LMFDB - Embedded newform 309.2.a.d.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [309,2,Mod(1,309)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(309, 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("309.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 309 = 3 \cdot 103 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	309.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:	$2.46737742246$
Analytic rank:	$0$
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} - x^{7} - 13x^{6} + 11x^{5} + 52x^{4} - 35x^{3} - 59x^{2} + 27x + 1 $ x^8 - x^7 - 13x^6 + 11x^5 + 52x^4 - 35x^3 - 59x^2 + 27x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$0.458525$ of defining polynomial
Character	$\chi$	$=$	309.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.458525 q^{2} +1.00000 q^{3} -1.78975 q^{4} -2.98439 q^{5} -0.458525 q^{6} +3.66803 q^{7} +1.73770 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-0.458525 q^{2} +1.00000 q^{3} -1.78975 q^{4} -2.98439 q^{5} -0.458525 q^{6} +3.66803 q^{7} +1.73770 q^{8} +1.00000 q^{9} +1.36842 q^{10} +2.13538 q^{11} -1.78975 q^{12} +2.66018 q^{13} -1.68188 q^{14} -2.98439 q^{15} +2.78273 q^{16} +3.80340 q^{17} -0.458525 q^{18} +4.96530 q^{19} +5.34133 q^{20} +3.66803 q^{21} -0.979123 q^{22} -3.09116 q^{23} +1.73770 q^{24} +3.90658 q^{25} -1.21976 q^{26} +1.00000 q^{27} -6.56487 q^{28} -2.37332 q^{29} +1.36842 q^{30} -10.3152 q^{31} -4.75135 q^{32} +2.13538 q^{33} -1.74396 q^{34} -10.9468 q^{35} -1.78975 q^{36} +10.3257 q^{37} -2.27672 q^{38} +2.66018 q^{39} -5.18597 q^{40} -0.478830 q^{41} -1.68188 q^{42} +0.253919 q^{43} -3.82180 q^{44} -2.98439 q^{45} +1.41738 q^{46} +9.38210 q^{47} +2.78273 q^{48} +6.45444 q^{49} -1.79127 q^{50} +3.80340 q^{51} -4.76108 q^{52} -2.55904 q^{53} -0.458525 q^{54} -6.37279 q^{55} +6.37393 q^{56} +4.96530 q^{57} +1.08822 q^{58} -8.80184 q^{59} +5.34133 q^{60} +3.80381 q^{61} +4.72980 q^{62} +3.66803 q^{63} -3.38685 q^{64} -7.93902 q^{65} -0.979123 q^{66} +8.54392 q^{67} -6.80716 q^{68} -3.09116 q^{69} +5.01940 q^{70} -10.5239 q^{71} +1.73770 q^{72} -0.419705 q^{73} -4.73460 q^{74} +3.90658 q^{75} -8.88668 q^{76} +7.83262 q^{77} -1.21976 q^{78} +1.62104 q^{79} -8.30476 q^{80} +1.00000 q^{81} +0.219556 q^{82} -2.65708 q^{83} -6.56487 q^{84} -11.3508 q^{85} -0.116428 q^{86} -2.37332 q^{87} +3.71064 q^{88} +11.6837 q^{89} +1.36842 q^{90} +9.75763 q^{91} +5.53243 q^{92} -10.3152 q^{93} -4.30193 q^{94} -14.8184 q^{95} -4.75135 q^{96} -10.1758 q^{97} -2.95952 q^{98} +2.13538 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - q^{2} + 8 q^{3} + 11 q^{4} - q^{5} - q^{6} + 6 q^{7} - 3 q^{8} + 8 q^{9}+O(q^{10}) $ 8 * q - q^2 + 8 * q^3 + 11 * q^4 - q^5 - q^6 + 6 * q^7 - 3 * q^8 + 8 * q^9	\( 8 q - q^{2} + 8 q^{3} + 11 q^{4} - q^{5} - q^{6} + 6 q^{7} - 3 q^{8} + 8 q^{9} + 3 q^{10} + 6 q^{11} + 11 q^{12} + 9 q^{13} - 6 q^{14} - q^{15} + 9 q^{16} - 4 q^{17} - q^{18} + 16 q^{19} - 23 q^{20} + 6 q^{21} - 4 q^{22} - 11 q^{23} - 3 q^{24} + 15 q^{25} - 14 q^{26} + 8 q^{27} - 5 q^{28} + 3 q^{30} + 17 q^{31} - 12 q^{32} + 6 q^{33} - 8 q^{34} + 4 q^{35} + 11 q^{36} - 6 q^{37} - 3 q^{38} + 9 q^{39} + 3 q^{40} + 12 q^{41} - 6 q^{42} + 9 q^{43} + 8 q^{44} - q^{45} - 30 q^{46} - 6 q^{47} + 9 q^{48} + 18 q^{49} - 36 q^{50} - 4 q^{51} + 23 q^{52} - 16 q^{53} - q^{54} - 10 q^{55} - 13 q^{56} + 16 q^{57} - 22 q^{58} + 11 q^{59} - 23 q^{60} + 5 q^{61} - 25 q^{62} + 6 q^{63} - 35 q^{64} - 41 q^{65} - 4 q^{66} + 5 q^{67} - 19 q^{68} - 11 q^{69} - 48 q^{70} - 10 q^{71} - 3 q^{72} + 14 q^{73} + 4 q^{74} + 15 q^{75} - 12 q^{76} - 40 q^{77} - 14 q^{78} + 14 q^{79} - 19 q^{80} + 8 q^{81} - 13 q^{82} - 23 q^{83} - 5 q^{84} - 4 q^{85} - 3 q^{86} - 30 q^{88} - 14 q^{89} + 3 q^{90} - 6 q^{91} - 21 q^{92} + 17 q^{93} + 22 q^{94} + 6 q^{95} - 12 q^{96} + 3 q^{97} - 18 q^{98} + 6 q^{99}+O(q^{100}) \) 8 * q - q^2 + 8 * q^3 + 11 * q^4 - q^5 - q^6 + 6 * q^7 - 3 * q^8 + 8 * q^9 + 3 * q^10 + 6 * q^11 + 11 * q^12 + 9 * q^13 - 6 * q^14 - q^15 + 9 * q^16 - 4 * q^17 - q^18 + 16 * q^19 - 23 * q^20 + 6 * q^21 - 4 * q^22 - 11 * q^23 - 3 * q^24 + 15 * q^25 - 14 * q^26 + 8 * q^27 - 5 * q^28 + 3 * q^30 + 17 * q^31 - 12 * q^32 + 6 * q^33 - 8 * q^34 + 4 * q^35 + 11 * q^36 - 6 * q^37 - 3 * q^38 + 9 * q^39 + 3 * q^40 + 12 * q^41 - 6 * q^42 + 9 * q^43 + 8 * q^44 - q^45 - 30 * q^46 - 6 * q^47 + 9 * q^48 + 18 * q^49 - 36 * q^50 - 4 * q^51 + 23 * q^52 - 16 * q^53 - q^54 - 10 * q^55 - 13 * q^56 + 16 * q^57 - 22 * q^58 + 11 * q^59 - 23 * q^60 + 5 * q^61 - 25 * q^62 + 6 * q^63 - 35 * q^64 - 41 * q^65 - 4 * q^66 + 5 * q^67 - 19 * q^68 - 11 * q^69 - 48 * q^70 - 10 * q^71 - 3 * q^72 + 14 * q^73 + 4 * q^74 + 15 * q^75 - 12 * q^76 - 40 * q^77 - 14 * q^78 + 14 * q^79 - 19 * q^80 + 8 * q^81 - 13 * q^82 - 23 * q^83 - 5 * q^84 - 4 * q^85 - 3 * q^86 - 30 * q^88 - 14 * q^89 + 3 * q^90 - 6 * q^91 - 21 * q^92 + 17 * q^93 + 22 * q^94 + 6 * q^95 - 12 * q^96 + 3 * q^97 - 18 * q^98 + 6 * 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 $

$2$	−0.458525	−0.324226	−0.162113	−	0.986772i	$-0.551831\pi$
$2$	−0.458525	−0.324226	−0.162113	+	0.986772i	$0.551831\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−1.78975	−0.894877
$5$	−2.98439	−1.33466	−0.667330	−	0.744762i	$-0.732563\pi$
$5$	−2.98439	−1.33466	−0.667330	+	0.744762i	$0.732563\pi$
$6$	−0.458525	−0.187192
$7$	3.66803	1.38638	0.693192	−	0.720753i	$-0.256204\pi$
$7$	3.66803	1.38638	0.693192	+	0.720753i	$0.256204\pi$
$8$	1.73770	0.614369
$9$	1.00000	0.333333
$10$	1.36842	0.432732
$11$	2.13538	0.643840	0.321920	−	0.946767i	$-0.395672\pi$
$11$	2.13538	0.643840	0.321920	+	0.946767i	$0.395672\pi$
$12$	−1.78975	−0.516658
$13$	2.66018	0.737802	0.368901	−	0.929469i	$-0.379734\pi$
$13$	2.66018	0.737802	0.368901	+	0.929469i	$0.379734\pi$
$14$	−1.68188	−0.449502
$15$	−2.98439	−0.770566
$16$	2.78273	0.695683
$17$	3.80340	0.922461	0.461231	−	0.887280i	$-0.347408\pi$
$17$	3.80340	0.922461	0.461231	+	0.887280i	$0.347408\pi$
$18$	−0.458525	−0.108075
$19$	4.96530	1.13912	0.569559	−	0.821950i	$-0.307114\pi$
$19$	4.96530	1.13912	0.569559	+	0.821950i	$0.307114\pi$
$20$	5.34133	1.19436
$21$	3.66803	0.800430
$22$	−0.979123	−0.208750
$23$	−3.09116	−0.644552	−0.322276	−	0.946646i	$-0.604448\pi$
$23$	−3.09116	−0.644552	−0.322276	+	0.946646i	$0.604448\pi$
$24$	1.73770	0.354706
$25$	3.90658	0.781316
$26$	−1.21976	−0.239215
$27$	1.00000	0.192450
$28$	−6.56487	−1.24064
$29$	−2.37332	−0.440714	−0.220357	−	0.975419i	$-0.570722\pi$
$29$	−2.37332	−0.440714	−0.220357	+	0.975419i	$0.570722\pi$
$30$	1.36842	0.249838
$31$	−10.3152	−1.85267	−0.926336	−	0.376697i	$-0.877060\pi$
$31$	−10.3152	−1.85267	−0.926336	+	0.376697i	$0.877060\pi$
$32$	−4.75135	−0.839927
$33$	2.13538	0.371721
$34$	−1.74396	−0.299086
$35$	−10.9468	−1.85035
$36$	−1.78975	−0.298292
$37$	10.3257	1.69754	0.848769	−	0.528765i	$-0.177345\pi$
$37$	10.3257	1.69754	0.848769	+	0.528765i	$0.177345\pi$
$38$	−2.27672	−0.369332
$39$	2.66018	0.425970
$40$	−5.18597	−0.819973
$41$	−0.478830	−0.0747807	−0.0373903	−	0.999301i	$-0.511904\pi$
$41$	−0.478830	−0.0747807	−0.0373903	+	0.999301i	$0.511904\pi$
$42$	−1.68188	−0.259520
$43$	0.253919	0.0387223	0.0193612	−	0.999813i	$-0.493837\pi$
$43$	0.253919	0.0387223	0.0193612	+	0.999813i	$0.493837\pi$
$44$	−3.82180	−0.576158
$45$	−2.98439	−0.444887
$46$	1.41738	0.208981
$47$	9.38210	1.36852	0.684260	−	0.729238i	$-0.260125\pi$
$47$	9.38210	1.36852	0.684260	+	0.729238i	$0.260125\pi$
$48$	2.78273	0.401653
$49$	6.45444	0.922063
$50$	−1.79127	−0.253323
$51$	3.80340	0.532583
$52$	−4.76108	−0.660242
$53$	−2.55904	−0.351511	−0.175756	−	0.984434i	$-0.556237\pi$
$53$	−2.55904	−0.351511	−0.175756	+	0.984434i	$0.556237\pi$
$54$	−0.458525	−0.0623974
$55$	−6.37279	−0.859307
$56$	6.37393	0.851752
$57$	4.96530	0.657671
$58$	1.08822	0.142891
$59$	−8.80184	−1.14590	−0.572951	−	0.819590i	$-0.694201\pi$
$59$	−8.80184	−1.14590	−0.572951	+	0.819590i	$0.694201\pi$
$60$	5.34133	0.689562
$61$	3.80381	0.487028	0.243514	−	0.969897i	$-0.421700\pi$
$61$	3.80381	0.487028	0.243514	+	0.969897i	$0.421700\pi$
$62$	4.72980	0.600685
$63$	3.66803	0.462128
$64$	−3.38685	−0.423356
$65$	−7.93902	−0.984715
$66$	−0.979123	−0.120522
$67$	8.54392	1.04381	0.521903	−	0.853005i	$-0.325222\pi$
$67$	8.54392	1.04381	0.521903	+	0.853005i	$0.325222\pi$
$68$	−6.80716	−0.825490
$69$	−3.09116	−0.372132
$70$	5.01940	0.599932
$71$	−10.5239	−1.24895	−0.624476	−	0.781044i	$-0.714687\pi$
$71$	−10.5239	−1.24895	−0.624476	+	0.781044i	$0.714687\pi$
$72$	1.73770	0.204790
$73$	−0.419705	−0.0491227	−0.0245614	−	0.999698i	$-0.507819\pi$
$73$	−0.419705	−0.0491227	−0.0245614	+	0.999698i	$0.507819\pi$
$74$	−4.73460	−0.550386
$75$	3.90658	0.451093
$76$	−8.88668	−1.01937
$77$	7.83262	0.892610
$78$	−1.21976	−0.138111
$79$	1.62104	0.182381	0.0911904	−	0.995833i	$-0.470933\pi$
$79$	1.62104	0.182381	0.0911904	+	0.995833i	$0.470933\pi$
$80$	−8.30476	−0.928500
$81$	1.00000	0.111111
$82$	0.219556	0.0242459
$83$	−2.65708	−0.291652	−0.145826	−	0.989310i	$-0.546584\pi$
$83$	−2.65708	−0.291652	−0.145826	+	0.989310i	$0.546584\pi$
$84$	−6.56487	−0.716286
$85$	−11.3508	−1.23117
$86$	−0.116428	−0.0125548
$87$	−2.37332	−0.254446
$88$	3.71064	0.395555
$89$	11.6837	1.23847	0.619233	−	0.785207i	$-0.287444\pi$
$89$	11.6837	1.23847	0.619233	+	0.785207i	$0.287444\pi$
$90$	1.36842	0.144244
$91$	9.75763	1.02288
$92$	5.53243	0.576795
$93$	−10.3152	−1.06964
$94$	−4.30193	−0.443710
$95$	−14.8184	−1.52034
$96$	−4.75135	−0.484932
$97$	−10.1758	−1.03320	−0.516598	−	0.856228i	$-0.672802\pi$
$97$	−10.1758	−1.03320	−0.516598	+	0.856228i	$0.672802\pi$
$98$	−2.95952	−0.298957
$99$	2.13538	0.214613
$100$	−6.99182	−0.699182
$101$	−19.2892	−1.91934	−0.959672	−	0.281122i	$-0.909293\pi$
$101$	−19.2892	−1.91934	−0.959672	+	0.281122i	$0.909293\pi$
$102$	−1.74396	−0.172677
$103$	−1.00000	−0.0985329
$103$	−1.00000	−0.0985329
$104$	4.62259	0.453283
$105$	−10.9468	−1.06830
$106$	1.17338	0.113969
$107$	6.54912	0.633127	0.316564	−	0.948571i	$-0.397471\pi$
$107$	6.54912	0.633127	0.316564	+	0.948571i	$0.397471\pi$
$108$	−1.78975	−0.172219
$109$	−8.37842	−0.802507	−0.401253	−	0.915967i	$-0.631425\pi$
$109$	−8.37842	−0.802507	−0.401253	+	0.915967i	$0.631425\pi$
$110$	2.92208	0.278610
$111$	10.3257	0.980074
$112$	10.2071	0.964484
$113$	7.37287	0.693582	0.346791	−	0.937943i	$-0.387271\pi$
$113$	7.37287	0.693582	0.346791	+	0.937943i	$0.387271\pi$
$114$	−2.27672	−0.213234
$115$	9.22524	0.860258
$116$	4.24765	0.394385
$117$	2.66018	0.245934
$118$	4.03586	0.371531
$119$	13.9510	1.27889
$120$	−5.18597	−0.473412
$121$	−6.44017	−0.585470
$122$	−1.74414	−0.157907
$123$	−0.478830	−0.0431746
$124$	18.4618	1.65792
$125$	3.26319	0.291868
$126$	−1.68188	−0.149834
$127$	−20.7486	−1.84114	−0.920571	−	0.390576i	$-0.872276\pi$
$127$	−20.7486	−1.84114	−0.920571	+	0.390576i	$0.872276\pi$
$128$	11.0557	0.977191
$129$	0.253919	0.0223564
$130$	3.64024	0.319270
$131$	20.3182	1.77521	0.887606	−	0.460603i	$-0.152367\pi$
$131$	20.3182	1.77521	0.887606	+	0.460603i	$0.152367\pi$
$132$	−3.82180	−0.332645
$133$	18.2129	1.57926
$134$	−3.91760	−0.338429
$135$	−2.98439	−0.256855
$136$	6.60917	0.566731
$137$	−1.87493	−0.160186	−0.0800931	−	0.996787i	$-0.525522\pi$
$137$	−1.87493	−0.160186	−0.0800931	+	0.996787i	$0.525522\pi$
$138$	1.41738	0.120655
$139$	−18.5684	−1.57495	−0.787476	−	0.616345i	$-0.788613\pi$
$139$	−18.5684	−1.57495	−0.787476	+	0.616345i	$0.788613\pi$
$140$	19.5921	1.65584
$141$	9.38210	0.790115
$142$	4.82545	0.404943
$143$	5.68049	0.475026
$144$	2.78273	0.231894
$145$	7.08290	0.588203
$146$	0.192445	0.0159269
$147$	6.45444	0.532353
$148$	−18.4805	−1.51909
$149$	4.22992	0.346529	0.173264	−	0.984875i	$-0.444569\pi$
$149$	4.22992	0.346529	0.173264	+	0.984875i	$0.444569\pi$
$150$	−1.79127	−0.146256
$151$	−4.34293	−0.353423	−0.176711	−	0.984263i	$-0.556546\pi$
$151$	−4.34293	−0.353423	−0.176711	+	0.984263i	$0.556546\pi$
$152$	8.62820	0.699839
$153$	3.80340	0.307487
$154$	−3.59145	−0.289407
$155$	30.7847	2.47269
$156$	−4.76108	−0.381191
$157$	−6.02039	−0.480479	−0.240240	−	0.970714i	$-0.577226\pi$
$157$	−6.02039	−0.480479	−0.240240	+	0.970714i	$0.577226\pi$
$158$	−0.743286	−0.0591326
$159$	−2.55904	−0.202945
$160$	14.1799	1.12102
$161$	−11.3385	−0.893598
$162$	−0.458525	−0.0360251
$163$	−18.6296	−1.45919	−0.729593	−	0.683882i	$-0.760290\pi$
$163$	−18.6296	−1.45919	−0.729593	+	0.683882i	$0.760290\pi$
$164$	0.856988	0.0669195
$165$	−6.37279	−0.496121
$166$	1.21834	0.0945612
$167$	19.3259	1.49549	0.747743	−	0.663988i	$-0.231137\pi$
$167$	19.3259	1.49549	0.747743	+	0.663988i	$0.231137\pi$
$168$	6.37393	0.491759
$169$	−5.92343	−0.455648
$170$	5.20464	0.399178
$171$	4.96530	0.379706
$172$	−0.454453	−0.0346518
$173$	−21.9441	−1.66838	−0.834191	−	0.551476i	$-0.814065\pi$
$173$	−21.9441	−1.66838	−0.834191	+	0.551476i	$0.814065\pi$
$174$	1.08822	0.0824981
$175$	14.3295	1.08321
$176$	5.94218	0.447908
$177$	−8.80184	−0.661587
$178$	−5.35725	−0.401543
$179$	24.8516	1.85749	0.928747	−	0.370714i	$-0.120887\pi$
$179$	24.8516	1.85749	0.928747	+	0.370714i	$0.120887\pi$
$180$	5.34133	0.398119
$181$	17.3523	1.28978	0.644892	−	0.764274i	$-0.276903\pi$
$181$	17.3523	1.28978	0.644892	+	0.764274i	$0.276903\pi$
$182$	−4.47412	−0.331644
$183$	3.80381	0.281185
$184$	−5.37151	−0.395993
$185$	−30.8160	−2.26563
$186$	4.72980	0.346806
$187$	8.12170	0.593917
$188$	−16.7917	−1.22466
$189$	3.66803	0.266810
$190$	6.79461	0.492933
$191$	−16.6672	−1.20600	−0.603000	−	0.797741i	$-0.706028\pi$
$191$	−16.6672	−1.20600	−0.603000	+	0.797741i	$0.706028\pi$
$192$	−3.38685	−0.244425
$193$	−16.7961	−1.20901	−0.604504	−	0.796602i	$-0.706629\pi$
$193$	−16.7961	−1.20901	−0.604504	+	0.796602i	$0.706629\pi$
$194$	4.66586	0.334989
$195$	−7.93902	−0.568525
$196$	−11.5519	−0.825133
$197$	−2.21629	−0.157904	−0.0789522	−	0.996878i	$-0.525157\pi$
$197$	−2.21629	−0.157904	−0.0789522	+	0.996878i	$0.525157\pi$
$198$	−0.979123	−0.0695832
$199$	11.4441	0.811248	0.405624	−	0.914040i	$-0.367054\pi$
$199$	11.4441	0.811248	0.405624	+	0.914040i	$0.367054\pi$
$200$	6.78846	0.480016
$201$	8.54392	0.602641
$202$	8.84457	0.622302
$203$	−8.70539	−0.610999
$204$	−6.80716	−0.476597
$205$	1.42902	0.0998067
$206$	0.458525	0.0319470
$207$	−3.09116	−0.214851
$208$	7.40258	0.513276
$209$	10.6028	0.733410
$210$	5.01940	0.346371
$211$	20.9436	1.44182	0.720909	−	0.693030i	$-0.243725\pi$
$211$	20.9436	1.44182	0.720909	+	0.693030i	$0.243725\pi$
$212$	4.58006	0.314560
$213$	−10.5239	−0.721083
$214$	−3.00293	−0.205276
$215$	−0.757794	−0.0516812
$216$	1.73770	0.118235
$217$	−37.8366	−2.56852
$218$	3.84171	0.260194
$219$	−0.419705	−0.0283610
$220$	11.4057	0.768974
$221$	10.1178	0.680594
$222$	−4.73460	−0.317765
$223$	−2.52865	−0.169331	−0.0846656	−	0.996409i	$-0.526982\pi$
$223$	−2.52865	−0.169331	−0.0846656	+	0.996409i	$0.526982\pi$
$224$	−17.4281	−1.16446
$225$	3.90658	0.260439
$226$	−3.38065	−0.224877
$227$	18.8238	1.24938	0.624689	−	0.780874i	$-0.285226\pi$
$227$	18.8238	1.24938	0.624689	+	0.780874i	$0.285226\pi$
$228$	−8.88668	−0.588535
$229$	14.3108	0.945682	0.472841	−	0.881148i	$-0.343229\pi$
$229$	14.3108	0.945682	0.472841	+	0.881148i	$0.343229\pi$
$230$	−4.23000	−0.278918
$231$	7.83262	0.515348
$232$	−4.12411	−0.270761
$233$	−11.3116	−0.741046	−0.370523	−	0.928823i	$-0.620821\pi$
$233$	−11.3116	−0.741046	−0.370523	+	0.928823i	$0.620821\pi$
$234$	−1.21976	−0.0797382
$235$	−27.9998	−1.82651
$236$	15.7531	1.02544
$237$	1.62104	0.105298
$238$	−6.39688	−0.414648
$239$	−2.31792	−0.149934	−0.0749670	−	0.997186i	$-0.523885\pi$
$239$	−2.31792	−0.149934	−0.0749670	+	0.997186i	$0.523885\pi$
$240$	−8.30476	−0.536070
$241$	−9.16521	−0.590383	−0.295192	−	0.955438i	$-0.595384\pi$
$241$	−9.16521	−0.590383	−0.295192	+	0.955438i	$0.595384\pi$
$242$	2.95298	0.189825
$243$	1.00000	0.0641500
$244$	−6.80788	−0.435830
$245$	−19.2626	−1.23064
$246$	0.219556	0.0139983
$247$	13.2086	0.840444
$248$	−17.9248	−1.13822
$249$	−2.65708	−0.168385
$250$	−1.49625	−0.0946314
$251$	9.03064	0.570009	0.285005	−	0.958526i	$-0.408005\pi$
$251$	9.03064	0.570009	0.285005	+	0.958526i	$0.408005\pi$
$252$	−6.56487	−0.413548
$253$	−6.60080	−0.414988
$254$	9.51376	0.596946
$255$	−11.3508	−0.710817
$256$	1.70441	0.106526
$257$	−7.77565	−0.485032	−0.242516	−	0.970147i	$-0.577973\pi$
$257$	−7.77565	−0.485032	−0.242516	+	0.970147i	$0.577973\pi$
$258$	−0.116428	−0.00724852
$259$	37.8750	2.35344
$260$	14.2089	0.881199
$261$	−2.37332	−0.146905
$262$	−9.31642	−0.575570
$263$	−28.3335	−1.74712	−0.873558	−	0.486720i	$-0.838193\pi$
$263$	−28.3335	−1.74712	−0.873558	+	0.486720i	$0.838193\pi$
$264$	3.71064	0.228374
$265$	7.63718	0.469148
$266$	−8.35106	−0.512037
$267$	11.6837	0.715029
$268$	−15.2915	−0.934078
$269$	−16.6902	−1.01762	−0.508810	−	0.860879i	$-0.669915\pi$
$269$	−16.6902	−1.01762	−0.508810	+	0.860879i	$0.669915\pi$
$270$	1.36842	0.0832792
$271$	−1.37569	−0.0835674	−0.0417837	−	0.999127i	$-0.513304\pi$
$271$	−1.37569	−0.0835674	−0.0417837	+	0.999127i	$0.513304\pi$
$272$	10.5839	0.641740
$273$	9.75763	0.590559
$274$	0.859703	0.0519365
$275$	8.34202	0.503043
$276$	5.53243	0.333013
$277$	11.4613	0.688643	0.344321	−	0.938852i	$-0.388109\pi$
$277$	11.4613	0.688643	0.344321	+	0.938852i	$0.388109\pi$
$278$	8.51409	0.510641
$279$	−10.3152	−0.617558
$280$	−19.0223	−1.13680
$281$	−32.5060	−1.93915	−0.969573	−	0.244802i	$-0.921277\pi$
$281$	−32.5060	−1.93915	−0.969573	+	0.244802i	$0.921277\pi$
$282$	−4.30193	−0.256176
$283$	−3.13052	−0.186090	−0.0930449	−	0.995662i	$-0.529660\pi$
$283$	−3.13052	−0.186090	−0.0930449	+	0.995662i	$0.529660\pi$
$284$	18.8351	1.11766
$285$	−14.8184	−0.877766
$286$	−2.60465	−0.154016
$287$	−1.75636	−0.103675
$288$	−4.75135	−0.279976
$289$	−2.53411	−0.149066
$290$	−3.24769	−0.190711
$291$	−10.1758	−0.596516
$292$	0.751169	0.0439588
$293$	11.0997	0.648453	0.324227	−	0.945979i	$-0.394896\pi$
$293$	11.0997	0.648453	0.324227	+	0.945979i	$0.394896\pi$
$294$	−2.95952	−0.172603
$295$	26.2681	1.52939
$296$	17.9430	1.04291
$297$	2.13538	0.123907
$298$	−1.93952	−0.112354
$299$	−8.22306	−0.475552
$300$	−6.99182	−0.403673
$301$	0.931384	0.0536841
$302$	1.99134	0.114589
$303$	−19.2892	−1.10813
$304$	13.8171	0.792466
$305$	−11.3520	−0.650016
$306$	−1.74396	−0.0996953
$307$	−34.2464	−1.95455	−0.977273	−	0.211985i	$-0.932007\pi$
$307$	−34.2464	−1.95455	−0.977273	+	0.211985i	$0.932007\pi$
$308$	−14.0185	−0.798776
$309$	−1.00000	−0.0568880
$310$	−14.1156	−0.801710
$311$	2.17838	0.123525	0.0617624	−	0.998091i	$-0.480328\pi$
$311$	2.17838	0.123525	0.0617624	+	0.998091i	$0.480328\pi$
$312$	4.62259	0.261703
$313$	21.0901	1.19208	0.596041	−	0.802954i	$-0.296740\pi$
$313$	21.0901	1.19208	0.596041	+	0.802954i	$0.296740\pi$
$314$	2.76050	0.155784
$315$	−10.9468	−0.616784
$316$	−2.90126	−0.163208
$317$	25.2402	1.41763	0.708816	−	0.705394i	$-0.249230\pi$
$317$	25.2402	1.41763	0.708816	+	0.705394i	$0.249230\pi$
$318$	1.17338	0.0658001
$319$	−5.06792	−0.283749
$320$	10.1077	0.565037
$321$	6.54912	0.365536
$322$	5.19898	0.289728
$323$	18.8851	1.05079
$324$	−1.78975	−0.0994308
$325$	10.3922	0.576457
$326$	8.54216	0.473106
$327$	−8.37842	−0.463328
$328$	−0.832062	−0.0459429
$329$	34.4138	1.89730
$330$	2.92208	0.160855
$331$	−1.31265	−0.0721495	−0.0360748	−	0.999349i	$-0.511485\pi$
$331$	−1.31265	−0.0721495	−0.0360748	+	0.999349i	$0.511485\pi$
$332$	4.75551	0.260993
$333$	10.3257	0.565846
$334$	−8.86143	−0.484876
$335$	−25.4984	−1.39312
$336$	10.2071	0.556845
$337$	−8.88390	−0.483937	−0.241968	−	0.970284i	$-0.577793\pi$
$337$	−8.88390	−0.483937	−0.241968	+	0.970284i	$0.577793\pi$
$338$	2.71604	0.147733
$339$	7.37287	0.400440
$340$	20.3152	1.10175
$341$	−22.0269	−1.19282
$342$	−2.27672	−0.123111
$343$	−2.00113	−0.108051
$344$	0.441235	0.0237898
$345$	9.22524	0.496670
$346$	10.0619	0.540933
$347$	23.8609	1.28092	0.640460	−	0.767991i	$-0.278744\pi$
$347$	23.8609	1.28092	0.640460	+	0.767991i	$0.278744\pi$
$348$	4.24765	0.227698
$349$	−23.3815	−1.25159	−0.625793	−	0.779989i	$-0.715224\pi$
$349$	−23.3815	−1.25159	−0.625793	+	0.779989i	$0.715224\pi$
$350$	−6.57041	−0.351203
$351$	2.66018	0.141990
$352$	−10.1459	−0.540779
$353$	−11.9228	−0.634589	−0.317294	−	0.948327i	$-0.602774\pi$
$353$	−11.9228	−0.634589	−0.317294	+	0.948327i	$0.602774\pi$
$354$	4.03586	0.214504
$355$	31.4073	1.66693
$356$	−20.9109	−1.10828
$357$	13.9510	0.738365
$358$	−11.3951	−0.602248
$359$	−17.3273	−0.914502	−0.457251	−	0.889338i	$-0.651166\pi$
$359$	−17.3273	−0.914502	−0.457251	+	0.889338i	$0.651166\pi$
$360$	−5.18597	−0.273324
$361$	5.65425	0.297592
$362$	−7.95645	−0.418181
$363$	−6.44017	−0.338021
$364$	−17.4638	−0.915350
$365$	1.25256	0.0655621
$366$	−1.74414	−0.0911677
$367$	−26.2428	−1.36986	−0.684931	−	0.728608i	$-0.740168\pi$
$367$	−26.2428	−1.36986	−0.684931	+	0.728608i	$0.740168\pi$
$368$	−8.60188	−0.448404
$369$	−0.478830	−0.0249269
$370$	14.1299	0.734578
$371$	−9.38664	−0.487330
$372$	18.4618	0.957198
$373$	5.57695	0.288763	0.144382	−	0.989522i	$-0.453881\pi$
$373$	5.57695	0.288763	0.144382	+	0.989522i	$0.453881\pi$
$374$	−3.72400	−0.192563
$375$	3.26319	0.168510
$376$	16.3033	0.840776
$377$	−6.31346	−0.325159
$378$	−1.68188	−0.0865067
$379$	−19.8014	−1.01713	−0.508564	−	0.861024i	$-0.669824\pi$
$379$	−19.8014	−1.01713	−0.508564	+	0.861024i	$0.669824\pi$
$380$	26.5213	1.36051
$381$	−20.7486	−1.06298
$382$	7.64235	0.391017
$383$	−28.2014	−1.44102	−0.720511	−	0.693444i	$-0.756093\pi$
$383$	−28.2014	−1.44102	−0.720511	+	0.693444i	$0.756093\pi$
$384$	11.0557	0.564181
$385$	−23.3756	−1.19133
$386$	7.70143	0.391992
$387$	0.253919	0.0129074
$388$	18.2122	0.924583
$389$	−33.8333	−1.71542	−0.857708	−	0.514137i	$-0.828112\pi$
$389$	−33.8333	−1.71542	−0.857708	+	0.514137i	$0.828112\pi$
$390$	3.64024	0.184331
$391$	−11.7569	−0.594574
$392$	11.2159	0.566487
$393$	20.3182	1.02492
$394$	1.01623	0.0511967
$395$	−4.83780	−0.243416
$396$	−3.82180	−0.192053
$397$	−33.7691	−1.69482	−0.847412	−	0.530936i	$-0.821840\pi$
$397$	−33.7691	−1.69482	−0.847412	+	0.530936i	$0.821840\pi$
$398$	−5.24739	−0.263028
$399$	18.2129	0.911785
$400$	10.8710	0.543548
$401$	19.0553	0.951575	0.475788	−	0.879560i	$-0.342163\pi$
$401$	19.0553	0.951575	0.475788	+	0.879560i	$0.342163\pi$
$402$	−3.91760	−0.195392
$403$	−27.4404	−1.36691
$404$	34.5229	1.71758
$405$	−2.98439	−0.148296
$406$	3.99164	0.198102
$407$	22.0493	1.09294
$408$	6.60917	0.327203
$409$	17.9740	0.888756	0.444378	−	0.895839i	$-0.353425\pi$
$409$	17.9740	0.888756	0.444378	+	0.895839i	$0.353425\pi$
$410$	−0.655239	−0.0323600
$411$	−1.87493	−0.0924835
$412$	1.78975	0.0881749
$413$	−32.2854	−1.58866
$414$	1.41738	0.0696602
$415$	7.92975	0.389256
$416$	−12.6395	−0.619700
$417$	−18.5684	−0.909299
$418$	−4.86164	−0.237791
$419$	26.6221	1.30057	0.650287	−	0.759688i	$-0.274649\pi$
$419$	26.6221	1.30057	0.650287	+	0.759688i	$0.274649\pi$
$420$	19.5921	0.955999
$421$	29.7737	1.45108	0.725541	−	0.688179i	$-0.241590\pi$
$421$	29.7737	1.45108	0.725541	+	0.688179i	$0.241590\pi$
$422$	−9.60317	−0.467475
$423$	9.38210	0.456173
$424$	−4.44684	−0.215958
$425$	14.8583	0.720734
$426$	4.82545	0.233794
$427$	13.9525	0.675208
$428$	−11.7213	−0.566571
$429$	5.68049	0.274257
$430$	0.347468	0.0167564
$431$	17.2890	0.832783	0.416392	−	0.909185i	$-0.363295\pi$
$431$	17.2890	0.832783	0.416392	+	0.909185i	$0.363295\pi$
$432$	2.78273	0.133884
$433$	7.38057	0.354688	0.177344	−	0.984149i	$-0.443250\pi$
$433$	7.38057	0.354688	0.177344	+	0.984149i	$0.443250\pi$
$434$	17.3490	0.832781
$435$	7.08290	0.339599
$436$	14.9953	0.718145
$437$	−15.3486	−0.734222
$438$	0.192445	0.00919539
$439$	26.0169	1.24172	0.620860	−	0.783922i	$-0.286784\pi$
$439$	26.0169	1.24172	0.620860	+	0.783922i	$0.286784\pi$
$440$	−11.0740	−0.527931
$441$	6.45444	0.307354
$442$	−4.63924	−0.220666
$443$	−4.22269	−0.200626	−0.100313	−	0.994956i	$-0.531984\pi$
$443$	−4.22269	−0.200626	−0.100313	+	0.994956i	$0.531984\pi$
$444$	−18.4805	−0.877046
$445$	−34.8686	−1.65293
$446$	1.15945	0.0549016
$447$	4.22992	0.200068
$448$	−12.4231	−0.586935
$449$	−8.37123	−0.395063	−0.197531	−	0.980297i	$-0.563292\pi$
$449$	−8.37123	−0.395063	−0.197531	+	0.980297i	$0.563292\pi$
$450$	−1.79127	−0.0844411
$451$	−1.02248	−0.0481468
$452$	−13.1956	−0.620671
$453$	−4.34293	−0.204049
$454$	−8.63117	−0.405081
$455$	−29.1206	−1.36519
$456$	8.62820	0.404052
$457$	0.606332	0.0283630	0.0141815	−	0.999899i	$-0.495486\pi$
$457$	0.606332	0.0283630	0.0141815	+	0.999899i	$0.495486\pi$
$458$	−6.56184	−0.306615
$459$	3.80340	0.177528
$460$	−16.5109	−0.769825
$461$	4.27053	0.198898	0.0994492	−	0.995043i	$-0.468292\pi$
$461$	4.27053	0.198898	0.0994492	+	0.995043i	$0.468292\pi$
$462$	−3.59145	−0.167089
$463$	2.98883	0.138903	0.0694513	−	0.997585i	$-0.477875\pi$
$463$	2.98883	0.138903	0.0694513	+	0.997585i	$0.477875\pi$
$464$	−6.60430	−0.306597
$465$	30.7847	1.42761
$466$	5.18664	0.240266
$467$	−0.816012	−0.0377606	−0.0188803	−	0.999822i	$-0.506010\pi$
$467$	−0.816012	−0.0377606	−0.0188803	+	0.999822i	$0.506010\pi$
$468$	−4.76108	−0.220081
$469$	31.3393	1.44712
$470$	12.8386	0.592202
$471$	−6.02039	−0.277405
$472$	−15.2949	−0.704006
$473$	0.542213	0.0249310
$474$	−0.743286	−0.0341402
$475$	19.3974	0.890012
$476$	−24.9689	−1.14445
$477$	−2.55904	−0.117170
$478$	1.06283	0.0486125
$479$	3.11421	0.142292	0.0711460	−	0.997466i	$-0.477334\pi$
$479$	3.11421	0.142292	0.0711460	+	0.997466i	$0.477334\pi$
$480$	14.1799	0.647220
$481$	27.4683	1.25245
$482$	4.20248	0.191418
$483$	−11.3385	−0.515919
$484$	11.5263	0.523924
$485$	30.3685	1.37896
$486$	−0.458525	−0.0207991
$487$	38.3012	1.73559	0.867797	−	0.496919i	$-0.165535\pi$
$487$	38.3012	1.73559	0.867797	+	0.496919i	$0.165535\pi$
$488$	6.60986	0.299215
$489$	−18.6296	−0.842461
$490$	8.83237	0.399006
$491$	21.0582	0.950345	0.475173	−	0.879893i	$-0.342386\pi$
$491$	21.0582	0.950345	0.475173	+	0.879893i	$0.342386\pi$
$492$	0.856988	0.0386360
$493$	−9.02668	−0.406541
$494$	−6.05648	−0.272494
$495$	−6.37279	−0.286436
$496$	−28.7046	−1.28887
$497$	−38.6018	−1.73153
$498$	1.21834	0.0545949
$499$	−12.6489	−0.566241	−0.283120	−	0.959084i	$-0.591370\pi$
$499$	−12.6489	−0.566241	−0.283120	+	0.959084i	$0.591370\pi$
$500$	−5.84030	−0.261186
$501$	19.3259	0.863419
$502$	−4.14078	−0.184812
$503$	32.7964	1.46232	0.731159	−	0.682207i	$-0.238980\pi$
$503$	32.7964	1.46232	0.731159	+	0.682207i	$0.238980\pi$
$504$	6.37393	0.283917
$505$	57.5664	2.56167
$506$	3.02663	0.134550
$507$	−5.92343	−0.263069
$508$	37.1349	1.64760
$509$	5.80334	0.257229	0.128614	−	0.991695i	$-0.458947\pi$
$509$	5.80334	0.257229	0.128614	+	0.991695i	$0.458947\pi$
$510$	5.20464	0.230466
$511$	−1.53949	−0.0681030
$512$	−22.8928	−1.01173
$513$	4.96530	0.219224
$514$	3.56533	0.157260
$515$	2.98439	0.131508
$516$	−0.454453	−0.0200062
$517$	20.0343	0.881108
$518$	−17.3666	−0.763047
$519$	−21.9441	−0.963240
$520$	−13.7956	−0.604978
$521$	26.2510	1.15008	0.575038	−	0.818127i	$-0.304987\pi$
$521$	26.2510	1.15008	0.575038	+	0.818127i	$0.304987\pi$
$522$	1.08822	0.0476303
$523$	36.2301	1.58423	0.792115	−	0.610372i	$-0.208980\pi$
$523$	36.2301	1.58423	0.792115	+	0.610372i	$0.208980\pi$
$524$	−36.3647	−1.58860
$525$	14.3295	0.625389
$526$	12.9916	0.566461
$527$	−39.2331	−1.70902
$528$	5.94218	0.258600
$529$	−13.4447	−0.584552
$530$	−3.50184	−0.152110
$531$	−8.80184	−0.381967
$532$	−32.5966	−1.41324
$533$	−1.27378	−0.0551733
$534$	−5.35725	−0.231831
$535$	−19.5451	−0.845009
$536$	14.8467	0.641281
$537$	24.8516	1.07242
$538$	7.65289	0.329939
$539$	13.7827	0.593661
$540$	5.34133	0.229854
$541$	−6.57953	−0.282876	−0.141438	−	0.989947i	$-0.545173\pi$
$541$	−6.57953	−0.282876	−0.141438	+	0.989947i	$0.545173\pi$
$542$	0.630790	0.0270947
$543$	17.3523	0.744657
$544$	−18.0713	−0.774800
$545$	25.0045	1.07107
$546$	−4.47412	−0.191475
$547$	3.50387	0.149814	0.0749072	−	0.997191i	$-0.476134\pi$
$547$	3.50387	0.149814	0.0749072	+	0.997191i	$0.476134\pi$
$548$	3.35567	0.143347
$549$	3.80381	0.162343
$550$	−3.82502	−0.163100
$551$	−11.7842	−0.502025
$552$	−5.37151	−0.228627
$553$	5.94601	0.252850
$554$	−5.25529	−0.223276
$555$	−30.8160	−1.30806
$556$	33.2329	1.40939
$557$	−5.39985	−0.228799	−0.114400	−	0.993435i	$-0.536494\pi$
$557$	−5.39985	−0.228799	−0.114400	+	0.993435i	$0.536494\pi$
$558$	4.72980	0.200228
$559$	0.675472	0.0285694
$560$	−30.4621	−1.28726
$561$	8.12170	0.342898
$562$	14.9048	0.628722
$563$	33.4179	1.40840	0.704198	−	0.710004i	$-0.251307\pi$
$563$	33.4179	1.40840	0.704198	+	0.710004i	$0.251307\pi$
$564$	−16.7917	−0.707056
$565$	−22.0035	−0.925695
$566$	1.43542	0.0603352
$567$	3.66803	0.154043
$568$	−18.2873	−0.767317
$569$	30.5161	1.27930	0.639651	−	0.768666i	$-0.279079\pi$
$569$	30.5161	1.27930	0.639651	+	0.768666i	$0.279079\pi$
$570$	6.79461	0.284595
$571$	14.3509	0.600566	0.300283	−	0.953850i	$-0.402919\pi$
$571$	14.3509	0.600566	0.300283	+	0.953850i	$0.402919\pi$
$572$	−10.1667	−0.425090
$573$	−16.6672	−0.696284
$574$	0.805336	0.0336141
$575$	−12.0759	−0.503599
$576$	−3.38685	−0.141119
$577$	20.5720	0.856423	0.428212	−	0.903678i	$-0.359144\pi$
$577$	20.5720	0.856423	0.428212	+	0.903678i	$0.359144\pi$
$578$	1.16195	0.0483310
$579$	−16.7961	−0.698022
$580$	−12.6767	−0.526369
$581$	−9.74623	−0.404342
$582$	4.66586	0.193406
$583$	−5.46451	−0.226317
$584$	−0.729320	−0.0301795
$585$	−7.93902	−0.328238
$586$	−5.08950	−0.210246
$587$	−10.9524	−0.452054	−0.226027	−	0.974121i	$-0.572574\pi$
$587$	−10.9524	−0.452054	−0.226027	+	0.974121i	$0.572574\pi$
$588$	−11.5519	−0.476391
$589$	−51.2183	−2.11041
$590$	−12.0446	−0.495868
$591$	−2.21629	−0.0911662
$592$	28.7337	1.18095
$593$	−36.2516	−1.48868	−0.744338	−	0.667803i	$-0.767235\pi$
$593$	−36.2516	−1.48868	−0.744338	+	0.667803i	$0.767235\pi$
$594$	−0.979123	−0.0401739
$595$	−41.6352	−1.70688
$596$	−7.57052	−0.310101
$597$	11.4441	0.468374
$598$	3.77048	0.154186
$599$	−21.5881	−0.882067	−0.441034	−	0.897491i	$-0.645388\pi$
$599$	−21.5881	−0.882067	−0.441034	+	0.897491i	$0.645388\pi$
$600$	6.78846	0.277138
$601$	29.9558	1.22192	0.610960	−	0.791661i	$-0.290783\pi$
$601$	29.9558	1.22192	0.610960	+	0.791661i	$0.290783\pi$
$602$	−0.427063	−0.0174058
$603$	8.54392	0.347935
$604$	7.77278	0.316270
$605$	19.2200	0.781404
$606$	8.84457	0.359286
$607$	−25.4515	−1.03304	−0.516522	−	0.856274i	$-0.672774\pi$
$607$	−25.4515	−1.03304	−0.516522	+	0.856274i	$0.672774\pi$
$608$	−23.5919	−0.956777
$609$	−8.70539	−0.352760
$610$	5.20519	0.210752
$611$	24.9581	1.00970
$612$	−6.80716	−0.275163
$613$	4.86213	0.196380	0.0981898	−	0.995168i	$-0.468695\pi$
$613$	4.86213	0.196380	0.0981898	+	0.995168i	$0.468695\pi$
$614$	15.7028	0.633715
$615$	1.42902	0.0576235
$616$	13.6107	0.548392
$617$	−22.0082	−0.886019	−0.443009	−	0.896517i	$-0.646089\pi$
$617$	−22.0082	−0.886019	−0.443009	+	0.896517i	$0.646089\pi$
$618$	0.458525	0.0184446
$619$	23.1972	0.932376	0.466188	−	0.884686i	$-0.345627\pi$
$619$	23.1972	0.932376	0.466188	+	0.884686i	$0.345627\pi$
$620$	−55.0971	−2.21275
$621$	−3.09116	−0.124044
$622$	−0.998844	−0.0400500
$623$	42.8560	1.71699
$624$	7.40258	0.296340
$625$	−29.2715	−1.17086
$626$	−9.67033	−0.386504
$627$	10.6028	0.423435
$628$	10.7750	0.429970
$629$	39.2729	1.56591
$630$	5.01940	0.199977
$631$	−12.6554	−0.503803	−0.251901	−	0.967753i	$-0.581056\pi$
$631$	−12.6554	−0.503803	−0.251901	+	0.967753i	$0.581056\pi$
$632$	2.81687	0.112049
$633$	20.9436	0.832434
$634$	−11.5733	−0.459633
$635$	61.9219	2.45730
$636$	4.58006	0.181611
$637$	17.1700	0.680300
$638$	2.32377	0.0919989
$639$	−10.5239	−0.416317
$640$	−32.9944	−1.30422
$641$	6.02243	0.237871	0.118936	−	0.992902i	$-0.462052\pi$
$641$	6.02243	0.237871	0.118936	+	0.992902i	$0.462052\pi$
$642$	−3.00293	−0.118516
$643$	−7.13315	−0.281304	−0.140652	−	0.990059i	$-0.544920\pi$
$643$	−7.13315	−0.281304	−0.140652	+	0.990059i	$0.544920\pi$
$644$	20.2931	0.799660
$645$	−0.757794	−0.0298381
$646$	−8.65927	−0.340695
$647$	−17.0613	−0.670748	−0.335374	−	0.942085i	$-0.608863\pi$
$647$	−17.0613	−0.670748	−0.335374	+	0.942085i	$0.608863\pi$
$648$	1.73770	0.0682632
$649$	−18.7952	−0.737777
$650$	−4.76509	−0.186902
$651$	−37.8366	−1.48293
$652$	33.3425	1.30579
$653$	−29.0458	−1.13665	−0.568325	−	0.822804i	$-0.692409\pi$
$653$	−29.0458	−1.13665	−0.568325	+	0.822804i	$0.692409\pi$
$654$	3.84171	0.150223
$655$	−60.6375	−2.36930
$656$	−1.33246	−0.0520236
$657$	−0.419705	−0.0163742
$658$	−15.7796	−0.615153
$659$	−1.07369	−0.0418251	−0.0209126	−	0.999781i	$-0.506657\pi$
$659$	−1.07369	−0.0418251	−0.0209126	+	0.999781i	$0.506657\pi$
$660$	11.4057	0.443968
$661$	−12.0339	−0.468064	−0.234032	−	0.972229i	$-0.575192\pi$
$661$	−12.0339	−0.468064	−0.234032	+	0.972229i	$0.575192\pi$
$662$	0.601881	0.0233928
$663$	10.1178	0.392941
$664$	−4.61719	−0.179182
$665$	−54.3543	−2.10777
$666$	−4.73460	−0.183462
$667$	7.33631	0.284063
$668$	−34.5887	−1.33828
$669$	−2.52865	−0.0977634
$670$	11.6916	0.451688
$671$	8.12255	0.313568
$672$	−17.4281	−0.672303
$673$	2.44879	0.0943939	0.0471969	−	0.998886i	$-0.484971\pi$
$673$	2.44879	0.0943939	0.0471969	+	0.998886i	$0.484971\pi$
$674$	4.07349	0.156905
$675$	3.90658	0.150364
$676$	10.6015	0.407749
$677$	−41.3634	−1.58973	−0.794863	−	0.606789i	$-0.792457\pi$
$677$	−41.3634	−1.58973	−0.794863	+	0.606789i	$0.792457\pi$
$678$	−3.38065	−0.129833
$679$	−37.3251	−1.43241
$680$	−19.7243	−0.756393
$681$	18.8238	0.721329
$682$	10.0999	0.386745
$683$	−12.3026	−0.470745	−0.235372	−	0.971905i	$-0.575631\pi$
$683$	−12.3026	−0.470745	−0.235372	+	0.971905i	$0.575631\pi$
$684$	−8.88668	−0.339791
$685$	5.59552	0.213794
$686$	0.917568	0.0350329
$687$	14.3108	0.545990
$688$	0.706590	0.0269385
$689$	−6.80752	−0.259346
$690$	−4.23000	−0.161033
$691$	44.6490	1.69853	0.849263	−	0.527969i	$-0.177046\pi$
$691$	44.6490	1.69853	0.849263	+	0.527969i	$0.177046\pi$
$692$	39.2746	1.49300
$693$	7.83262	0.297537
$694$	−10.9408	−0.415308
$695$	55.4154	2.10203
$696$	−4.12411	−0.156324
$697$	−1.82118	−0.0689823
$698$	10.7210	0.405797
$699$	−11.3116	−0.427843
$700$	−25.6462	−0.969336
$701$	−7.73853	−0.292280	−0.146140	−	0.989264i	$-0.546685\pi$
$701$	−7.73853	−0.292280	−0.146140	+	0.989264i	$0.546685\pi$
$702$	−1.21976	−0.0460369
$703$	51.2703	1.93370
$704$	−7.23220	−0.272574
$705$	−27.9998	−1.05454
$706$	5.46692	0.205750
$707$	−70.7532	−2.66095
$708$	15.7531	0.592039
$709$	−8.53625	−0.320585	−0.160293	−	0.987070i	$-0.551244\pi$
$709$	−8.53625	−0.320585	−0.160293	+	0.987070i	$0.551244\pi$
$710$	−14.4010	−0.540461
$711$	1.62104	0.0607936
$712$	20.3027	0.760875
$713$	31.8861	1.19414
$714$	−6.39688	−0.239397
$715$	−16.9528	−0.633998
$716$	−44.4782	−1.66223
$717$	−2.31792	−0.0865645
$718$	7.94502	0.296505
$719$	38.5134	1.43631	0.718154	−	0.695884i	$-0.244987\pi$
$719$	38.5134	1.43631	0.718154	+	0.695884i	$0.244987\pi$
$720$	−8.30476	−0.309500
$721$	−3.66803	−0.136605
$722$	−2.59261	−0.0964871
$723$	−9.16521	−0.340858
$724$	−31.0563	−1.15420
$725$	−9.27155	−0.344337
$726$	2.95298	0.109595
$727$	27.0036	1.00151	0.500754	−	0.865590i	$-0.333056\pi$
$727$	27.0036	1.00151	0.500754	+	0.865590i	$0.333056\pi$
$728$	16.9558	0.628424
$729$	1.00000	0.0370370
$730$	−0.574331	−0.0212570
$731$	0.965758	0.0357199
$732$	−6.80788	−0.251627
$733$	16.5282	0.610483	0.305241	−	0.952275i	$-0.401263\pi$
$733$	16.5282	0.610483	0.305241	+	0.952275i	$0.401263\pi$
$734$	12.0330	0.444145
$735$	−19.2626	−0.710510
$736$	14.6872	0.541377
$737$	18.2445	0.672043
$738$	0.219556	0.00808195
$739$	−39.0406	−1.43613	−0.718065	−	0.695976i	$-0.754972\pi$
$739$	−39.0406	−1.43613	−0.718065	+	0.695976i	$0.754972\pi$
$740$	55.1530	2.02746
$741$	13.2086	0.485231
$742$	4.30401	0.158005
$743$	−14.7926	−0.542688	−0.271344	−	0.962482i	$-0.587468\pi$
$743$	−14.7926	−0.542688	−0.271344	+	0.962482i	$0.587468\pi$
$744$	−17.9248	−0.657154
$745$	−12.6237	−0.462498
$746$	−2.55717	−0.0936246
$747$	−2.65708	−0.0972173
$748$	−14.5358	−0.531483
$749$	24.0224	0.877758
$750$	−1.49625	−0.0546354
$751$	−34.3035	−1.25175	−0.625877	−	0.779922i	$-0.715259\pi$
$751$	−34.3035	−1.25175	−0.625877	+	0.779922i	$0.715259\pi$
$752$	26.1079	0.952056
$753$	9.03064	0.329095
$754$	2.89488	0.105425
$755$	12.9610	0.471699
$756$	−6.56487	−0.238762
$757$	20.2105	0.734562	0.367281	−	0.930110i	$-0.380289\pi$
$757$	20.2105	0.734562	0.367281	+	0.930110i	$0.380289\pi$
$758$	9.07943	0.329780
$759$	−6.60080	−0.239594
$760$	−25.7499	−0.934047
$761$	43.8610	1.58996	0.794980	−	0.606636i	$-0.207481\pi$
$761$	43.8610	1.58996	0.794980	+	0.606636i	$0.207481\pi$
$762$	9.51376	0.344647
$763$	−30.7323	−1.11258
$764$	29.8303	1.07922
$765$	−11.3508	−0.410391
$766$	12.9310	0.467217
$767$	−23.4145	−0.845449
$768$	1.70441	0.0615026
$769$	−20.1466	−0.726504	−0.363252	−	0.931691i	$-0.618334\pi$
$769$	−20.1466	−0.726504	−0.363252	+	0.931691i	$0.618334\pi$
$770$	10.7183	0.386260
$771$	−7.77565	−0.280033
$772$	30.0609	1.08191
$773$	27.2315	0.979449	0.489724	−	0.871877i	$-0.337097\pi$
$773$	27.2315	0.979449	0.489724	+	0.871877i	$0.337097\pi$
$774$	−0.116428	−0.00418493
$775$	−40.2973	−1.44752
$776$	−17.6825	−0.634763
$777$	37.8750	1.35876
$778$	15.5134	0.556183
$779$	−2.37754	−0.0851841
$780$	14.2089	0.508760
$781$	−22.4724	−0.804125
$782$	5.39086	0.192777
$783$	−2.37332	−0.0848154
$784$	17.9610	0.641463
$785$	17.9672	0.641276
$786$	−9.31642	−0.332306
$787$	20.6687	0.736760	0.368380	−	0.929675i	$-0.379913\pi$
$787$	20.6687	0.736760	0.368380	+	0.929675i	$0.379913\pi$
$788$	3.96662	0.141305
$789$	−28.3335	−1.00870
$790$	2.21825	0.0789219
$791$	27.0439	0.961571
$792$	3.71064	0.131852
$793$	10.1188	0.359330
$794$	15.4840	0.549506
$795$	7.63718	0.270863
$796$	−20.4821	−0.725967
$797$	−21.6345	−0.766334	−0.383167	−	0.923679i	$-0.625167\pi$
$797$	−21.6345	−0.766334	−0.383167	+	0.923679i	$0.625167\pi$
$798$	−8.35106	−0.295624
$799$	35.6839	1.26241
$800$	−18.5615	−0.656249
$801$	11.6837	0.412822
$802$	−8.73732	−0.308526
$803$	−0.896227	−0.0316272
$804$	−15.2915	−0.539290
$805$	33.8384	1.19265
$806$	12.5821	0.443187
$807$	−16.6902	−0.587524
$808$	−33.5187	−1.17919
$809$	−26.6456	−0.936809	−0.468405	−	0.883514i	$-0.655171\pi$
$809$	−26.6456	−0.936809	−0.468405	+	0.883514i	$0.655171\pi$
$810$	1.36842	0.0480813
$811$	−17.4183	−0.611639	−0.305819	−	0.952090i	$-0.598930\pi$
$811$	−17.4183	−0.611639	−0.305819	+	0.952090i	$0.598930\pi$
$812$	15.5805	0.546769
$813$	−1.37569	−0.0482477
$814$	−10.1101	−0.354360
$815$	55.5981	1.94752
$816$	10.5839	0.370509
$817$	1.26079	0.0441094
$818$	−8.24151	−0.288158
$819$	9.75763	0.340959
$820$	−2.55759	−0.0893148
$821$	−31.1893	−1.08851	−0.544257	−	0.838919i	$-0.683188\pi$
$821$	−31.1893	−1.08851	−0.544257	+	0.838919i	$0.683188\pi$
$822$	0.859703	0.0299856
$823$	−24.1120	−0.840492	−0.420246	−	0.907410i	$-0.638056\pi$
$823$	−24.1120	−0.840492	−0.420246	+	0.907410i	$0.638056\pi$
$824$	−1.73770	−0.0605356
$825$	8.34202	0.290432
$826$	14.8037	0.515086
$827$	−10.7875	−0.375118	−0.187559	−	0.982253i	$-0.560058\pi$
$827$	−10.7875	−0.375118	−0.187559	+	0.982253i	$0.560058\pi$
$828$	5.53243	0.192265
$829$	27.5658	0.957400	0.478700	−	0.877979i	$-0.341108\pi$
$829$	27.5658	0.957400	0.478700	+	0.877979i	$0.341108\pi$
$830$	−3.63599	−0.126207
$831$	11.4613	0.397588
$832$	−9.00965	−0.312353
$833$	24.5488	0.850567
$834$	8.51409	0.294819
$835$	−57.6761	−1.99597
$836$	−18.9764	−0.656312
$837$	−10.3152	−0.356547
$838$	−12.2069	−0.421680
$839$	18.7755	0.648202	0.324101	−	0.946023i	$-0.394938\pi$
$839$	18.7755	0.648202	0.324101	+	0.946023i	$0.394938\pi$
$840$	−19.0223	−0.656331
$841$	−23.3674	−0.805771
$842$	−13.6520	−0.470479
$843$	−32.5060	−1.11957
$844$	−37.4839	−1.29025
$845$	17.6778	0.608135
$846$	−4.30193	−0.147903
$847$	−23.6227	−0.811687
$848$	−7.12112	−0.244540
$849$	−3.13052	−0.107439
$850$	−6.81291	−0.233681
$851$	−31.9185	−1.09415
$852$	18.8351	0.645281
$853$	10.5985	0.362885	0.181442	−	0.983402i	$-0.441923\pi$
$853$	10.5985	0.362885	0.181442	+	0.983402i	$0.441923\pi$
$854$	−6.39756	−0.218920
$855$	−14.8184	−0.506779
$856$	11.3804	0.388974
$857$	−5.93635	−0.202782	−0.101391	−	0.994847i	$-0.532329\pi$
$857$	−5.93635	−0.202782	−0.101391	+	0.994847i	$0.532329\pi$
$858$	−2.60465	−0.0889212
$859$	45.7851	1.56217	0.781084	−	0.624426i	$-0.214667\pi$
$859$	45.7851	1.56217	0.781084	+	0.624426i	$0.214667\pi$
$860$	1.35627	0.0462483
$861$	−1.75636	−0.0598567
$862$	−7.92745	−0.270010
$863$	−8.99269	−0.306115	−0.153057	−	0.988217i	$-0.548912\pi$
$863$	−8.99269	−0.306115	−0.153057	+	0.988217i	$0.548912\pi$
$864$	−4.75135	−0.161644
$865$	65.4898	2.22672
$866$	−3.38418	−0.114999
$867$	−2.53411	−0.0860630
$868$	67.7183	2.29851
$869$	3.46152	0.117424
$870$	−3.24769	−0.110107
$871$	22.7284	0.770122
$872$	−14.5592	−0.493035
$873$	−10.1758	−0.344398
$874$	7.03770	0.238054
$875$	11.9695	0.404642
$876$	0.751169	0.0253796
$877$	34.2472	1.15645	0.578223	−	0.815879i	$-0.303746\pi$
$877$	34.2472	1.15645	0.578223	+	0.815879i	$0.303746\pi$
$878$	−11.9294	−0.402598
$879$	11.0997	0.374385
$880$	−17.7338	−0.597805
$881$	18.4648	0.622095	0.311048	−	0.950394i	$-0.399320\pi$
$881$	18.4648	0.622095	0.311048	+	0.950394i	$0.399320\pi$
$882$	−2.95952	−0.0996523
$883$	−2.91705	−0.0981666	−0.0490833	−	0.998795i	$-0.515630\pi$
$883$	−2.91705	−0.0981666	−0.0490833	+	0.998795i	$0.515630\pi$
$884$	−18.1083	−0.609048
$885$	26.2681	0.882993
$886$	1.93621	0.0650483
$887$	33.1871	1.11432	0.557158	−	0.830407i	$-0.311892\pi$
$887$	33.1871	1.11432	0.557158	+	0.830407i	$0.311892\pi$
$888$	17.9430	0.602127
$889$	−76.1065	−2.55253
$890$	15.9881	0.535923
$891$	2.13538	0.0715378
$892$	4.52567	0.151531
$893$	46.5850	1.55891
$894$	−1.93952	−0.0648674
$895$	−74.1668	−2.47912
$896$	40.5525	1.35476
$897$	−8.22306	−0.274560
$898$	3.83842	0.128090
$899$	24.4813	0.816498
$900$	−6.99182	−0.233061
$901$	−9.73307	−0.324256
$902$	0.468833	0.0156104
$903$	0.931384	0.0309945
$904$	12.8118	0.426115
$905$	−51.7859	−1.72142
$906$	1.99134	0.0661579
$907$	−27.4103	−0.910145	−0.455073	−	0.890454i	$-0.650387\pi$
$907$	−27.4103	−0.910145	−0.455073	+	0.890454i	$0.650387\pi$
$908$	−33.6899	−1.11804
$909$	−19.2892	−0.639781
$910$	13.3525	0.442631
$911$	20.8301	0.690133	0.345066	−	0.938578i	$-0.387856\pi$
$911$	20.8301	0.690133	0.345066	+	0.938578i	$0.387856\pi$
$912$	13.8171	0.457530
$913$	−5.67385	−0.187777
$914$	−0.278018	−0.00919603
$915$	−11.3520	−0.375287
$916$	−25.6128	−0.846269
$917$	74.5279	2.46113
$918$	−1.74396	−0.0575591
$919$	55.1338	1.81870	0.909349	−	0.416035i	$-0.136581\pi$
$919$	55.1338	1.81870	0.909349	+	0.416035i	$0.136581\pi$
$920$	16.0307	0.528516
$921$	−34.2464	−1.12846
$922$	−1.95815	−0.0644881
$923$	−27.9954	−0.921480
$924$	−14.0185	−0.461174
$925$	40.3382	1.32631
$926$	−1.37045	−0.0450359
$927$	−1.00000	−0.0328443
$928$	11.2764	0.370168
$929$	−3.29062	−0.107962	−0.0539808	−	0.998542i	$-0.517191\pi$
$929$	−3.29062	−0.107962	−0.0539808	+	0.998542i	$0.517191\pi$
$930$	−14.1156	−0.462868
$931$	32.0483	1.05034
$932$	20.2449	0.663145
$933$	2.17838	0.0713171
$934$	0.374162	0.0122430
$935$	−24.2383	−0.792677
$936$	4.62259	0.151094
$937$	9.60541	0.313795	0.156898	−	0.987615i	$-0.449851\pi$
$937$	9.60541	0.313795	0.156898	+	0.987615i	$0.449851\pi$
$938$	−14.3699	−0.469193
$939$	21.0901	0.688249
$940$	50.1129	1.63450
$941$	−54.1022	−1.76368	−0.881841	−	0.471547i	$-0.843696\pi$
$941$	−54.1022	−1.76368	−0.881841	+	0.471547i	$0.843696\pi$
$942$	2.76050	0.0899419
$943$	1.48014	0.0482001
$944$	−24.4932	−0.797184
$945$	−10.9468	−0.356100
$946$	−0.248618	−0.00808328
$947$	15.7142	0.510643	0.255322	−	0.966856i	$-0.417819\pi$
$947$	15.7142	0.510643	0.255322	+	0.966856i	$0.417819\pi$
$948$	−2.90126	−0.0942284
$949$	−1.11649	−0.0362429
$950$	−8.89418	−0.288565
$951$	25.2402	0.818470
$952$	24.2426	0.785708
$953$	18.2052	0.589724	0.294862	−	0.955540i	$-0.404726\pi$
$953$	18.2052	0.589724	0.294862	+	0.955540i	$0.404726\pi$
$954$	1.17338	0.0379897
$955$	49.7416	1.60960
$956$	4.14852	0.134173
$957$	−5.06792	−0.163823
$958$	−1.42794	−0.0461348
$959$	−6.87730	−0.222080
$960$	10.1077	0.326224
$961$	75.4043	2.43240
$962$	−12.5949	−0.406076
$963$	6.54912	0.211042
$964$	16.4035	0.528321
$965$	50.1261	1.61362
$966$	5.19898	0.167274
$967$	−4.96023	−0.159510	−0.0797552	−	0.996814i	$-0.525414\pi$
$967$	−4.96023	−0.159510	−0.0797552	+	0.996814i	$0.525414\pi$
$968$	−11.1911	−0.359695
$969$	18.8851	0.606676
$970$	−13.9247	−0.447096
$971$	61.6703	1.97909	0.989546	−	0.144215i	$-0.0460656\pi$
$971$	61.6703	1.97909	0.989546	+	0.144215i	$0.0460656\pi$
$972$	−1.78975	−0.0574064
$973$	−68.1095	−2.18349
$974$	−17.5621	−0.562725
$975$	10.3922	0.332817
$976$	10.5850	0.338817
$977$	15.0777	0.482378	0.241189	−	0.970478i	$-0.422463\pi$
$977$	15.0777	0.482378	0.241189	+	0.970478i	$0.422463\pi$
$978$	8.54216	0.273148
$979$	24.9490	0.797374
$980$	34.4753	1.10127
$981$	−8.37842	−0.267502
$982$	−9.65573	−0.308127
$983$	−24.0848	−0.768185	−0.384092	−	0.923295i	$-0.625486\pi$
$983$	−24.0848	−0.768185	−0.384092	+	0.923295i	$0.625486\pi$
$984$	−0.832062	−0.0265252
$985$	6.61429	0.210749
$986$	4.13896	0.131811
$987$	34.4138	1.09540
$988$	−23.6402	−0.752095
$989$	−0.784907	−0.0249586
$990$	2.92208	0.0928699
$991$	24.1852	0.768268	0.384134	−	0.923277i	$-0.374500\pi$
$991$	24.1852	0.768268	0.384134	+	0.923277i	$0.374500\pi$
$992$	49.0113	1.55611
$993$	−1.31265	−0.0416556
$994$	17.6999	0.561407
$995$	−34.1535	−1.08274
$996$	4.75551	0.150684
$997$	5.16916	0.163709	0.0818545	−	0.996644i	$-0.473916\pi$
$997$	5.16916	0.163709	0.0818545	+	0.996644i	$0.473916\pi$
$998$	5.79982	0.183590
$999$	10.3257	0.326691

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	309.2.a.d.1.4	✓	8
3.2	odd	2		927.2.a.g.1.5		8
4.3	odd	2		4944.2.a.bf.1.2		8
5.4	even	2		7725.2.a.z.1.5		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
309.2.a.d.1.4	✓	8	1.1	even	1	trivial
927.2.a.g.1.5		8	3.2	odd	2
4944.2.a.bf.1.2		8	4.3	odd	2
7725.2.a.z.1.5		8	5.4	even	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:	\( N \)	\(=\)	\( 309 = 3 \cdot 103 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	309.a (trivial)

\(f(q)\)	\(=\)	\(q-0.458525 q^{2} +1.00000 q^{3} -1.78975 q^{4} -2.98439 q^{5} -0.458525 q^{6} +3.66803 q^{7} +1.73770 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-0.458525 q^{2} +1.00000 q^{3} -1.78975 q^{4} -2.98439 q^{5} -0.458525 q^{6} +3.66803 q^{7} +1.73770 q^{8} +1.00000 q^{9} +1.36842 q^{10} +2.13538 q^{11} -1.78975 q^{12} +2.66018 q^{13} -1.68188 q^{14} -2.98439 q^{15} +2.78273 q^{16} +3.80340 q^{17} -0.458525 q^{18} +4.96530 q^{19} +5.34133 q^{20} +3.66803 q^{21} -0.979123 q^{22} -3.09116 q^{23} +1.73770 q^{24} +3.90658 q^{25} -1.21976 q^{26} +1.00000 q^{27} -6.56487 q^{28} -2.37332 q^{29} +1.36842 q^{30} -10.3152 q^{31} -4.75135 q^{32} +2.13538 q^{33} -1.74396 q^{34} -10.9468 q^{35} -1.78975 q^{36} +10.3257 q^{37} -2.27672 q^{38} +2.66018 q^{39} -5.18597 q^{40} -0.478830 q^{41} -1.68188 q^{42} +0.253919 q^{43} -3.82180 q^{44} -2.98439 q^{45} +1.41738 q^{46} +9.38210 q^{47} +2.78273 q^{48} +6.45444 q^{49} -1.79127 q^{50} +3.80340 q^{51} -4.76108 q^{52} -2.55904 q^{53} -0.458525 q^{54} -6.37279 q^{55} +6.37393 q^{56} +4.96530 q^{57} +1.08822 q^{58} -8.80184 q^{59} +5.34133 q^{60} +3.80381 q^{61} +4.72980 q^{62} +3.66803 q^{63} -3.38685 q^{64} -7.93902 q^{65} -0.979123 q^{66} +8.54392 q^{67} -6.80716 q^{68} -3.09116 q^{69} +5.01940 q^{70} -10.5239 q^{71} +1.73770 q^{72} -0.419705 q^{73} -4.73460 q^{74} +3.90658 q^{75} -8.88668 q^{76} +7.83262 q^{77} -1.21976 q^{78} +1.62104 q^{79} -8.30476 q^{80} +1.00000 q^{81} +0.219556 q^{82} -2.65708 q^{83} -6.56487 q^{84} -11.3508 q^{85} -0.116428 q^{86} -2.37332 q^{87} +3.71064 q^{88} +11.6837 q^{89} +1.36842 q^{90} +9.75763 q^{91} +5.53243 q^{92} -10.3152 q^{93} -4.30193 q^{94} -14.8184 q^{95} -4.75135 q^{96} -10.1758 q^{97} -2.95952 q^{98} +2.13538 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - q^{2} + 8 q^{3} + 11 q^{4} - q^{5} - q^{6} + 6 q^{7} - 3 q^{8} + 8 q^{9}+O(q^{10}) \) 8 * q - q^2 + 8 * q^3 + 11 * q^4 - q^5 - q^6 + 6 * q^7 - 3 * q^8 + 8 * q^9	\( 8 q - q^{2} + 8 q^{3} + 11 q^{4} - q^{5} - q^{6} + 6 q^{7} - 3 q^{8} + 8 q^{9} + 3 q^{10} + 6 q^{11} + 11 q^{12} + 9 q^{13} - 6 q^{14} - q^{15} + 9 q^{16} - 4 q^{17} - q^{18} + 16 q^{19} - 23 q^{20} + 6 q^{21} - 4 q^{22} - 11 q^{23} - 3 q^{24} + 15 q^{25} - 14 q^{26} + 8 q^{27} - 5 q^{28} + 3 q^{30} + 17 q^{31} - 12 q^{32} + 6 q^{33} - 8 q^{34} + 4 q^{35} + 11 q^{36} - 6 q^{37} - 3 q^{38} + 9 q^{39} + 3 q^{40} + 12 q^{41} - 6 q^{42} + 9 q^{43} + 8 q^{44} - q^{45} - 30 q^{46} - 6 q^{47} + 9 q^{48} + 18 q^{49} - 36 q^{50} - 4 q^{51} + 23 q^{52} - 16 q^{53} - q^{54} - 10 q^{55} - 13 q^{56} + 16 q^{57} - 22 q^{58} + 11 q^{59} - 23 q^{60} + 5 q^{61} - 25 q^{62} + 6 q^{63} - 35 q^{64} - 41 q^{65} - 4 q^{66} + 5 q^{67} - 19 q^{68} - 11 q^{69} - 48 q^{70} - 10 q^{71} - 3 q^{72} + 14 q^{73} + 4 q^{74} + 15 q^{75} - 12 q^{76} - 40 q^{77} - 14 q^{78} + 14 q^{79} - 19 q^{80} + 8 q^{81} - 13 q^{82} - 23 q^{83} - 5 q^{84} - 4 q^{85} - 3 q^{86} - 30 q^{88} - 14 q^{89} + 3 q^{90} - 6 q^{91} - 21 q^{92} + 17 q^{93} + 22 q^{94} + 6 q^{95} - 12 q^{96} + 3 q^{97} - 18 q^{98} + 6 q^{99}+O(q^{100}) \) 8 * q - q^2 + 8 * q^3 + 11 * q^4 - q^5 - q^6 + 6 * q^7 - 3 * q^8 + 8 * q^9 + 3 * q^10 + 6 * q^11 + 11 * q^12 + 9 * q^13 - 6 * q^14 - q^15 + 9 * q^16 - 4 * q^17 - q^18 + 16 * q^19 - 23 * q^20 + 6 * q^21 - 4 * q^22 - 11 * q^23 - 3 * q^24 + 15 * q^25 - 14 * q^26 + 8 * q^27 - 5 * q^28 + 3 * q^30 + 17 * q^31 - 12 * q^32 + 6 * q^33 - 8 * q^34 + 4 * q^35 + 11 * q^36 - 6 * q^37 - 3 * q^38 + 9 * q^39 + 3 * q^40 + 12 * q^41 - 6 * q^42 + 9 * q^43 + 8 * q^44 - q^45 - 30 * q^46 - 6 * q^47 + 9 * q^48 + 18 * q^49 - 36 * q^50 - 4 * q^51 + 23 * q^52 - 16 * q^53 - q^54 - 10 * q^55 - 13 * q^56 + 16 * q^57 - 22 * q^58 + 11 * q^59 - 23 * q^60 + 5 * q^61 - 25 * q^62 + 6 * q^63 - 35 * q^64 - 41 * q^65 - 4 * q^66 + 5 * q^67 - 19 * q^68 - 11 * q^69 - 48 * q^70 - 10 * q^71 - 3 * q^72 + 14 * q^73 + 4 * q^74 + 15 * q^75 - 12 * q^76 - 40 * q^77 - 14 * q^78 + 14 * q^79 - 19 * q^80 + 8 * q^81 - 13 * q^82 - 23 * q^83 - 5 * q^84 - 4 * q^85 - 3 * q^86 - 30 * q^88 - 14 * q^89 + 3 * q^90 - 6 * q^91 - 21 * q^92 + 17 * q^93 + 22 * q^94 + 6 * q^95 - 12 * q^96 + 3 * q^97 - 18 * q^98 + 6 * q^99

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