999	0.139000	−	0.990292i	$-0.455611\pi$
$839$	8.05238	0.277999	0.139000	+	0.990292i	$0.455611\pi$
$840$	−60.3346	−2.08174
$841$	50.8517	1.75351
$842$	−12.4095	−0.427658
$843$	−20.3945	−0.702425
$844$	0.935918	0.0322156
$845$	−3.66529	−0.126090
$846$	8.11595	0.279032
$847$	−31.8313	−1.09374
$848$	2.83962	0.0975130
$849$	28.7786	0.987680
$850$	23.6500	0.811188
$851$	24.6050	0.843449
$852$	16.6665	0.570984
$853$	23.2057	0.794548	0.397274	−	0.917700i	$-0.369956\pi$
$853$	23.2057	0.794548	0.397274	+	0.917700i	$0.369956\pi$
$854$	10.9066	0.373216
$855$	−26.3165	−0.900007
$856$	4.17480	0.142692
$857$	15.2277	0.520169	0.260084	−	0.965586i	$-0.416250\pi$
$857$	15.2277	0.520169	0.260084	+	0.965586i	$0.416250\pi$
$858$	−5.79331	−0.197780
$859$	−50.4652	−1.72185	−0.860926	−	0.508730i	$-0.830115\pi$
$859$	−50.4652	−1.72185	−0.860926	+	0.508730i	$0.830115\pi$
$860$	4.10501	0.139980
$861$	1.46189	0.0498212
$862$	8.02175	0.273222
$863$	11.6005	0.394886	0.197443	−	0.980314i	$-0.436736\pi$
$863$	11.6005	0.394886	0.197443	+	0.980314i	$0.436736\pi$
$864$	−30.5744	−1.04016
$865$	59.3323	2.01736
$866$	−3.81340	−0.129585
$867$	−11.3411	−0.385165
$868$	−4.47384	−0.151852
$869$	19.1988	0.651273
$870$	43.4909	1.47448
$871$	−2.75239	−0.0932611
$872$	−5.61211	−0.190050
$873$	−7.19547	−0.243530
$874$	−49.1886	−1.66383
$875$	49.8664	1.68579
$876$	−2.06360	−0.0697226
$877$	41.1752	1.39039	0.695193	−	0.718823i	$-0.255319\pi$
$877$	41.1752	1.39039	0.695193	+	0.718823i	$0.255319\pi$
$878$	32.3030	1.09017
$879$	22.1488	0.747060
$880$	7.45111	0.251177
$881$	−44.4875	−1.49882	−0.749411	−	0.662105i	$-0.769663\pi$
$881$	−44.4875	−1.49882	−0.749411	+	0.662105i	$0.769663\pi$
$882$	−7.90826	−0.266285
$883$	2.12372	0.0714689	0.0357345	−	0.999361i	$-0.488623\pi$
$883$	2.12372	0.0714689	0.0357345	+	0.999361i	$0.488623\pi$
$884$	3.39372	0.114143
$885$	44.5405	1.49721
$886$	37.3398	1.25446
$887$	−43.5843	−1.46342	−0.731709	−	0.681617i	$-0.761277\pi$
$887$	−43.5843	−1.46342	−0.731709	+	0.681617i	$0.761277\pi$
$888$	−14.2837	−0.479331
$889$	−18.1304	−0.608073
$890$	−12.6366	−0.423579
$891$	−22.3593	−0.749063
$892$	26.4088	0.884231
$893$	65.7061	2.19877
$894$	4.89173	0.163604
$895$	−38.0375	−1.27145
$896$	20.7371	0.692779
$897$	−10.1860	−0.340101
$898$	3.46676	0.115687
$899$	8.93598	0.298032
$900$	9.28616	0.309539
$901$	−18.3139	−0.610124
$902$	−1.05570	−0.0351509
$903$	−5.59069	−0.186046
$904$	10.7225	0.356626
$905$	90.1954	2.99820
$906$	21.9068	0.727805
$907$	−7.93233	−0.263389	−0.131694	−	0.991290i	$-0.542042\pi$
$907$	−7.93233	−0.263389	−0.131694	+	0.991290i	$0.542042\pi$
$908$	24.5405	0.814406
$909$	−4.34191	−0.144012
$910$	−13.5486	−0.449130
$911$	−43.3097	−1.43491	−0.717457	−	0.696603i	$-0.754694\pi$
$911$	−43.3097	−1.43491	−0.717457	+	0.696603i	$0.754694\pi$
$912$	4.88330	0.161702
$913$	−49.0674	−1.62389
$914$	−38.0509	−1.25861
$915$	15.3898	0.508772
$916$	0.485450	0.0160397
$917$	−79.1868	−2.61498
$918$	−15.8610	−0.523491
$919$	−9.14057	−0.301519	−0.150760	−	0.988570i	$-0.548172\pi$
$919$	−9.14057	−0.301519	−0.150760	+	0.988570i	$0.548172\pi$
$920$	76.6069	2.52565
$921$	−11.9111	−0.392483
$922$	−27.9935	−0.921916
$923$	10.3705	0.341349
$924$	27.7767	0.913787
$925$	28.9929	0.953282
$926$	27.2221	0.894573
$927$	9.20698	0.302397
$928$	−48.3004	−1.58554
$929$	−22.0784	−0.724369	−0.362185	−	0.932106i	$-0.617969\pi$
$929$	−22.0784	−0.724369	−0.362185	+	0.932106i	$0.617969\pi$
$930$	4.86694	0.159593
$931$	−64.0246	−2.09832
$932$	−30.8123	−1.00929
$933$	17.1409	0.561166
$934$	−26.4162	−0.864365
$935$	−48.0553	−1.57158
$936$	−2.84670	−0.0930474
$937$	20.3221	0.663895	0.331947	−	0.943298i	$-0.392294\pi$
$937$	20.3221	0.663895	0.331947	+	0.943298i	$0.392294\pi$
$938$	−10.1741	−0.332195
$939$	16.9848	0.554279
$940$	−36.9299	−1.20452
$941$	45.9575	1.49817	0.749085	−	0.662474i	$-0.230493\pi$
$941$	45.9575	1.49817	0.749085	+	0.662474i	$0.230493\pi$
$942$	14.4566	0.471023
$943$	−1.85617	−0.0604451
$944$	3.97887	0.129501
$945$	−82.1327	−2.67178
$946$	4.03728	0.131263
$947$	−36.8726	−1.19820	−0.599100	−	0.800674i	$-0.704475\pi$
$947$	−36.8726	−1.19820	−0.599100	+	0.800674i	$0.704475\pi$
$948$	−7.07194	−0.229686
$949$	−1.28405	−0.0416820
$950$	−57.9606	−1.88049
$951$	−38.1881	−1.23833
$952$	34.7609	1.12661
$953$	−6.64980	−0.215408	−0.107704	−	0.994183i	$-0.534350\pi$
$953$	−6.64980	−0.215408	−0.107704	+	0.994183i	$0.534350\pi$
$954$	5.54391	0.179491
$955$	72.0933	2.33288
$956$	−16.4366	−0.531598
$957$	−55.4808	−1.79344
$958$	6.56767	0.212192
$959$	29.9701	0.967785
$960$	−31.1673	−1.00592
$961$	1.00000	0.0322581
$962$	−3.20751	−0.103414
$963$	1.39387	0.0449168
$964$	−5.97236	−0.192357
$965$	58.1406	1.87161
$966$	−37.6521	−1.21144
$967$	18.6536	0.599858	0.299929	−	0.953961i	$-0.403037\pi$
$967$	18.6536	0.599858	0.299929	+	0.953961i	$0.403037\pi$
$968$	−23.4625	−0.754112
$969$	−31.4944	−1.01175
$970$	−25.2423	−0.810481
$971$	−18.7748	−0.602513	−0.301257	−	0.953543i	$-0.597406\pi$
$971$	−18.7748	−0.602513	−0.301257	+	0.953543i	$0.597406\pi$
$972$	−10.9282	−0.350521
$973$	−44.6459	−1.43128
$974$	−14.4592	−0.463304
$975$	−12.0025	−0.384388
$976$	1.37480	0.0440062
$977$	−6.81092	−0.217901	−0.108950	−	0.994047i	$-0.534749\pi$
$977$	−6.81092	−0.217901	−0.108950	+	0.994047i	$0.534749\pi$
$978$	−1.28326	−0.0410341
$979$	16.1203	0.515207
$980$	35.9849	1.14949
$981$	−1.87376	−0.0598244
$982$	14.2254	0.453951
$983$	−60.6078	−1.93309	−0.966544	−	0.256502i	$-0.917430\pi$
$983$	−60.6078	−1.93309	−0.966544	+	0.256502i	$0.917430\pi$
$984$	1.07754	0.0343509
$985$	7.05489	0.224787
$986$	−25.0566	−0.797966
$987$	50.2956	1.60093
$988$	−8.31722	−0.264606
$989$	7.09849	0.225719
$990$	14.5471	0.462338
$991$	−13.9611	−0.443488	−0.221744	−	0.975105i	$-0.571175\pi$
$991$	−13.9611	−0.443488	−0.221744	+	0.975105i	$0.571175\pi$
$992$	−5.40516	−0.171614
$993$	25.4500	0.807631
$994$	38.3341	1.21588
$995$	35.0349	1.11068
$996$	18.0742	0.572702
$997$	13.6336	0.431781	0.215891	−	0.976418i	$-0.430735\pi$
$997$	13.6336	0.431781	0.215891	+	0.976418i	$0.430735\pi$
$998$	−4.82562	−0.152752
$999$	−19.4443	−0.615189

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.a.d.1.6	✓	8
3.2	odd	2		3627.2.a.q.1.3		8
4.3	odd	2		6448.2.a.bf.1.2		8
13.12	even	2		5239.2.a.j.1.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.a.d.1.6	✓	8	1.1	even	1	trivial
3627.2.a.q.1.3		8	3.2	odd	2
5239.2.a.j.1.3		8	13.12	even	2
6448.2.a.bf.1.2		8	4.3	odd	2

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:	\( N \)	\(=\)	\( 403 = 13 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	403.a (trivial)

\(f(q)\)	\(=\)	\(q+0.933096 q^{2} +1.42306 q^{3} -1.12933 q^{4} -3.66529 q^{5} +1.32785 q^{6} -3.96149 q^{7} -2.91997 q^{8} -0.974910 q^{9} +O(q^{10})\)	\(q+0.933096 q^{2} +1.42306 q^{3} -1.12933 q^{4} -3.66529 q^{5} +1.32785 q^{6} -3.96149 q^{7} -2.91997 q^{8} -0.974910 q^{9} -3.42006 q^{10} +4.36293 q^{11} -1.60710 q^{12} -1.00000 q^{13} -3.69645 q^{14} -5.21591 q^{15} -0.465945 q^{16} +3.00507 q^{17} -0.909684 q^{18} -7.36472 q^{19} +4.13933 q^{20} -5.63743 q^{21} +4.07103 q^{22} +7.15784 q^{23} -4.15528 q^{24} +8.43433 q^{25} -0.933096 q^{26} -5.65652 q^{27} +4.47384 q^{28} -8.93598 q^{29} -4.86694 q^{30} -1.00000 q^{31} +5.40516 q^{32} +6.20870 q^{33} +2.80402 q^{34} +14.5200 q^{35} +1.10100 q^{36} +3.43749 q^{37} -6.87199 q^{38} -1.42306 q^{39} +10.7025 q^{40} -0.259319 q^{41} -5.26026 q^{42} +0.991709 q^{43} -4.92720 q^{44} +3.57332 q^{45} +6.67895 q^{46} -8.92173 q^{47} -0.663066 q^{48} +8.69341 q^{49} +7.87003 q^{50} +4.27638 q^{51} +1.12933 q^{52} -6.09433 q^{53} -5.27808 q^{54} -15.9914 q^{55} +11.5674 q^{56} -10.4804 q^{57} -8.33812 q^{58} -8.53935 q^{59} +5.89049 q^{60} -2.95055 q^{61} -0.933096 q^{62} +3.86210 q^{63} +5.97543 q^{64} +3.66529 q^{65} +5.79331 q^{66} +2.75239 q^{67} -3.39372 q^{68} +10.1860 q^{69} +13.5486 q^{70} -10.3705 q^{71} +2.84670 q^{72} +1.28405 q^{73} +3.20751 q^{74} +12.0025 q^{75} +8.31722 q^{76} -17.2837 q^{77} -1.32785 q^{78} +4.40043 q^{79} +1.70782 q^{80} -5.12482 q^{81} -0.241970 q^{82} -11.2464 q^{83} +6.36653 q^{84} -11.0144 q^{85} +0.925360 q^{86} -12.7164 q^{87} -12.7396 q^{88} +3.69483 q^{89} +3.33425 q^{90} +3.96149 q^{91} -8.08358 q^{92} -1.42306 q^{93} -8.32483 q^{94} +26.9938 q^{95} +7.69185 q^{96} +7.38065 q^{97} +8.11179 q^{98} -4.25346 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 5 q^{2} - 3 q^{3} + 9 q^{4} - 15 q^{5} - 4 q^{7} - 15 q^{8} + 9 q^{9}+O(q^{10}) \) 8 * q - 5 * q^2 - 3 * q^3 + 9 * q^4 - 15 * q^5 - 4 * q^7 - 15 * q^8 + 9 * q^9	\( 8 q - 5 q^{2} - 3 q^{3} + 9 q^{4} - 15 q^{5} - 4 q^{7} - 15 q^{8} + 9 q^{9} + 9 q^{10} - 5 q^{11} - 9 q^{12} - 8 q^{13} + 2 q^{14} + 2 q^{15} + 3 q^{16} - 11 q^{17} - 30 q^{18} - 9 q^{19} - 31 q^{20} - 16 q^{21} - 2 q^{22} + 13 q^{24} + 19 q^{25} + 5 q^{26} - 9 q^{27} + 16 q^{28} - 12 q^{29} - 7 q^{30} - 8 q^{31} - 25 q^{32} - 14 q^{33} + 22 q^{34} + 7 q^{35} + 37 q^{36} - 9 q^{37} - 9 q^{38} + 3 q^{39} + 55 q^{40} - 25 q^{41} - 3 q^{42} + 7 q^{43} - 26 q^{44} - 45 q^{45} + 5 q^{46} - 17 q^{47} - 9 q^{48} - 11 q^{50} - 10 q^{51} - 9 q^{52} - 15 q^{53} + 54 q^{54} + 7 q^{55} - 14 q^{56} - 7 q^{57} - 5 q^{58} - 15 q^{59} + 61 q^{60} + 11 q^{61} + 5 q^{62} - 21 q^{63} + 47 q^{64} + 15 q^{65} + 83 q^{66} + 18 q^{67} - 16 q^{68} - 15 q^{69} - 24 q^{70} - 7 q^{71} - 21 q^{72} + 24 q^{73} + 48 q^{74} - 17 q^{75} - 3 q^{76} - 49 q^{77} + 33 q^{79} - 16 q^{80} + 20 q^{81} - q^{82} - 13 q^{83} - 6 q^{84} + q^{85} + 19 q^{86} + 18 q^{87} + 37 q^{88} - 23 q^{89} + 117 q^{90} + 4 q^{91} + 22 q^{92} + 3 q^{93} + 10 q^{94} + 43 q^{95} + 46 q^{96} - 17 q^{97} + 52 q^{98} + 51 q^{99}+O(q^{100}) \) 8 * q - 5 * q^2 - 3 * q^3 + 9 * q^4 - 15 * q^5 - 4 * q^7 - 15 * q^8 + 9 * q^9 + 9 * q^10 - 5 * q^11 - 9 * q^12 - 8 * q^13 + 2 * q^14 + 2 * q^15 + 3 * q^16 - 11 * q^17 - 30 * q^18 - 9 * q^19 - 31 * q^20 - 16 * q^21 - 2 * q^22 + 13 * q^24 + 19 * q^25 + 5 * q^26 - 9 * q^27 + 16 * q^28 - 12 * q^29 - 7 * q^30 - 8 * q^31 - 25 * q^32 - 14 * q^33 + 22 * q^34 + 7 * q^35 + 37 * q^36 - 9 * q^37 - 9 * q^38 + 3 * q^39 + 55 * q^40 - 25 * q^41 - 3 * q^42 + 7 * q^43 - 26 * q^44 - 45 * q^45 + 5 * q^46 - 17 * q^47 - 9 * q^48 - 11 * q^50 - 10 * q^51 - 9 * q^52 - 15 * q^53 + 54 * q^54 + 7 * q^55 - 14 * q^56 - 7 * q^57 - 5 * q^58 - 15 * q^59 + 61 * q^60 + 11 * q^61 + 5 * q^62 - 21 * q^63 + 47 * q^64 + 15 * q^65 + 83 * q^66 + 18 * q^67 - 16 * q^68 - 15 * q^69 - 24 * q^70 - 7 * q^71 - 21 * q^72 + 24 * q^73 + 48 * q^74 - 17 * q^75 - 3 * q^76 - 49 * q^77 + 33 * q^79 - 16 * q^80 + 20 * q^81 - q^82 - 13 * q^83 - 6 * q^84 + q^85 + 19 * q^86 + 18 * q^87 + 37 * q^88 - 23 * q^89 + 117 * q^90 + 4 * q^91 + 22 * q^92 + 3 * q^93 + 10 * q^94 + 43 * q^95 + 46 * q^96 - 17 * q^97 + 52 * q^98 + 51 * q^99

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