LMFDB - Embedded newform 403.2.a.e.1.2

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:	$0$
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} - 11x^{6} + 10x^{5} + 37x^{4} - 33x^{3} - 36x^{2} + 33x - 4 $ x^8 - x^7 - 11x^6 + 10x^5 + 37x^4 - 33x^3 - 36x^2 + 33x - 4
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			1.2
Root			$1.71590$ 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)

$f(q)$	$=$	$q-1.71590 q^{2} -1.27835 q^{3} +0.944315 q^{4} -2.00831 q^{5} +2.19352 q^{6} -3.76625 q^{7} +1.81145 q^{8} -1.36583 q^{9} +O(q^{10})$	\(q-1.71590 q^{2} -1.27835 q^{3} +0.944315 q^{4} -2.00831 q^{5} +2.19352 q^{6} -3.76625 q^{7} +1.81145 q^{8} -1.36583 q^{9} +3.44606 q^{10} -2.15365 q^{11} -1.20716 q^{12} +1.00000 q^{13} +6.46251 q^{14} +2.56732 q^{15} -4.99690 q^{16} +0.496780 q^{17} +2.34363 q^{18} -2.24448 q^{19} -1.89648 q^{20} +4.81457 q^{21} +3.69545 q^{22} +5.01638 q^{23} -2.31566 q^{24} -0.966689 q^{25} -1.71590 q^{26} +5.58104 q^{27} -3.55653 q^{28} +2.58766 q^{29} -4.40526 q^{30} -1.00000 q^{31} +4.95128 q^{32} +2.75311 q^{33} -0.852424 q^{34} +7.56380 q^{35} -1.28977 q^{36} +0.294584 q^{37} +3.85130 q^{38} -1.27835 q^{39} -3.63795 q^{40} +10.0822 q^{41} -8.26133 q^{42} +1.81940 q^{43} -2.03373 q^{44} +2.74301 q^{45} -8.60761 q^{46} -0.507742 q^{47} +6.38777 q^{48} +7.18464 q^{49} +1.65874 q^{50} -0.635057 q^{51} +0.944315 q^{52} +11.8424 q^{53} -9.57652 q^{54} +4.32520 q^{55} -6.82238 q^{56} +2.86922 q^{57} -4.44017 q^{58} -9.27755 q^{59} +2.42436 q^{60} +0.950004 q^{61} +1.71590 q^{62} +5.14405 q^{63} +1.49789 q^{64} -2.00831 q^{65} -4.72407 q^{66} -3.55834 q^{67} +0.469116 q^{68} -6.41267 q^{69} -12.9787 q^{70} -14.3034 q^{71} -2.47413 q^{72} -0.201282 q^{73} -0.505477 q^{74} +1.23576 q^{75} -2.11949 q^{76} +8.11119 q^{77} +2.19352 q^{78} -8.22770 q^{79} +10.0353 q^{80} -3.03702 q^{81} -17.3000 q^{82} +8.35070 q^{83} +4.54648 q^{84} -0.997688 q^{85} -3.12191 q^{86} -3.30793 q^{87} -3.90123 q^{88} +10.6914 q^{89} -4.70673 q^{90} -3.76625 q^{91} +4.73704 q^{92} +1.27835 q^{93} +0.871234 q^{94} +4.50761 q^{95} -6.32946 q^{96} -11.7424 q^{97} -12.3281 q^{98} +2.94152 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - q^{2} + 7 q^{3} + 7 q^{4} + 11 q^{5} - 2 q^{7} + 9 q^{9}+O(q^{10}) $ 8 * q - q^2 + 7 * q^3 + 7 * q^4 + 11 * q^5 - 2 * q^7 + 9 * q^9	\( 8 q - q^{2} + 7 q^{3} + 7 q^{4} + 11 q^{5} - 2 q^{7} + 9 q^{9} - 2 q^{10} - 2 q^{11} + 19 q^{12} + 8 q^{13} + 3 q^{14} + 6 q^{15} - 7 q^{16} + 7 q^{17} - 12 q^{18} - 5 q^{19} + 6 q^{20} + 8 q^{21} - 4 q^{22} + 14 q^{23} - 17 q^{24} + 17 q^{25} - q^{26} + 7 q^{27} - 9 q^{28} + 12 q^{29} - 9 q^{30} - 8 q^{31} - 21 q^{32} + 10 q^{33} - 12 q^{34} - q^{35} + 11 q^{36} + 2 q^{37} + 24 q^{38} + 7 q^{39} - 19 q^{40} + 13 q^{41} - 27 q^{42} - 5 q^{43} + 22 q^{44} + 19 q^{45} + 17 q^{46} + 23 q^{47} + 3 q^{48} + 26 q^{49} - 26 q^{50} + 18 q^{51} + 7 q^{52} + 25 q^{53} - 36 q^{54} - 17 q^{55} + 8 q^{56} - 35 q^{57} - 29 q^{58} - 5 q^{59} + 71 q^{60} - 9 q^{61} + q^{62} - 37 q^{63} - 14 q^{64} + 11 q^{65} - 41 q^{66} + 22 q^{67} - 6 q^{68} - 7 q^{69} - 29 q^{70} - 17 q^{71} - 34 q^{72} - 27 q^{73} + 14 q^{74} - 33 q^{75} - 36 q^{76} + 31 q^{77} - 23 q^{79} + 9 q^{80} - 12 q^{81} + 18 q^{82} - 25 q^{83} - 62 q^{84} + 13 q^{85} + 11 q^{86} + 26 q^{87} + 5 q^{88} + 2 q^{89} - 14 q^{90} - 2 q^{91} - 20 q^{92} - 7 q^{93} - 38 q^{94} + 3 q^{95} - 52 q^{96} - 15 q^{97} + 39 q^{98} - 8 q^{99}+O(q^{100}) \) 8 * q - q^2 + 7 * q^3 + 7 * q^4 + 11 * q^5 - 2 * q^7 + 9 * q^9 - 2 * q^10 - 2 * q^11 + 19 * q^12 + 8 * q^13 + 3 * q^14 + 6 * q^15 - 7 * q^16 + 7 * q^17 - 12 * q^18 - 5 * q^19 + 6 * q^20 + 8 * q^21 - 4 * q^22 + 14 * q^23 - 17 * q^24 + 17 * q^25 - q^26 + 7 * q^27 - 9 * q^28 + 12 * q^29 - 9 * q^30 - 8 * q^31 - 21 * q^32 + 10 * q^33 - 12 * q^34 - q^35 + 11 * q^36 + 2 * q^37 + 24 * q^38 + 7 * q^39 - 19 * q^40 + 13 * q^41 - 27 * q^42 - 5 * q^43 + 22 * q^44 + 19 * q^45 + 17 * q^46 + 23 * q^47 + 3 * q^48 + 26 * q^49 - 26 * q^50 + 18 * q^51 + 7 * q^52 + 25 * q^53 - 36 * q^54 - 17 * q^55 + 8 * q^56 - 35 * q^57 - 29 * q^58 - 5 * q^59 + 71 * q^60 - 9 * q^61 + q^62 - 37 * q^63 - 14 * q^64 + 11 * q^65 - 41 * q^66 + 22 * q^67 - 6 * q^68 - 7 * q^69 - 29 * q^70 - 17 * q^71 - 34 * q^72 - 27 * q^73 + 14 * q^74 - 33 * q^75 - 36 * q^76 + 31 * q^77 - 23 * q^79 + 9 * q^80 - 12 * q^81 + 18 * q^82 - 25 * q^83 - 62 * q^84 + 13 * q^85 + 11 * q^86 + 26 * q^87 + 5 * q^88 + 2 * q^89 - 14 * q^90 - 2 * q^91 - 20 * q^92 - 7 * q^93 - 38 * q^94 + 3 * q^95 - 52 * q^96 - 15 * q^97 + 39 * q^98 - 8 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−1.71590	−1.21333	−0.606663	−	0.794960i	$-0.707492\pi$
$2$	−1.71590	−1.21333	−0.606663	+	0.794960i	$0.707492\pi$
$3$	−1.27835	−0.738054	−0.369027	−	0.929419i	$-0.620309\pi$
$3$	−1.27835	−0.738054	−0.369027	+	0.929419i	$0.620309\pi$
$4$	0.944315	0.472158
$5$	−2.00831	−0.898144	−0.449072	−	0.893496i	$-0.648245\pi$
$5$	−2.00831	−0.898144	−0.449072	+	0.893496i	$0.648245\pi$
$6$	2.19352	0.895499
$7$	−3.76625	−1.42351	−0.711754	−	0.702428i	$-0.752099\pi$
$7$	−3.76625	−1.42351	−0.711754	+	0.702428i	$0.752099\pi$
$8$	1.81145	0.640444
$9$	−1.36583	−0.455276
$10$	3.44606	1.08974
$11$	−2.15365	−0.649350	−0.324675	−	0.945826i	$-0.605255\pi$
$11$	−2.15365	−0.649350	−0.324675	+	0.945826i	$0.605255\pi$
$12$	−1.20716	−0.348478
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	6.46251	1.72718
$15$	2.56732	0.662879
$16$	−4.99690	−1.24922
$17$	0.496780	0.120487	0.0602434	−	0.998184i	$-0.480812\pi$
$17$	0.496780	0.120487	0.0602434	+	0.998184i	$0.480812\pi$
$18$	2.34363	0.552398
$19$	−2.24448	−0.514918	−0.257459	−	0.966289i	$-0.582885\pi$
$19$	−2.24448	−0.514918	−0.257459	+	0.966289i	$0.582885\pi$
$20$	−1.89648	−0.424065
$21$	4.81457	1.05063
$22$	3.69545	0.787873
$23$	5.01638	1.04599	0.522994	−	0.852337i	$-0.324815\pi$
$23$	5.01638	1.04599	0.522994	+	0.852337i	$0.324815\pi$
$24$	−2.31566	−0.472683
$25$	−0.966689	−0.193338
$26$	−1.71590	−0.336516
$27$	5.58104	1.07407
$28$	−3.55653	−0.672120
$29$	2.58766	0.480517	0.240259	−	0.970709i	$-0.422768\pi$
$29$	2.58766	0.480517	0.240259	+	0.970709i	$0.422768\pi$
$30$	−4.40526	−0.804287
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	4.95128	0.875271
$33$	2.75311	0.479256
$34$	−0.852424	−0.146190
$35$	7.56380	1.27852
$36$	−1.28977	−0.214962
$37$	0.294584	0.0484293	0.0242147	−	0.999707i	$-0.492291\pi$
$37$	0.294584	0.0484293	0.0242147	+	0.999707i	$0.492291\pi$
$38$	3.85130	0.624763
$39$	−1.27835	−0.204699
$40$	−3.63795	−0.575211
$41$	10.0822	1.57457	0.787285	−	0.616589i	$-0.211486\pi$
$41$	10.0822	1.57457	0.787285	+	0.616589i	$0.211486\pi$
$42$	−8.26133	−1.27475
$43$	1.81940	0.277456	0.138728	−	0.990331i	$-0.455699\pi$
$43$	1.81940	0.277456	0.138728	+	0.990331i	$0.455699\pi$
$44$	−2.03373	−0.306596
$45$	2.74301	0.408904
$46$	−8.60761	−1.26912
$47$	−0.507742	−0.0740617	−0.0370309	−	0.999314i	$-0.511790\pi$
$47$	−0.507742	−0.0740617	−0.0370309	+	0.999314i	$0.511790\pi$
$48$	6.38777	0.921995
$49$	7.18464	1.02638
$50$	1.65874	0.234582
$51$	−0.635057	−0.0889257
$52$	0.944315	0.130953
$53$	11.8424	1.62669	0.813343	−	0.581785i	$-0.197645\pi$
$53$	11.8424	1.62669	0.813343	+	0.581785i	$0.197645\pi$
$54$	−9.57652	−1.30320
$55$	4.32520	0.583210
$56$	−6.82238	−0.911678
$57$	2.86922	0.380038
$58$	−4.44017	−0.583024
$59$	−9.27755	−1.20783	−0.603917	−	0.797047i	$-0.706394\pi$
$59$	−9.27755	−1.20783	−0.603917	+	0.797047i	$0.706394\pi$
$60$	2.42436	0.312983
$61$	0.950004	0.121636	0.0608178	−	0.998149i	$-0.480629\pi$
$61$	0.950004	0.121636	0.0608178	+	0.998149i	$0.480629\pi$
$62$	1.71590	0.217920
$63$	5.14405	0.648090
$64$	1.49789	0.187236
$65$	−2.00831	−0.249100
$66$	−4.72407	−0.581493
$67$	−3.55834	−0.434721	−0.217360	−	0.976091i	$-0.569745\pi$
$67$	−3.55834	−0.434721	−0.217360	+	0.976091i	$0.569745\pi$
$68$	0.469116	0.0568887
$69$	−6.41267	−0.771995
$70$	−12.9787	−1.55125
$71$	−14.3034	−1.69751	−0.848753	−	0.528790i	$-0.822646\pi$
$71$	−14.3034	−1.69751	−0.848753	+	0.528790i	$0.822646\pi$
$72$	−2.47413	−0.291579
$73$	−0.201282	−0.0235583	−0.0117791	−	0.999931i	$-0.503750\pi$
$73$	−0.201282	−0.0235583	−0.0117791	+	0.999931i	$0.503750\pi$
$74$	−0.505477	−0.0587605
$75$	1.23576	0.142694
$76$	−2.11949	−0.243123
$77$	8.11119	0.924356
$78$	2.19352	0.248367
$79$	−8.22770	−0.925689	−0.462844	−	0.886440i	$-0.653171\pi$
$79$	−8.22770	−0.925689	−0.462844	+	0.886440i	$0.653171\pi$
$80$	10.0353	1.12198
$81$	−3.03702	−0.337447
$82$	−17.3000	−1.91047
$83$	8.35070	0.916608	0.458304	−	0.888795i	$-0.348457\pi$
$83$	8.35070	0.916608	0.458304	+	0.888795i	$0.348457\pi$
$84$	4.54648	0.496061
$85$	−0.997688	−0.108214
$86$	−3.12191	−0.336644
$87$	−3.30793	−0.354648
$88$	−3.90123	−0.415873
$89$	10.6914	1.13328	0.566642	−	0.823964i	$-0.308242\pi$
$89$	10.6914	1.13328	0.566642	+	0.823964i	$0.308242\pi$
$90$	−4.70673	−0.496133
$91$	−3.76625	−0.394810
$92$	4.73704	0.493871
$93$	1.27835	0.132558
$94$	0.871234	0.0898610
$95$	4.50761	0.462471
$96$	−6.32946	−0.645997
$97$	−11.7424	−1.19226	−0.596131	−	0.802887i	$-0.703296\pi$
$97$	−11.7424	−1.19226	−0.596131	+	0.802887i	$0.703296\pi$
$98$	−12.3281	−1.24533
$99$	2.94152	0.295634
$100$	−0.912859	−0.0912859
$101$	−16.4019	−1.63205	−0.816024	−	0.578018i	$-0.803826\pi$
$101$	−16.4019	−1.63205	−0.816024	+	0.578018i	$0.803826\pi$
$102$	1.08969	0.107896
$103$	0.871093	0.0858314	0.0429157	−	0.999079i	$-0.486335\pi$
$103$	0.871093	0.0858314	0.0429157	+	0.999079i	$0.486335\pi$
$104$	1.81145	0.177627
$105$	−9.66916	−0.943613
$106$	−20.3205	−1.97370
$107$	3.04576	0.294445	0.147223	−	0.989103i	$-0.452967\pi$
$107$	3.04576	0.294445	0.147223	+	0.989103i	$0.452967\pi$
$108$	5.27026	0.507131
$109$	−9.34045	−0.894653	−0.447326	−	0.894371i	$-0.647624\pi$
$109$	−9.34045	−0.894653	−0.447326	+	0.894371i	$0.647624\pi$
$110$	−7.42162	−0.707623
$111$	−0.376580	−0.0357434
$112$	18.8196	1.77828
$113$	10.0672	0.947038	0.473519	−	0.880784i	$-0.342983\pi$
$113$	10.0672	0.947038	0.473519	+	0.880784i	$0.342983\pi$
$114$	−4.92330	−0.461109
$115$	−10.0744	−0.939447
$116$	2.44357	0.226880
$117$	−1.36583	−0.126271
$118$	15.9193	1.46549
$119$	−1.87100	−0.171514
$120$	4.65057	0.424537
$121$	−6.36178	−0.578344
$122$	−1.63011	−0.147584
$123$	−12.8885	−1.16212
$124$	−0.944315	−0.0848020
$125$	11.9830	1.07179
$126$	−8.82669	−0.786344
$127$	1.74953	0.155246	0.0776229	−	0.996983i	$-0.475267\pi$
$127$	1.74953	0.155246	0.0776229	+	0.996983i	$0.475267\pi$
$128$	−12.4728	−1.10245
$129$	−2.32582	−0.204777
$130$	3.44606	0.302240
$131$	6.38062	0.557477	0.278739	−	0.960367i	$-0.410084\pi$
$131$	6.38062	0.557477	0.278739	+	0.960367i	$0.410084\pi$
$132$	2.59981	0.226284
$133$	8.45326	0.732991
$134$	6.10577	0.527458
$135$	−11.2085	−0.964672
$136$	0.899892	0.0771651
$137$	9.98467	0.853048	0.426524	−	0.904476i	$-0.359738\pi$
$137$	9.98467	0.853048	0.426524	+	0.904476i	$0.359738\pi$
$138$	11.0035	0.936681
$139$	13.7642	1.16746	0.583732	−	0.811946i	$-0.301592\pi$
$139$	13.7642	1.16746	0.583732	+	0.811946i	$0.301592\pi$
$140$	7.14261	0.603661
$141$	0.649070	0.0546616
$142$	24.5433	2.05963
$143$	−2.15365	−0.180097
$144$	6.82491	0.568743
$145$	−5.19683	−0.431574
$146$	0.345380	0.0285839
$147$	−9.18446	−0.757522
$148$	0.278180	0.0228663
$149$	−1.03876	−0.0850988	−0.0425494	−	0.999094i	$-0.513548\pi$
$149$	−1.03876	−0.0850988	−0.0425494	+	0.999094i	$0.513548\pi$
$150$	−2.12045	−0.173134
$151$	12.8960	1.04946	0.524732	−	0.851267i	$-0.324166\pi$
$151$	12.8960	1.04946	0.524732	+	0.851267i	$0.324166\pi$
$152$	−4.06576	−0.329777
$153$	−0.678516	−0.0548548
$154$	−13.9180	−1.12154
$155$	2.00831	0.161311
$156$	−1.20716	−0.0966503
$157$	17.4874	1.39565	0.697823	−	0.716270i	$-0.254152\pi$
$157$	17.4874	1.39565	0.697823	+	0.716270i	$0.254152\pi$
$158$	14.1179	1.12316
$159$	−15.1388	−1.20058
$160$	−9.94371	−0.786119
$161$	−18.8929	−1.48897
$162$	5.21123	0.409433
$163$	−4.53010	−0.354825	−0.177412	−	0.984137i	$-0.556773\pi$
$163$	−4.53010	−0.354825	−0.177412	+	0.984137i	$0.556773\pi$
$164$	9.52075	0.743445
$165$	−5.52911	−0.430440
$166$	−14.3290	−1.11214
$167$	−1.92843	−0.149226	−0.0746130	−	0.997213i	$-0.523772\pi$
$167$	−1.92843	−0.149226	−0.0746130	+	0.997213i	$0.523772\pi$
$168$	8.72136	0.672868
$169$	1.00000	0.0769231
$170$	1.71193	0.131299
$171$	3.06557	0.234430
$172$	1.71809	0.131003
$173$	25.0188	1.90215	0.951073	−	0.308965i	$-0.0999827\pi$
$173$	25.0188	1.90215	0.951073	+	0.308965i	$0.0999827\pi$
$174$	5.67608	0.430303
$175$	3.64079	0.275218
$176$	10.7616	0.811185
$177$	11.8599	0.891446
$178$	−18.3453	−1.37504
$179$	−2.94357	−0.220013	−0.110006	−	0.993931i	$-0.535087\pi$
$179$	−2.94357	−0.220013	−0.110006	+	0.993931i	$0.535087\pi$
$180$	2.59026	0.193067
$181$	−6.31674	−0.469519	−0.234760	−	0.972053i	$-0.575430\pi$
$181$	−6.31674	−0.469519	−0.234760	+	0.972053i	$0.575430\pi$
$182$	6.46251	0.479033
$183$	−1.21444	−0.0897736
$184$	9.08692	0.669897
$185$	−0.591616	−0.0434965
$186$	−2.19352	−0.160836
$187$	−1.06989	−0.0782381
$188$	−0.479468	−0.0349688
$189$	−21.0196	−1.52895
$190$	−7.73461	−0.561127
$191$	−4.83994	−0.350206	−0.175103	−	0.984550i	$-0.556026\pi$
$191$	−4.83994	−0.350206	−0.175103	+	0.984550i	$0.556026\pi$
$192$	−1.91482	−0.138190
$193$	5.22331	0.375982	0.187991	−	0.982171i	$-0.439802\pi$
$193$	5.22331	0.375982	0.187991	+	0.982171i	$0.439802\pi$
$194$	20.1488	1.44660
$195$	2.56732	0.183849
$196$	6.78456	0.484612
$197$	26.6868	1.90136	0.950678	−	0.310178i	$-0.100389\pi$
$197$	26.6868	1.90136	0.950678	+	0.310178i	$0.100389\pi$
$198$	−5.04736	−0.358700
$199$	24.0970	1.70819	0.854096	−	0.520115i	$-0.174111\pi$
$199$	24.0970	1.70819	0.854096	+	0.520115i	$0.174111\pi$
$200$	−1.75111	−0.123822
$201$	4.54880	0.320848
$202$	28.1440	1.98020
$203$	−9.74579	−0.684020
$204$	−0.599694	−0.0419870
$205$	−20.2481	−1.41419
$206$	−1.49471	−0.104141
$207$	−6.85152	−0.476213
$208$	−4.99690	−0.346473
$209$	4.83382	0.334362
$210$	16.5913	1.14491
$211$	19.7623	1.36049	0.680245	−	0.732985i	$-0.261873\pi$
$211$	19.7623	1.36049	0.680245	+	0.732985i	$0.261873\pi$
$212$	11.1830	0.768052
$213$	18.2848	1.25285
$214$	−5.22623	−0.357258
$215$	−3.65392	−0.249195
$216$	10.1098	0.687884
$217$	3.76625	0.255670
$218$	16.0273	1.08550
$219$	0.257308	0.0173873
$220$	4.08435	0.275367
$221$	0.496780	0.0334170
$222$	0.646175	0.0433684
$223$	−23.2605	−1.55764	−0.778819	−	0.627249i	$-0.784181\pi$
$223$	−23.2605	−1.55764	−0.778819	+	0.627249i	$0.784181\pi$
$224$	−18.6478	−1.24596
$225$	1.32033	0.0880221
$226$	−17.2742	−1.14907
$227$	−20.0781	−1.33263	−0.666316	−	0.745669i	$-0.732130\pi$
$227$	−20.0781	−1.33263	−0.666316	+	0.745669i	$0.732130\pi$
$228$	2.70945	0.179438
$229$	26.2646	1.73561	0.867806	−	0.496903i	$-0.165529\pi$
$229$	26.2646	1.73561	0.867806	+	0.496903i	$0.165529\pi$
$230$	17.2867	1.13985
$231$	−10.3689	−0.682225
$232$	4.68743	0.307745
$233$	−15.8613	−1.03911	−0.519553	−	0.854438i	$-0.673902\pi$
$233$	−15.8613	−1.03911	−0.519553	+	0.854438i	$0.673902\pi$
$234$	2.34363	0.153208
$235$	1.01970	0.0665181
$236$	−8.76093	−0.570288
$237$	10.5179	0.683208
$238$	3.21044	0.208102
$239$	4.39630	0.284373	0.142187	−	0.989840i	$-0.454587\pi$
$239$	4.39630	0.284373	0.142187	+	0.989840i	$0.454587\pi$
$240$	−12.8286	−0.828084
$241$	−1.02299	−0.0658965	−0.0329482	−	0.999457i	$-0.510490\pi$
$241$	−1.02299	−0.0658965	−0.0329482	+	0.999457i	$0.510490\pi$
$242$	10.9162	0.701719
$243$	−12.8608	−0.825018
$244$	0.897104	0.0574312
$245$	−14.4290	−0.921834
$246$	22.1154	1.41003
$247$	−2.24448	−0.142813
$248$	−1.81145	−0.115027
$249$	−10.6751	−0.676506
$250$	−20.5616	−1.30043
$251$	−13.9792	−0.882362	−0.441181	−	0.897418i	$-0.645440\pi$
$251$	−13.9792	−0.882362	−0.441181	+	0.897418i	$0.645440\pi$
$252$	4.85761	0.306001
$253$	−10.8035	−0.679212
$254$	−3.00202	−0.188364
$255$	1.27539	0.0798681
$256$	18.4063	1.15039
$257$	0.257748	0.0160779	0.00803895	−	0.999968i	$-0.497441\pi$
$257$	0.257748	0.0160779	0.00803895	+	0.999968i	$0.497441\pi$
$258$	3.99088	0.248462
$259$	−1.10948	−0.0689395
$260$	−1.89648	−0.117615
$261$	−3.53431	−0.218768
$262$	−10.9485	−0.676401
$263$	8.76897	0.540718	0.270359	−	0.962760i	$-0.412858\pi$
$263$	8.76897	0.540718	0.270359	+	0.962760i	$0.412858\pi$
$264$	4.98713	0.306937
$265$	−23.7833	−1.46100
$266$	−14.5050	−0.889356
$267$	−13.6673	−0.836425
$268$	−3.36020	−0.205257
$269$	21.8161	1.33015	0.665075	−	0.746777i	$-0.268400\pi$
$269$	21.8161	1.33015	0.665075	+	0.746777i	$0.268400\pi$
$270$	19.2326	1.17046
$271$	−9.87577	−0.599910	−0.299955	−	0.953953i	$-0.596972\pi$
$271$	−9.87577	−0.599910	−0.299955	+	0.953953i	$0.596972\pi$
$272$	−2.48236	−0.150515
$273$	4.81457	0.291391
$274$	−17.1327	−1.03502
$275$	2.08191	0.125544
$276$	−6.05558	−0.364503
$277$	26.6465	1.60103	0.800516	−	0.599311i	$-0.204559\pi$
$277$	26.6465	1.60103	0.800516	+	0.599311i	$0.204559\pi$
$278$	−23.6180	−1.41651
$279$	1.36583	0.0817700
$280$	13.7014	0.818818
$281$	14.9758	0.893383	0.446692	−	0.894688i	$-0.352602\pi$
$281$	14.9758	0.893383	0.446692	+	0.894688i	$0.352602\pi$
$282$	−1.11374	−0.0663222
$283$	−0.617256	−0.0366921	−0.0183460	−	0.999832i	$-0.505840\pi$
$283$	−0.617256	−0.0366921	−0.0183460	+	0.999832i	$0.505840\pi$
$284$	−13.5070	−0.801490
$285$	−5.76229	−0.341328
$286$	3.69545	0.218517
$287$	−37.9720	−2.24141
$288$	−6.76260	−0.398490
$289$	−16.7532	−0.985483
$290$	8.91725	0.523639
$291$	15.0109	0.879954
$292$	−0.190074	−0.0111232
$293$	−10.0331	−0.586140	−0.293070	−	0.956091i	$-0.594677\pi$
$293$	−10.0331	−0.586140	−0.293070	+	0.956091i	$0.594677\pi$
$294$	15.7596	0.919120
$295$	18.6322	1.08481
$296$	0.533624	0.0310163
$297$	−12.0196	−0.697449
$298$	1.78241	0.103252
$299$	5.01638	0.290105
$300$	1.16695	0.0673739
$301$	−6.85232	−0.394961
$302$	−22.1283	−1.27334
$303$	20.9673	1.20454
$304$	11.2154	0.643249
$305$	−1.90790	−0.109246
$306$	1.16427	0.0665567
$307$	15.1634	0.865420	0.432710	−	0.901533i	$-0.357557\pi$
$307$	15.1634	0.865420	0.432710	+	0.901533i	$0.357557\pi$
$308$	7.65952	0.436442
$309$	−1.11356	−0.0633482
$310$	−3.44606	−0.195723
$311$	1.74794	0.0991166	0.0495583	−	0.998771i	$-0.484219\pi$
$311$	1.74794	0.0991166	0.0495583	+	0.998771i	$0.484219\pi$
$312$	−2.31566	−0.131099
$313$	25.0943	1.41841	0.709207	−	0.705001i	$-0.249053\pi$
$313$	25.0943	1.41841	0.709207	+	0.705001i	$0.249053\pi$
$314$	−30.0066	−1.69337
$315$	−10.3309	−0.582078
$316$	−7.76954	−0.437071
$317$	−2.47716	−0.139131	−0.0695657	−	0.997577i	$-0.522161\pi$
$317$	−2.47716	−0.139131	−0.0695657	+	0.997577i	$0.522161\pi$
$318$	25.9766	1.45670
$319$	−5.57293	−0.312024
$320$	−3.00823	−0.168165
$321$	−3.89354	−0.217316
$322$	32.4184	1.80661
$323$	−1.11501	−0.0620408
$324$	−2.86791	−0.159328
$325$	−0.966689	−0.0536222
$326$	7.77320	0.430518
$327$	11.9403	0.660302
$328$	18.2634	1.00842
$329$	1.91228	0.105428
$330$	9.48740	0.522264
$331$	−0.633422	−0.0348160	−0.0174080	−	0.999848i	$-0.505541\pi$
$331$	−0.633422	−0.0348160	−0.0174080	+	0.999848i	$0.505541\pi$
$332$	7.88569	0.432784
$333$	−0.402351	−0.0220487
$334$	3.30899	0.181060
$335$	7.14626	0.390442
$336$	−24.0579	−1.31247
$337$	−36.2828	−1.97645	−0.988225	−	0.153009i	$-0.951104\pi$
$337$	−36.2828	−1.97645	−0.988225	+	0.153009i	$0.951104\pi$
$338$	−1.71590	−0.0933327
$339$	−12.8693	−0.698965
$340$	−0.942132	−0.0510943
$341$	2.15365	0.116627
$342$	−5.26022	−0.284440
$343$	−0.695401	−0.0375481
$344$	3.29575	0.177695
$345$	12.8786	0.693362
$346$	−42.9298	−2.30792
$347$	−33.1151	−1.77771	−0.888855	−	0.458189i	$-0.848498\pi$
$347$	−33.1151	−1.77771	−0.888855	+	0.458189i	$0.848498\pi$
$348$	−3.12373	−0.167450
$349$	22.8673	1.22406	0.612031	−	0.790834i	$-0.290353\pi$
$349$	22.8673	1.22406	0.612031	+	0.790834i	$0.290353\pi$
$350$	−6.24724	−0.333929
$351$	5.58104	0.297894
$352$	−10.6633	−0.568358
$353$	−30.1364	−1.60400	−0.802000	−	0.597324i	$-0.796230\pi$
$353$	−30.1364	−1.60400	−0.802000	+	0.597324i	$0.796230\pi$
$354$	−20.3505	−1.08161
$355$	28.7257	1.52460
$356$	10.0960	0.535089
$357$	2.39178	0.126587
$358$	5.05088	0.266947
$359$	−37.0241	−1.95405	−0.977027	−	0.213114i	$-0.931640\pi$
$359$	−37.0241	−1.95405	−0.977027	+	0.213114i	$0.931640\pi$
$360$	4.96882	0.261880
$361$	−13.9623	−0.734859
$362$	10.8389	0.569679
$363$	8.13257	0.426849
$364$	−3.55653	−0.186413
$365$	0.404237	0.0211587
$366$	2.08385	0.108925
$367$	−6.20543	−0.323921	−0.161960	−	0.986797i	$-0.551782\pi$
$367$	−6.20543	−0.323921	−0.161960	+	0.986797i	$0.551782\pi$
$368$	−25.0663	−1.30667
$369$	−13.7705	−0.716865
$370$	1.01515	0.0527754
$371$	−44.6016	−2.31560
$372$	1.20716	0.0625885
$373$	13.7337	0.711104	0.355552	−	0.934656i	$-0.384293\pi$
$373$	13.7337	0.711104	0.355552	+	0.934656i	$0.384293\pi$
$374$	1.83583	0.0949283
$375$	−15.3184	−0.791038
$376$	−0.919749	−0.0474324
$377$	2.58766	0.133271
$378$	36.0676	1.85512
$379$	−25.8930	−1.33003	−0.665017	−	0.746828i	$-0.731576\pi$
$379$	−25.8930	−1.33003	−0.665017	+	0.746828i	$0.731576\pi$
$380$	4.25660	0.218359
$381$	−2.23651	−0.114580
$382$	8.30486	0.424914
$383$	28.5609	1.45939	0.729696	−	0.683771i	$-0.239661\pi$
$383$	28.5609	1.45939	0.729696	+	0.683771i	$0.239661\pi$
$384$	15.9446	0.813667
$385$	−16.2898	−0.830205
$386$	−8.96268	−0.456188
$387$	−2.48499	−0.126319
$388$	−11.0885	−0.562936
$389$	20.3418	1.03137	0.515686	−	0.856778i	$-0.327537\pi$
$389$	20.3418	1.03137	0.515686	+	0.856778i	$0.327537\pi$
$390$	−4.40526	−0.223069
$391$	2.49203	0.126028
$392$	13.0146	0.657338
$393$	−8.15665	−0.411448
$394$	−45.7919	−2.30696
$395$	16.5238	0.831401
$396$	2.77772	0.139586
$397$	33.1898	1.66575	0.832875	−	0.553461i	$-0.186693\pi$
$397$	33.1898	1.66575	0.832875	+	0.553461i	$0.186693\pi$
$398$	−41.3481	−2.07259
$399$	−10.8062	−0.540987
$400$	4.83045	0.241522
$401$	20.3121	1.01434	0.507170	−	0.861846i	$-0.330692\pi$
$401$	20.3121	1.01434	0.507170	+	0.861846i	$0.330692\pi$
$402$	−7.80529	−0.389292
$403$	−1.00000	−0.0498135
$404$	−15.4885	−0.770584
$405$	6.09929	0.303076
$406$	16.7228	0.829939
$407$	−0.634431	−0.0314476
$408$	−1.15037	−0.0569520
$409$	31.8817	1.57645	0.788224	−	0.615388i	$-0.211001\pi$
$409$	31.8817	1.57645	0.788224	+	0.615388i	$0.211001\pi$
$410$	34.7438	1.71587
$411$	−12.7639	−0.629596
$412$	0.822586	0.0405259
$413$	34.9416	1.71936
$414$	11.7565	0.577801
$415$	−16.7708	−0.823246
$416$	4.95128	0.242757
$417$	−17.5954	−0.861651
$418$	−8.29436	−0.405690
$419$	−6.96481	−0.340253	−0.170127	−	0.985422i	$-0.554418\pi$
$419$	−6.96481	−0.340253	−0.170127	+	0.985422i	$0.554418\pi$
$420$	−9.13073	−0.445534
$421$	−14.3546	−0.699602	−0.349801	−	0.936824i	$-0.613751\pi$
$421$	−14.3546	−0.699602	−0.349801	+	0.936824i	$0.613751\pi$
$422$	−33.9101	−1.65072
$423$	0.693488	0.0337186
$424$	21.4520	1.04180
$425$	−0.480231	−0.0232946
$426$	−31.3748	−1.52012
$427$	−3.57795	−0.173149
$428$	2.87616	0.139024
$429$	2.75311	0.132922
$430$	6.26976	0.302355
$431$	30.0120	1.44563	0.722814	−	0.691043i	$-0.242848\pi$
$431$	30.0120	1.44563	0.722814	+	0.691043i	$0.242848\pi$
$432$	−27.8879	−1.34176
$433$	−0.206096	−0.00990434	−0.00495217	−	0.999988i	$-0.501576\pi$
$433$	−0.206096	−0.00990434	−0.00495217	+	0.999988i	$0.501576\pi$
$434$	−6.46251	−0.310210
$435$	6.64336	0.318525
$436$	−8.82033	−0.422417
$437$	−11.2591	−0.538598
$438$	−0.441516	−0.0210964
$439$	−7.25666	−0.346341	−0.173171	−	0.984892i	$-0.555401\pi$
$439$	−7.25666	−0.346341	−0.173171	+	0.984892i	$0.555401\pi$
$440$	7.83489	0.373514
$441$	−9.81299	−0.467285
$442$	−0.852424	−0.0405457
$443$	−24.6429	−1.17082	−0.585409	−	0.810738i	$-0.699066\pi$
$443$	−24.6429	−1.17082	−0.585409	+	0.810738i	$0.699066\pi$
$444$	−0.355611	−0.0168765
$445$	−21.4716	−1.01785
$446$	39.9127	1.88992
$447$	1.32790	0.0628075
$448$	−5.64143	−0.266532
$449$	−16.4348	−0.775607	−0.387803	−	0.921742i	$-0.626766\pi$
$449$	−16.4348	−0.775607	−0.387803	+	0.921742i	$0.626766\pi$
$450$	−2.26556	−0.106799
$451$	−21.7135	−1.02245
$452$	9.50657	0.447151
$453$	−16.4856	−0.774561
$454$	34.4521	1.61692
$455$	7.56380	0.354596
$456$	5.19745	0.243393
$457$	−15.1179	−0.707184	−0.353592	−	0.935400i	$-0.615040\pi$
$457$	−15.1179	−0.707184	−0.353592	+	0.935400i	$0.615040\pi$
$458$	−45.0674	−2.10586
$459$	2.77255	0.129411
$460$	−9.51345	−0.443567
$461$	5.84665	0.272306	0.136153	−	0.990688i	$-0.456526\pi$
$461$	5.84665	0.272306	0.136153	+	0.990688i	$0.456526\pi$
$462$	17.7920	0.827760
$463$	17.1752	0.798199	0.399099	−	0.916908i	$-0.369323\pi$
$463$	17.1752	0.798199	0.399099	+	0.916908i	$0.369323\pi$
$464$	−12.9303	−0.600274
$465$	−2.56732	−0.119057
$466$	27.2164	1.26077
$467$	30.6084	1.41639	0.708194	−	0.706018i	$-0.249510\pi$
$467$	30.6084	1.41639	0.708194	+	0.706018i	$0.249510\pi$
$468$	−1.28977	−0.0596198
$469$	13.4016	0.618829
$470$	−1.74971	−0.0807081
$471$	−22.3550	−1.03006
$472$	−16.8058	−0.773550
$473$	−3.91835	−0.180166
$474$	−18.0476	−0.828954
$475$	2.16971	0.0995532
$476$	−1.76681	−0.0809816
$477$	−16.1748	−0.740591
$478$	−7.54361	−0.345037
$479$	42.3293	1.93408	0.967038	−	0.254632i	$-0.0819542\pi$
$479$	42.3293	1.93408	0.967038	+	0.254632i	$0.0819542\pi$
$480$	12.7115	0.580199
$481$	0.294584	0.0134319
$482$	1.75535	0.0799538
$483$	24.1517	1.09894
$484$	−6.00753	−0.273070
$485$	23.5824	1.07082
$486$	22.0678	1.00102
$487$	10.5479	0.477970	0.238985	−	0.971023i	$-0.423185\pi$
$487$	10.5479	0.477970	0.238985	+	0.971023i	$0.423185\pi$
$488$	1.72089	0.0779008
$489$	5.79104	0.261880
$490$	24.7587	1.11848
$491$	0.275205	0.0124198	0.00620992	−	0.999981i	$-0.498023\pi$
$491$	0.275205	0.0124198	0.00620992	+	0.999981i	$0.498023\pi$
$492$	−12.1708	−0.548703
$493$	1.28550	0.0578960
$494$	3.85130	0.173278
$495$	−5.90749	−0.265522
$496$	4.99690	0.224367
$497$	53.8703	2.41641
$498$	18.3174	0.820822
$499$	8.13461	0.364155	0.182078	−	0.983284i	$-0.441718\pi$
$499$	8.13461	0.364155	0.182078	+	0.983284i	$0.441718\pi$
$500$	11.3157	0.506053
$501$	2.46520	0.110137
$502$	23.9870	1.07059
$503$	30.3823	1.35468	0.677339	−	0.735671i	$-0.263133\pi$
$503$	30.3823	1.35468	0.677339	+	0.735671i	$0.263133\pi$
$504$	9.31820	0.415066
$505$	32.9401	1.46581
$506$	18.5378	0.824105
$507$	−1.27835	−0.0567734
$508$	1.65211	0.0733005
$509$	25.1419	1.11439	0.557197	−	0.830380i	$-0.311877\pi$
$509$	25.1419	1.11439	0.557197	+	0.830380i	$0.311877\pi$
$510$	−2.18844	−0.0969059
$511$	0.758079	0.0335354
$512$	−6.63779	−0.293352
$513$	−12.5265	−0.553060
$514$	−0.442271	−0.0195077
$515$	−1.74943	−0.0770889
$516$	−2.19631	−0.0966872
$517$	1.09350	0.0480920
$518$	1.90375	0.0836461
$519$	−31.9827	−1.40389
$520$	−3.63795	−0.159535
$521$	13.2486	0.580433	0.290216	−	0.956961i	$-0.406273\pi$
$521$	13.2486	0.580433	0.290216	+	0.956961i	$0.406273\pi$
$522$	6.06452	0.265437
$523$	−6.95375	−0.304066	−0.152033	−	0.988375i	$-0.548582\pi$
$523$	−6.95375	−0.304066	−0.152033	+	0.988375i	$0.548582\pi$
$524$	6.02532	0.263217
$525$	−4.65419	−0.203126
$526$	−15.0467	−0.656066
$527$	−0.496780	−0.0216401
$528$	−13.7570	−0.598698
$529$	2.16405	0.0940891
$530$	40.8098	1.77266
$531$	12.6715	0.549898
$532$	7.98254	0.346087
$533$	10.0822	0.436707
$534$	23.4517	1.01486
$535$	−6.11684	−0.264454
$536$	−6.44576	−0.278415
$537$	3.76291	0.162381
$538$	−37.4342	−1.61390
$539$	−15.4732	−0.666478
$540$	−10.5843	−0.455477
$541$	−6.35465	−0.273208	−0.136604	−	0.990626i	$-0.543619\pi$
$541$	−6.35465	−0.273208	−0.136604	+	0.990626i	$0.543619\pi$
$542$	16.9458	0.727886
$543$	8.07498	0.346531
$544$	2.45970	0.105459
$545$	18.7585	0.803527
$546$	−8.26133	−0.353552
$547$	36.6787	1.56827	0.784134	−	0.620592i	$-0.213108\pi$
$547$	36.6787	1.56827	0.784134	+	0.620592i	$0.213108\pi$
$548$	9.42868	0.402773
$549$	−1.29754	−0.0553778
$550$	−3.57235	−0.152326
$551$	−5.80795	−0.247427
$552$	−11.6162	−0.494420
$553$	30.9876	1.31773
$554$	−45.7227	−1.94257
$555$	0.756291	0.0321027
$556$	12.9977	0.551227
$557$	−9.71684	−0.411716	−0.205858	−	0.978582i	$-0.565998\pi$
$557$	−9.71684	−0.411716	−0.205858	+	0.978582i	$0.565998\pi$
$558$	−2.34363	−0.0992136
$559$	1.81940	0.0769524
$560$	−37.7955	−1.59715
$561$	1.36769	0.0577439
$562$	−25.6970	−1.08396
$563$	4.11024	0.173226	0.0866129	−	0.996242i	$-0.472396\pi$
$563$	4.11024	0.173226	0.0866129	+	0.996242i	$0.472396\pi$
$564$	0.612927	0.0258089
$565$	−20.2180	−0.850577
$566$	1.05915	0.0445194
$567$	11.4382	0.480359
$568$	−25.9100	−1.08716
$569$	−21.2242	−0.889765	−0.444883	−	0.895589i	$-0.646755\pi$
$569$	−21.2242	−0.889765	−0.444883	+	0.895589i	$0.646755\pi$
$570$	9.88751	0.414142
$571$	−25.2319	−1.05592	−0.527962	−	0.849268i	$-0.677044\pi$
$571$	−25.2319	−1.05592	−0.527962	+	0.849268i	$0.677044\pi$
$572$	−2.03373	−0.0850343
$573$	6.18713	0.258471
$574$	65.1561	2.71956
$575$	−4.84928	−0.202229
$576$	−2.04586	−0.0852443
$577$	−17.2195	−0.716858	−0.358429	−	0.933557i	$-0.616687\pi$
$577$	−17.2195	−0.716858	−0.358429	+	0.933557i	$0.616687\pi$
$578$	28.7468	1.19571
$579$	−6.67720	−0.277495
$580$	−4.90745	−0.203771
$581$	−31.4508	−1.30480
$582$	−25.7572	−1.06767
$583$	−25.5045	−1.05629
$584$	−0.364613	−0.0150878
$585$	2.74301	0.113409
$586$	17.2158	0.711179
$587$	27.5779	1.13826	0.569132	−	0.822246i	$-0.307279\pi$
$587$	27.5779	1.13826	0.569132	+	0.822246i	$0.307279\pi$
$588$	−8.67303	−0.357670
$589$	2.24448	0.0924821
$590$	−31.9710	−1.31623
$591$	−34.1150	−1.40330
$592$	−1.47201	−0.0604991
$593$	−27.7199	−1.13832	−0.569160	−	0.822227i	$-0.692731\pi$
$593$	−27.7199	−1.13832	−0.569160	+	0.822227i	$0.692731\pi$
$594$	20.6245	0.846233
$595$	3.75754	0.154044
$596$	−0.980919	−0.0401800
$597$	−30.8044	−1.26074
$598$	−8.60761	−0.351991
$599$	6.37470	0.260463	0.130232	−	0.991484i	$-0.458428\pi$
$599$	6.37470	0.260463	0.130232	+	0.991484i	$0.458428\pi$
$600$	2.23852	0.0913874
$601$	−18.0626	−0.736791	−0.368395	−	0.929669i	$-0.620093\pi$
$601$	−18.0626	−0.736791	−0.368395	+	0.929669i	$0.620093\pi$
$602$	11.7579	0.479216
$603$	4.86009	0.197918
$604$	12.1779	0.495512
$605$	12.7764	0.519436
$606$	−35.9778	−1.46150
$607$	−40.4233	−1.64073	−0.820365	−	0.571840i	$-0.806230\pi$
$607$	−40.4233	−1.64073	−0.820365	+	0.571840i	$0.806230\pi$
$608$	−11.1130	−0.450693
$609$	12.4585	0.504844
$610$	3.27377	0.132551
$611$	−0.507742	−0.0205410
$612$	−0.640733	−0.0259001
$613$	2.27446	0.0918647	0.0459324	−	0.998945i	$-0.485374\pi$
$613$	2.27446	0.0918647	0.0459324	+	0.998945i	$0.485374\pi$
$614$	−26.0189	−1.05004
$615$	25.8841	1.04375
$616$	14.6930	0.591999
$617$	−8.66707	−0.348923	−0.174461	−	0.984664i	$-0.555818\pi$
$617$	−8.66707	−0.348923	−0.174461	+	0.984664i	$0.555818\pi$
$618$	1.91076	0.0768619
$619$	23.1997	0.932476	0.466238	−	0.884659i	$-0.345609\pi$
$619$	23.1997	0.932476	0.466238	+	0.884659i	$0.345609\pi$
$620$	1.89648	0.0761644
$621$	27.9966	1.12347
$622$	−2.99929	−0.120261
$623$	−40.2664	−1.61324
$624$	6.38777	0.255715
$625$	−19.2321	−0.769283
$626$	−43.0593	−1.72100
$627$	−6.17930	−0.246778
$628$	16.5136	0.658965
$629$	0.146343	0.00583509
$630$	17.7267	0.706250
$631$	−5.82893	−0.232046	−0.116023	−	0.993247i	$-0.537015\pi$
$631$	−5.82893	−0.232046	−0.116023	+	0.993247i	$0.537015\pi$
$632$	−14.9041	−0.592852
$633$	−25.2630	−1.00411
$634$	4.25057	0.168812
$635$	−3.51360	−0.139433
$636$	−14.2958	−0.566864
$637$	7.18464	0.284666
$638$	9.56259	0.378587
$639$	19.5361	0.772834
$640$	25.0492	0.990158
$641$	−20.2507	−0.799853	−0.399927	−	0.916547i	$-0.630964\pi$
$641$	−20.2507	−0.799853	−0.399927	+	0.916547i	$0.630964\pi$
$642$	6.68093	0.263675
$643$	45.8218	1.80704	0.903519	−	0.428549i	$-0.140975\pi$
$643$	45.8218	1.80704	0.903519	+	0.428549i	$0.140975\pi$
$644$	−17.8409	−0.703029
$645$	4.67098	0.183920
$646$	1.91325	0.0752757
$647$	−11.3964	−0.448039	−0.224019	−	0.974585i	$-0.571918\pi$
$647$	−11.3964	−0.448039	−0.224019	+	0.974585i	$0.571918\pi$
$648$	−5.50142	−0.216116
$649$	19.9806	0.784307
$650$	1.65874	0.0650612
$651$	−4.81457	−0.188698
$652$	−4.27784	−0.167533
$653$	46.6256	1.82460	0.912299	−	0.409524i	$-0.134305\pi$
$653$	46.6256	1.82460	0.912299	+	0.409524i	$0.134305\pi$
$654$	−20.4884	−0.801161
$655$	−12.8143	−0.500695
$656$	−50.3796	−1.96699
$657$	0.274917	0.0107255
$658$	−3.28129	−0.127918
$659$	−7.37477	−0.287280	−0.143640	−	0.989630i	$-0.545881\pi$
$659$	−7.37477	−0.287280	−0.143640	+	0.989630i	$0.545881\pi$
$660$	−5.22122	−0.203236
$661$	−21.1144	−0.821256	−0.410628	−	0.911803i	$-0.634691\pi$
$661$	−21.1144	−0.821256	−0.410628	+	0.911803i	$0.634691\pi$
$662$	1.08689	0.0422432
$663$	−0.635057	−0.0246636
$664$	15.1269	0.587037
$665$	−16.9768	−0.658331
$666$	0.690395	0.0267523
$667$	12.9807	0.502615
$668$	−1.82104	−0.0704582
$669$	29.7350	1.14962
$670$	−12.2623	−0.473733
$671$	−2.04598	−0.0789841
$672$	23.8383	0.919583
$673$	−41.1497	−1.58620	−0.793102	−	0.609089i	$-0.791535\pi$
$673$	−41.1497	−1.58620	−0.793102	+	0.609089i	$0.791535\pi$
$674$	62.2577	2.39808
$675$	−5.39513	−0.207659
$676$	0.944315	0.0363198
$677$	−22.9776	−0.883100	−0.441550	−	0.897237i	$-0.645571\pi$
$677$	−22.9776	−0.883100	−0.441550	+	0.897237i	$0.645571\pi$
$678$	22.0825	0.848072
$679$	44.2249	1.69720
$680$	−1.80726	−0.0693053
$681$	25.6668	0.983555
$682$	−3.69545	−0.141506
$683$	32.9875	1.26223	0.631115	−	0.775689i	$-0.282598\pi$
$683$	32.9875	1.26223	0.631115	+	0.775689i	$0.282598\pi$
$684$	2.89487	0.110688
$685$	−20.0523	−0.766160
$686$	1.19324	0.0455580
$687$	−33.5753	−1.28098
$688$	−9.09136	−0.346605
$689$	11.8424	0.451161
$690$	−22.0985	−0.841274
$691$	45.3400	1.72482	0.862408	−	0.506214i	$-0.168955\pi$
$691$	45.3400	1.72482	0.862408	+	0.506214i	$0.168955\pi$
$692$	23.6257	0.898113
$693$	−11.0785	−0.420837
$694$	56.8221	2.15694
$695$	−27.6428	−1.04855
$696$	−5.99216	−0.227132
$697$	5.00862	0.189715
$698$	−39.2381	−1.48518
$699$	20.2762	0.766917
$700$	3.43805	0.129946
$701$	−6.92157	−0.261424	−0.130712	−	0.991420i	$-0.541726\pi$
$701$	−6.92157	−0.261424	−0.130712	+	0.991420i	$0.541726\pi$
$702$	−9.57652	−0.361442
$703$	−0.661187	−0.0249371
$704$	−3.22593	−0.121582
$705$	−1.30353	−0.0490939
$706$	51.7111	1.94617
$707$	61.7736	2.32323
$708$	11.1995	0.420903
$709$	13.6882	0.514071	0.257035	−	0.966402i	$-0.417254\pi$
$709$	13.6882	0.514071	0.257035	+	0.966402i	$0.417254\pi$
$710$	−49.2905	−1.84984
$711$	11.2376	0.421444
$712$	19.3669	0.725805
$713$	−5.01638	−0.187865
$714$	−4.10406	−0.153591
$715$	4.32520	0.161753
$716$	−2.77966	−0.103881
$717$	−5.62000	−0.209883
$718$	63.5296	2.37090
$719$	−11.2725	−0.420394	−0.210197	−	0.977659i	$-0.567411\pi$
$719$	−11.2725	−0.420394	−0.210197	+	0.977659i	$0.567411\pi$
$720$	−13.7065	−0.510813
$721$	−3.28075	−0.122182
$722$	23.9580	0.891623
$723$	1.30773	0.0486351
$724$	−5.96499	−0.221687
$725$	−2.50147	−0.0929021
$726$	−13.9547	−0.517907
$727$	−12.3867	−0.459398	−0.229699	−	0.973262i	$-0.573774\pi$
$727$	−12.3867	−0.459398	−0.229699	+	0.973262i	$0.573774\pi$
$728$	−6.82238	−0.252854
$729$	25.5516	0.946355
$730$	−0.693631	−0.0256724
$731$	0.903841	0.0334298
$732$	−1.14681	−0.0423873
$733$	−39.7813	−1.46935	−0.734677	−	0.678417i	$-0.762666\pi$
$733$	−39.7813	−1.46935	−0.734677	+	0.678417i	$0.762666\pi$
$734$	10.6479	0.393021
$735$	18.4453	0.680363
$736$	24.8375	0.915523
$737$	7.66343	0.282286
$738$	23.6288	0.869790
$739$	36.9890	1.36066	0.680331	−	0.732905i	$-0.261836\pi$
$739$	36.9890	1.36066	0.680331	+	0.732905i	$0.261836\pi$
$740$	−0.558672	−0.0205372
$741$	2.86922	0.105403
$742$	76.5319	2.80958
$743$	−27.4834	−1.00827	−0.504134	−	0.863626i	$-0.668188\pi$
$743$	−27.4834	−1.00827	−0.504134	+	0.863626i	$0.668188\pi$
$744$	2.31566	0.0848963
$745$	2.08616	0.0764309
$746$	−23.5657	−0.862800
$747$	−11.4056	−0.417310
$748$	−1.01031	−0.0369407
$749$	−11.4711	−0.419145
$750$	26.2848	0.959786
$751$	−20.1318	−0.734621	−0.367310	−	0.930098i	$-0.619721\pi$
$751$	−20.1318	−0.734621	−0.367310	+	0.930098i	$0.619721\pi$
$752$	2.53713	0.0925198
$753$	17.8703	0.651231
$754$	−4.44017	−0.161702
$755$	−25.8992	−0.942570
$756$	−19.8491	−0.721906
$757$	16.9336	0.615462	0.307731	−	0.951473i	$-0.400430\pi$
$757$	16.9336	0.615462	0.307731	+	0.951473i	$0.400430\pi$
$758$	44.4298	1.61376
$759$	13.8107	0.501295
$760$	8.16531	0.296187
$761$	−28.7763	−1.04314	−0.521570	−	0.853209i	$-0.674653\pi$
$761$	−28.7763	−1.04314	−0.521570	+	0.853209i	$0.674653\pi$
$762$	3.83763	0.139023
$763$	35.1785	1.27355
$764$	−4.57043	−0.165352
$765$	1.36267	0.0492675
$766$	−49.0076	−1.77072
$767$	−9.27755	−0.334993
$768$	−23.5296	−0.849053
$769$	−0.250334	−0.00902729	−0.00451364	−	0.999990i	$-0.501437\pi$
$769$	−0.250334	−0.00902729	−0.00451364	+	0.999990i	$0.501437\pi$
$770$	27.9517	1.00731
$771$	−0.329492	−0.0118664
$772$	4.93245	0.177523
$773$	26.2845	0.945389	0.472695	−	0.881226i	$-0.343281\pi$
$773$	26.2845	0.945389	0.472695	+	0.881226i	$0.343281\pi$
$774$	4.26399	0.153266
$775$	0.966689	0.0347245
$776$	−21.2708	−0.763578
$777$	1.41830	0.0508811
$778$	−34.9045	−1.25139
$779$	−22.6292	−0.810775
$780$	2.42436	0.0868059
$781$	30.8046	1.10228
$782$	−4.27608	−0.152912
$783$	14.4419	0.516110
$784$	−35.9009	−1.28218
$785$	−35.1201	−1.25349
$786$	13.9960	0.499221
$787$	0.930025	0.0331518	0.0165759	−	0.999863i	$-0.494723\pi$
$787$	0.930025	0.0331518	0.0165759	+	0.999863i	$0.494723\pi$
$788$	25.2008	0.897740
$789$	−11.2098	−0.399079
$790$	−28.3532	−1.00876
$791$	−37.9154	−1.34812
$792$	5.32842	0.189337
$793$	0.950004	0.0337356
$794$	−56.9504	−2.02110
$795$	30.4033	1.07829
$796$	22.7552	0.806536
$797$	−32.6076	−1.15502	−0.577511	−	0.816383i	$-0.695976\pi$
$797$	−32.6076	−1.15502	−0.577511	+	0.816383i	$0.695976\pi$
$798$	18.5424	0.656393
$799$	−0.252236	−0.00892346
$800$	−4.78635	−0.169223
$801$	−14.6026	−0.515957
$802$	−34.8536	−1.23072
$803$	0.433492	0.0152976
$804$	4.29550	0.151491
$805$	37.9429	1.33731
$806$	1.71590	0.0604400
$807$	−27.8885	−0.981722
$808$	−29.7112	−1.04524
$809$	5.91439	0.207939	0.103969	−	0.994580i	$-0.466846\pi$
$809$	5.91439	0.207939	0.103969	+	0.994580i	$0.466846\pi$
$810$	−10.4658	−0.367730
$811$	33.8322	1.18801	0.594004	−	0.804462i	$-0.297546\pi$
$811$	33.8322	1.18801	0.594004	+	0.804462i	$0.297546\pi$
$812$	−9.20310	−0.322965
$813$	12.6247	0.442766
$814$	1.08862	0.0381561
$815$	9.09785	0.318684
$816$	3.17331	0.111088
$817$	−4.08360	−0.142867
$818$	−54.7058	−1.91274
$819$	5.14405	0.179748
$820$	−19.1206	−0.667721
$821$	−48.4170	−1.68977	−0.844883	−	0.534951i	$-0.820330\pi$
$821$	−48.4170	−1.68977	−0.844883	+	0.534951i	$0.820330\pi$
$822$	21.9015	0.763904
$823$	−25.1197	−0.875617	−0.437808	−	0.899068i	$-0.644245\pi$
$823$	−25.1197	−0.875617	−0.437808	+	0.899068i	$0.644245\pi$
$824$	1.57794	0.0549702
$825$	−2.66140	−0.0926582
$826$	−59.9562	−2.08614
$827$	−5.94407	−0.206696	−0.103348	−	0.994645i	$-0.532955\pi$
$827$	−5.94407	−0.206696	−0.103348	+	0.994645i	$0.532955\pi$
$828$	−6.46999	−0.224848
$829$	−53.5934	−1.86138	−0.930688	−	0.365814i	$-0.880791\pi$
$829$	−53.5934	−1.86138	−0.930688	+	0.365814i	$0.880791\pi$
$830$	28.7770	0.998865
$831$	−34.0634	−1.18165
$832$	1.49789	0.0519300
$833$	3.56918	0.123665
$834$	30.1920	1.04546
$835$	3.87288	0.134026
$836$	4.56465	0.157872
$837$	−5.58104	−0.192909
$838$	11.9509	0.412838
$839$	38.6051	1.33280	0.666399	−	0.745596i	$-0.267835\pi$
$839$	38.6051	1.33280	0.666399	+	0.745596i	$0.267835\pi$
$840$	−17.5152	−0.604332
$841$	−22.3040	−0.769103
$842$	24.6311	0.848845
$843$	−19.1443	−0.659365
$844$	18.6618	0.642366
$845$	−2.00831	−0.0690880
$846$	−1.18996	−0.0409116
$847$	23.9601	0.823278
$848$	−59.1755	−2.03210
$849$	0.789067	0.0270807
$850$	0.824029	0.0282640
$851$	1.47774	0.0506564
$852$	17.2666	0.591543
$853$	−22.1076	−0.756948	−0.378474	−	0.925612i	$-0.623551\pi$
$853$	−22.1076	−0.756948	−0.378474	+	0.925612i	$0.623551\pi$
$854$	6.13941	0.210086
$855$	−6.15662	−0.210552
$856$	5.51725	0.188576
$857$	−7.36019	−0.251419	−0.125710	−	0.992067i	$-0.540121\pi$
$857$	−7.36019	−0.251419	−0.125710	+	0.992067i	$0.540121\pi$
$858$	−4.72407	−0.161277
$859$	−3.66077	−0.124904	−0.0624519	−	0.998048i	$-0.519892\pi$
$859$	−3.66077	−0.124904	−0.0624519	+	0.998048i	$0.519892\pi$
$860$	−3.45045	−0.117659
$861$	48.5414	1.65429
$862$	−51.4976	−1.75402
$863$	54.7738	1.86452	0.932261	−	0.361785i	$-0.117833\pi$
$863$	54.7738	1.86452	0.932261	+	0.361785i	$0.117833\pi$
$864$	27.6333	0.940105
$865$	−50.2456	−1.70840
$866$	0.353640	0.0120172
$867$	21.4164	0.727340
$868$	3.55653	0.120716
$869$	17.7196	0.601096
$870$	−11.3993	−0.386474
$871$	−3.55834	−0.120570
$872$	−16.9198	−0.572975
$873$	16.0381	0.542809
$874$	19.3196	0.653495
$875$	−45.1308	−1.52570
$876$	0.242980	0.00820954
$877$	49.1389	1.65930	0.829652	−	0.558282i	$-0.188539\pi$
$877$	49.1389	1.65930	0.829652	+	0.558282i	$0.188539\pi$
$878$	12.4517	0.420225
$879$	12.8258	0.432603
$880$	−21.6126	−0.728560
$881$	−22.1103	−0.744916	−0.372458	−	0.928049i	$-0.621485\pi$
$881$	−22.1103	−0.744916	−0.372458	+	0.928049i	$0.621485\pi$
$882$	16.8381	0.566969
$883$	−14.6523	−0.493089	−0.246544	−	0.969132i	$-0.579295\pi$
$883$	−14.6523	−0.493089	−0.246544	+	0.969132i	$0.579295\pi$
$884$	0.469116	0.0157781
$885$	−23.8184	−0.800647
$886$	42.2847	1.42058
$887$	−2.43421	−0.0817326	−0.0408663	−	0.999165i	$-0.513012\pi$
$887$	−2.43421	−0.0817326	−0.0408663	+	0.999165i	$0.513012\pi$
$888$	−0.682157	−0.0228917
$889$	−6.58918	−0.220994
$890$	36.8432	1.23499
$891$	6.54069	0.219121
$892$	−21.9652	−0.735450
$893$	1.13961	0.0381358
$894$	−2.27854	−0.0762059
$895$	5.91161	0.197603
$896$	46.9757	1.56935
$897$	−6.41267	−0.214113
$898$	28.2005	0.941063
$899$	−2.58766	−0.0863034
$900$	1.24681	0.0415603
$901$	5.88309	0.195994
$902$	37.2582	1.24056
$903$	8.75964	0.291502
$904$	18.2362	0.606525
$905$	12.6860	0.421696
$906$	28.2877	0.939794
$907$	−20.1766	−0.669953	−0.334976	−	0.942227i	$-0.608728\pi$
$907$	−20.1766	−0.669953	−0.334976	+	0.942227i	$0.608728\pi$
$908$	−18.9601	−0.629213
$909$	22.4022	0.743033
$910$	−12.9787	−0.430241
$911$	2.05680	0.0681448	0.0340724	−	0.999419i	$-0.489152\pi$
$911$	2.05680	0.0681448	0.0340724	+	0.999419i	$0.489152\pi$
$912$	−14.3372	−0.474752
$913$	−17.9845	−0.595200
$914$	25.9408	0.858044
$915$	2.43896	0.0806296
$916$	24.8020	0.819483
$917$	−24.0310	−0.793574
$918$	−4.75742	−0.157018
$919$	−55.2635	−1.82297	−0.911487	−	0.411329i	$-0.865065\pi$
$919$	−55.2635	−1.82297	−0.911487	+	0.411329i	$0.865065\pi$
$920$	−18.2494	−0.601663
$921$	−19.3841	−0.638727
$922$	−10.0323	−0.330395
$923$	−14.3034	−0.470803
$924$	−9.79152	−0.322117
$925$	−0.284771	−0.00936321
$926$	−29.4709	−0.968475
$927$	−1.18976	−0.0390770
$928$	12.8123	0.420583
$929$	14.1140	0.463067	0.231533	−	0.972827i	$-0.425626\pi$
$929$	14.1140	0.463067	0.231533	+	0.972827i	$0.425626\pi$
$930$	4.40526	0.144454
$931$	−16.1258	−0.528501
$932$	−14.9780	−0.490622
$933$	−2.23447	−0.0731534
$934$	−52.5210	−1.71854
$935$	2.14867	0.0702691
$936$	−2.47413	−0.0808695
$937$	11.3267	0.370027	0.185013	−	0.982736i	$-0.440767\pi$
$937$	11.3267	0.370027	0.185013	+	0.982736i	$0.440767\pi$
$938$	−22.9958	−0.750841
$939$	−32.0792	−1.04687
$940$	0.962921	0.0314070
$941$	21.3415	0.695714	0.347857	−	0.937548i	$-0.386910\pi$
$941$	21.3415	0.695714	0.347857	+	0.937548i	$0.386910\pi$
$942$	38.3589	1.24980
$943$	50.5760	1.64698
$944$	46.3590	1.50886
$945$	42.2139	1.37322
$946$	6.72351	0.218600
$947$	−18.4829	−0.600613	−0.300307	−	0.953843i	$-0.597089\pi$
$947$	−18.4829	−0.600613	−0.300307	+	0.953843i	$0.597089\pi$
$948$	9.93217	0.322582
$949$	−0.201282	−0.00653389
$950$	−3.72301	−0.120790
$951$	3.16668	0.102686
$952$	−3.38922	−0.109845
$953$	24.3280	0.788061	0.394030	−	0.919097i	$-0.371081\pi$
$953$	24.3280	0.788061	0.394030	+	0.919097i	$0.371081\pi$
$954$	27.7543	0.898578
$955$	9.72011	0.314535
$956$	4.15149	0.134269
$957$	7.12413	0.230291
$958$	−72.6329	−2.34666
$959$	−37.6048	−1.21432
$960$	3.84556	0.124115
$961$	1.00000	0.0322581
$962$	−0.505477	−0.0162972
$963$	−4.15999	−0.134054
$964$	−0.966023	−0.0311135
$965$	−10.4900	−0.337686
$966$	−41.4420	−1.33337
$967$	1.26661	0.0407314	0.0203657	−	0.999793i	$-0.493517\pi$
$967$	1.26661	0.0407314	0.0203657	+	0.999793i	$0.493517\pi$
$968$	−11.5241	−0.370397
$969$	1.42537	0.0457895
$970$	−40.4651	−1.29926
$971$	59.8957	1.92214	0.961072	−	0.276298i	$-0.0891076\pi$
$971$	59.8957	1.92214	0.961072	+	0.276298i	$0.0891076\pi$
$972$	−12.1446	−0.389539
$973$	−51.8394	−1.66190
$974$	−18.0991	−0.579933
$975$	1.23576	0.0395761
$976$	−4.74708	−0.151950
$977$	5.32423	0.170337	0.0851685	−	0.996367i	$-0.472857\pi$
$977$	5.32423	0.170337	0.0851685	+	0.996367i	$0.472857\pi$
$978$	−9.93685	−0.317746
$979$	−23.0255	−0.735898
$980$	−13.6255	−0.435251
$981$	12.7575	0.407314
$982$	−0.472225	−0.0150693
$983$	27.1096	0.864663	0.432332	−	0.901715i	$-0.357691\pi$
$983$	27.1096	0.864663	0.432332	+	0.901715i	$0.357691\pi$
$984$	−23.3469	−0.744272
$985$	−53.5954	−1.70769
$986$	−2.20579	−0.0702466
$987$	−2.44456	−0.0778112
$988$	−2.11949	−0.0674301
$989$	9.12680	0.290215
$990$	10.1367	0.322164
$991$	43.8434	1.39273	0.696366	−	0.717687i	$-0.254799\pi$
$991$	43.8434	1.39273	0.696366	+	0.717687i	$0.254799\pi$
$992$	−4.95128	−0.157203
$993$	0.809734	0.0256961
$994$	−92.4361	−2.93190
$995$	−48.3943	−1.53420
$996$	−10.0806	−0.319418
$997$	39.3311	1.24563	0.622813	−	0.782370i	$-0.285990\pi$
$997$	39.3311	1.24563	0.622813	+	0.782370i	$0.285990\pi$
$998$	−13.9582	−0.441838
$999$	1.64409	0.0520166

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.a.e.1.2	✓	8
3.2	odd	2		3627.2.a.p.1.7		8
4.3	odd	2		6448.2.a.bd.1.7		8
13.12	even	2		5239.2.a.i.1.7		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.a.e.1.2	✓	8	1.1	even	1	trivial
3627.2.a.p.1.7		8	3.2	odd	2
5239.2.a.i.1.7		8	13.12	even	2
6448.2.a.bd.1.7		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)

\(f(q)\)	\(=\)	\(q-1.71590 q^{2} -1.27835 q^{3} +0.944315 q^{4} -2.00831 q^{5} +2.19352 q^{6} -3.76625 q^{7} +1.81145 q^{8} -1.36583 q^{9} +O(q^{10})\)	\(q-1.71590 q^{2} -1.27835 q^{3} +0.944315 q^{4} -2.00831 q^{5} +2.19352 q^{6} -3.76625 q^{7} +1.81145 q^{8} -1.36583 q^{9} +3.44606 q^{10} -2.15365 q^{11} -1.20716 q^{12} +1.00000 q^{13} +6.46251 q^{14} +2.56732 q^{15} -4.99690 q^{16} +0.496780 q^{17} +2.34363 q^{18} -2.24448 q^{19} -1.89648 q^{20} +4.81457 q^{21} +3.69545 q^{22} +5.01638 q^{23} -2.31566 q^{24} -0.966689 q^{25} -1.71590 q^{26} +5.58104 q^{27} -3.55653 q^{28} +2.58766 q^{29} -4.40526 q^{30} -1.00000 q^{31} +4.95128 q^{32} +2.75311 q^{33} -0.852424 q^{34} +7.56380 q^{35} -1.28977 q^{36} +0.294584 q^{37} +3.85130 q^{38} -1.27835 q^{39} -3.63795 q^{40} +10.0822 q^{41} -8.26133 q^{42} +1.81940 q^{43} -2.03373 q^{44} +2.74301 q^{45} -8.60761 q^{46} -0.507742 q^{47} +6.38777 q^{48} +7.18464 q^{49} +1.65874 q^{50} -0.635057 q^{51} +0.944315 q^{52} +11.8424 q^{53} -9.57652 q^{54} +4.32520 q^{55} -6.82238 q^{56} +2.86922 q^{57} -4.44017 q^{58} -9.27755 q^{59} +2.42436 q^{60} +0.950004 q^{61} +1.71590 q^{62} +5.14405 q^{63} +1.49789 q^{64} -2.00831 q^{65} -4.72407 q^{66} -3.55834 q^{67} +0.469116 q^{68} -6.41267 q^{69} -12.9787 q^{70} -14.3034 q^{71} -2.47413 q^{72} -0.201282 q^{73} -0.505477 q^{74} +1.23576 q^{75} -2.11949 q^{76} +8.11119 q^{77} +2.19352 q^{78} -8.22770 q^{79} +10.0353 q^{80} -3.03702 q^{81} -17.3000 q^{82} +8.35070 q^{83} +4.54648 q^{84} -0.997688 q^{85} -3.12191 q^{86} -3.30793 q^{87} -3.90123 q^{88} +10.6914 q^{89} -4.70673 q^{90} -3.76625 q^{91} +4.73704 q^{92} +1.27835 q^{93} +0.871234 q^{94} +4.50761 q^{95} -6.32946 q^{96} -11.7424 q^{97} -12.3281 q^{98} +2.94152 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - q^{2} + 7 q^{3} + 7 q^{4} + 11 q^{5} - 2 q^{7} + 9 q^{9}+O(q^{10}) \) 8 * q - q^2 + 7 * q^3 + 7 * q^4 + 11 * q^5 - 2 * q^7 + 9 * q^9	\( 8 q - q^{2} + 7 q^{3} + 7 q^{4} + 11 q^{5} - 2 q^{7} + 9 q^{9} - 2 q^{10} - 2 q^{11} + 19 q^{12} + 8 q^{13} + 3 q^{14} + 6 q^{15} - 7 q^{16} + 7 q^{17} - 12 q^{18} - 5 q^{19} + 6 q^{20} + 8 q^{21} - 4 q^{22} + 14 q^{23} - 17 q^{24} + 17 q^{25} - q^{26} + 7 q^{27} - 9 q^{28} + 12 q^{29} - 9 q^{30} - 8 q^{31} - 21 q^{32} + 10 q^{33} - 12 q^{34} - q^{35} + 11 q^{36} + 2 q^{37} + 24 q^{38} + 7 q^{39} - 19 q^{40} + 13 q^{41} - 27 q^{42} - 5 q^{43} + 22 q^{44} + 19 q^{45} + 17 q^{46} + 23 q^{47} + 3 q^{48} + 26 q^{49} - 26 q^{50} + 18 q^{51} + 7 q^{52} + 25 q^{53} - 36 q^{54} - 17 q^{55} + 8 q^{56} - 35 q^{57} - 29 q^{58} - 5 q^{59} + 71 q^{60} - 9 q^{61} + q^{62} - 37 q^{63} - 14 q^{64} + 11 q^{65} - 41 q^{66} + 22 q^{67} - 6 q^{68} - 7 q^{69} - 29 q^{70} - 17 q^{71} - 34 q^{72} - 27 q^{73} + 14 q^{74} - 33 q^{75} - 36 q^{76} + 31 q^{77} - 23 q^{79} + 9 q^{80} - 12 q^{81} + 18 q^{82} - 25 q^{83} - 62 q^{84} + 13 q^{85} + 11 q^{86} + 26 q^{87} + 5 q^{88} + 2 q^{89} - 14 q^{90} - 2 q^{91} - 20 q^{92} - 7 q^{93} - 38 q^{94} + 3 q^{95} - 52 q^{96} - 15 q^{97} + 39 q^{98} - 8 q^{99}+O(q^{100}) \) 8 * q - q^2 + 7 * q^3 + 7 * q^4 + 11 * q^5 - 2 * q^7 + 9 * q^9 - 2 * q^10 - 2 * q^11 + 19 * q^12 + 8 * q^13 + 3 * q^14 + 6 * q^15 - 7 * q^16 + 7 * q^17 - 12 * q^18 - 5 * q^19 + 6 * q^20 + 8 * q^21 - 4 * q^22 + 14 * q^23 - 17 * q^24 + 17 * q^25 - q^26 + 7 * q^27 - 9 * q^28 + 12 * q^29 - 9 * q^30 - 8 * q^31 - 21 * q^32 + 10 * q^33 - 12 * q^34 - q^35 + 11 * q^36 + 2 * q^37 + 24 * q^38 + 7 * q^39 - 19 * q^40 + 13 * q^41 - 27 * q^42 - 5 * q^43 + 22 * q^44 + 19 * q^45 + 17 * q^46 + 23 * q^47 + 3 * q^48 + 26 * q^49 - 26 * q^50 + 18 * q^51 + 7 * q^52 + 25 * q^53 - 36 * q^54 - 17 * q^55 + 8 * q^56 - 35 * q^57 - 29 * q^58 - 5 * q^59 + 71 * q^60 - 9 * q^61 + q^62 - 37 * q^63 - 14 * q^64 + 11 * q^65 - 41 * q^66 + 22 * q^67 - 6 * q^68 - 7 * q^69 - 29 * q^70 - 17 * q^71 - 34 * q^72 - 27 * q^73 + 14 * q^74 - 33 * q^75 - 36 * q^76 + 31 * q^77 - 23 * q^79 + 9 * q^80 - 12 * q^81 + 18 * q^82 - 25 * q^83 - 62 * q^84 + 13 * q^85 + 11 * q^86 + 26 * q^87 + 5 * q^88 + 2 * q^89 - 14 * q^90 - 2 * q^91 - 20 * q^92 - 7 * q^93 - 38 * q^94 + 3 * q^95 - 52 * q^96 - 15 * q^97 + 39 * q^98 - 8 * q^99

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